,
DAVID F. ROGERS J. ALAN ADAMS
MATHEMATICAL ELEMENTS FOR
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THIS 8001( BFLONGS I 0
MIKE HAYDEN ~~n£1YATICAL
ElB£Nrs FOR
CO'RJTER G£\1\PH l cs
•
MATHEMATICAL ELEMENTS FOR COMPUTER GRAPHICS DAVID F. mRS
.Aerospace Engineering oepar tt1ent United States Naval h:al:lemy
J.
~
PIWf>
Engineering oepart:ment United States Naval Academy
l~cal
ftf.Mw-HILL BooK Ca1>ANY St. Louis san Francisoo Auckland DUsseldorf Johannesburg Kua la Llr'r(:lUr I..ooda'l
New York t~oo
M:lnt:real
S3o Paulo
New
Delhi
Panazra
Paris
Singapore Sydney Tokyo Toronto
-
Copyright@l976 by ~w-Hill ' Inc. All rights reserved. Printed in the united States of Merica. The prtXJian ~ of this publicatioo may be reproduced. No other part of this publication may be reproduced , stored in a retrieval system, or transmitted , in any form or by any rre.ans, electraric, ~ . ~t:ocnpy.ing , recording, or ot:heiwise, without the prior written permission of the publisher.
3 4 5 6 7 8 9 0
FGRFGR
7 9
Fairfield Graphics was printer and bi nder . Ll.brary of Qn]ress catalogirq m Publicat.ioo Data ~.
Oa\nd F 1937r.\"lt:hesratica.l elencnts for o:xt~ter graphics .
Incl1rles blbliographical references and index. t. Cl.:rput.er graphics. I. Admns, Janes Alan 1936-, jowt autl-or. II. Tt tle ·;)85.R6 001.6'443 75-29930 ISBN o-o7-o53527-2
CONTENTS
X
XIII
PIUAC£ QW'lER 1
nmax.x:TIOO
ro Cll
~
1
3
1-2
Overview of catp.lter Gra[il:ics Iepresent.i.ng Pictures to be Presented
1-3
Preparing Pictures for Presentation
3
1-4
Presenting Previously Prepared Pictures
5
1-5
lnteract.in
8
1-6
Descript.ioo of scrne Typical Graphics Dcvi oes
ll
1-7
Classificatioo of Graphics Devices
16
References
23
1-1
OW'T£R 2 romrs 2-l 2-2 2-3 2-4 2-5 2-6 2- 7 2-8
liND LINES
Introcb::tioo Representation of Points
Transformat:.ioos and t-'.atrices Tranformation of Points
3
24 24 24 25 25
Parallel Lines
27 28 29
Intersect.ing Lines
30
Transformation of Straight Lines
t-tidr:oint Transformation
2-9
~t.Jtlcn
11
2-10
ReflectJ.on scaling
31
2-11
32
2-12 canbined Operations
33
2-13 TransformatScn of a Unit Square 2·14 Arbitrary 2 X 2 ~tati.on Mat rix
34
2-15 '1\.Q-Dlr.W'm.Si.ooa.l 'J.'ranS1at.ioos and Hcnogeneous Ox>rd.inates
36 41
2-16
Points at Infinity
2-17
'1\.o-DllT'enSl.OOal Jbtation about
35
an Arbitrary Axis
44
References
CW\PJER 3
43
'ruREE-DINENSICN\L 'I'Rl\NSFOR>'.ATlQ.:S 1\ND PRml:TICN;
46
3-1
Introduction
46
3-2
'lbme-Dinensional Scaling
3-3
'nlrec-Dirrensional Shearing
3-4
'furee-DiJrensional IOtations
47 49 49
3-5
Reflection in 'I'hree Dimensions
51
3-6
Translation in Three Dilrensions
54
3-7
Three-D~ional
3-8
Elerrcnts for the General JOtation
3-9
Affire and Perspective
lbtations about
an Arbitrary Axis
54
r-~trix
55
Gect~etry
59
3-10 Axooc:rretric Projections
60
3-11
Perspective TranSfonMt.ions
3-12
Techniques for Generating Perspective Views
3-13
Pomts at Infinity
66 73 77
3-14
Reoa\stn.ction of 'Ihree-Dirrensional Information
3-15
Stereograriric Projection References
QW1fER £t
78 84 87 89
PLANE aJRVES
89 90
4-l
Introduction
4-2
Nonpara.rrctric
4-3
Param:>t.ric CUr\tes
4-4 4- S 4-6
Nonpa.ram:?tric
Parar-etric Representation of Conic Sections
102
4-7
103
4-8
Pa.razmt.ric Reprcsentatioo of a Circle Para~TW!tric ~presentation of an Ellipse
4-9
Pararrctr ic Rcprescnt.:ltion of a Parabola
curves ~resentation
92
of Conic Sections
Nonparanetric Circular Arcs
4-10 Paranetric RaprcSUltation of a Hyperbola
94 98
104 106 108
4-ll A Prooodure for the Use of Conic secti.als 4-U Circular Axe Inte.q:olation
5-.l 5-2 5-3 5-4 5-5 5-6 5-7 5-8
ll3
us
References
OW'TER 5
ill
Bourdary COnditions
116 ll6 116 119 123 124
Paratolic Blending
133
Bezier CUrves a-spline CUrves
139
SPACE a.JRVES
Int.rodu:tion
o.uves
Representation of Space
CUbic Splines N
144 155
References
ffiAPTER 6
S'llRFACE IESCFUPI'IClil 1\ND GENERATIClil
6-12 Generalized Q)ons Surfaces 6-13 COOC1usions References
.157 157 158 162 164 165 166 168 170 175 176 180 181 185 186
APffNDIX A CCI.fiVI'ER GRAPHICS
188
6-1 6- 2 6-3 6-4 6-5 6-6 6-7 6-8 6-9 6-10
Int.rodllct ioo Spherical surfaces PlanE: 'Jurfaoes CUrved SUrface Ieprescntation
Bilinear Surface IDfted or Ruled Surfaces Linear COons surfaces
Bicubic Surface Patch 'Ibe F-Patch Bezier Surfaces
6-ll &-spline SUrfaces
A-1 A-2 A-3
SOFniARE
Carp.Iter GraJirics Primitives
Catplter Graphics Elarents
canonical Space
IWENDIX B
NM'RIX
B-1
Te.oninology
B-2
Mdi tion and Subtraction
B-3 B-4
Multiplicatioo ~t
of a Square Matrix
.189 191 194 196 196 197 197 198
a-s C-1
Im-erse ot a
Square t-1iltrix
199 200 200 201 202 202 203 204 204
C-2
A"' Al.godtm for '1\.o-Oim.msi.onal Translations A '1\.o-Dirrensl.CnAl. Sealing Algorittm
C- 3
h '1\.o-OiJrensl.onal Peflection Algorithm
C-4
A General. '1\.o-Di.m:nsl.ooa.l lbtation Alqorit:ml
c-s
A 'Ihree-Dilrensional Scaling Algoritml
C-6
A.'1 Algoritl'm for 'lbree-O.urensional lbtaticn About the x- Axis
C-7
r.n
C-8
An Algorittm for 'Ihree-D.inensional
C-9
An Algorittm for 'Ihree-DJ.SrenSional Reflectioos
206
C-10
An Algorithm for 'llu:ee-Dilrensional Translation
207
e-ll
An Algorithm for 'lhree-Di.nensional ~taticn
•
Al.CJ.Xl.tml for 'Ihree-Dirrensioocll lbtaticn About the y-Axis ~taticn
About the z-Axis
about 'Any Arbitrary Axis in Spoce
205
C-13
h Dirrctric Projective Algorithm
207 208 209
C-14
An Isaret.ric Projective Algoritl'm
210
c-12 An Axooaretric Projective Algoritlm
c -15 An Algorittm for Perspective Transfonnaticns C-16 'lbroe-Dimmsialal C-17
·en Algoritbns
A Stereo Algorithm
C-18 An Algorithm for a Nonparametric Circle C-19
An Algont:ml for a Paran-etnc Circle
c-20
rar~ic
Ellipse Al.gon.tm&
c - 21 An Algorittm for a Paranetric Parabola C-22
Algorithms for Pa.ra.rrot.nc HypeJ::bolas
c-23
An Algorittm for a Circle through Three Points
c-24
An Algoritl'm for Generating CUbic Splines
C-25
An Algorithm for Paralx>lic Blending
c-26 A Bczier CUrve Algorithm
c - 27 A a-spline CUrve Algorithm C-28 1\n Algorithm for a Bilinear Surface Patch C- 29
An Algorittm for a Linear Coons Surface
C-30 An Algoritml for a Bicubic Surface Patch C-31
INIIX
lktLer SUrface Generatioo Algorittm
210
211 213
214 215
216
217 217 218 220 223 225 226 228 228
229 230 232
FOREWORD
•
Since its inception rore than a decade ago, the field of carput.er
gr~
has
captured thl! iiragi.naticn and technical interest of rapidly increasing nuttlers of
individuals fran nany disciplines.
A high percentage of the g:ro.fing rc:&Ilks of
cx:xtpUter graphics professionals has given primary attentioo to catputer-oriented
problems
1.n
progianming, S'Jstem design , hardware, etc.
'lh.is was pointed out by
Dr. Ivan Sutherland in his introduction to Mr . Prince' s book , "Interactive
Grafhlcs for Ccrp..lt.er- Aided Design," in 1971 and it is still true t.oday.
I
believe that an inadequate balance of attentioo has been given to applicatiooorientcd problem; .
There has been a dearth of producticn of useful information
t:Mt bears directly oo the develO(Jtent and iltplerrentatioo of truly productive applications.
Understanding the practical aspects of carputer grat:*\ics with
regard to both the nature and use of applications r epresents an essentja] and ultiJMte requitetettt in the devel o p tent of practical catputer graphic syst.enS . ~.athenatical
tcdurlques , especially principles of geanet.ry and transfozntttioos,
are indiCJenOus to nost
~
gra{ilic appl.icatioos .
Yet, l arge ni.Bbcrs of
graphic prograrmcrs and analysts struggle over or gloss over the basic as the catplex problems of the mat:.hem:ltical eleuents . operational potential of catputer graphics is often
as~
rurthemore, the full
~izcd
whenever the
rMt:he'natic.u relationships , constraints, and options are inadequately expl oited. By the.u: authorship of this text, D:rs . Jt>gcrs and 1\dams have rea:xpuzed the
valuable relevance of their back9XOU'ld to these przlctical consl.derntioos.
'lbUr text is oaJCi~, is ~ive, and is written in a style unusually
oonducive to ease of rcacUng,
, iUlCl use.
I t exmplifi.es the
rare
type of '-Ork that rrost practit.iooers stould wish to place in a prominent locatioo
within their library since it should prove to be an invaluable ready rcfarcnce
for rrost disciplines. It is also well suited as the b
autn:>rs in producing an excellent and needed text,
"Mat:hernat ical Elecrents for Cotp..lter Graphics."
s.
H. "Olas" 0\ascn
Locl
•
PREFACE
A new and .rapidly expanding fjeld called "CXJlPUter gra~" is errerging.
'Ihis field cxxrbines roth the old and the new:
the age old art of graptucal
cx:mtU'licatioo and the re11 technology of catpUters. AlJrost everyone can expect to be affected by this rapidly expanding techoology. A new era in the usc of CXITpUter graphics,
not just by the large mrpanies and agencies \\ho ml'.lde rra.ny
of the initial advances in software and hardware, but by the general user, is beginning.
Lc.w-cost graphics terminals, ~ sharing, plus advances in mini-
and miCJ:O
'lbday,
practical , reliable, CXlSt effective , and readily availAble. The purpose of
this lxlok is to present an introdtx:ti.on to the rro.thanatical
theory underlying cx:.op.rtcr graphics techniques in a unified manner.
Al.trough
new ways of pt:eSentlng material are given, no actual ''new" mathematical TMterial
is presented .
All the material in this boolc exists scattered throughout the
technical literature. •
111 CTM
'Ibis lxx>k atterrpts to bring it all together in one place
•
notatJoo.
In selecting material, ~ dlose tecmiques \Jtich
were fund.alrentally nath-
ematical in nature rather than t.rose \olhi.dl
First-year college rratheratics is a sufficient
prerequisite for the major part of the text.
After a discuss:i.on of c:urront cxnpoter graphics tochnol.ogy in Olapter 1, t.~
manipulatIon of graphical elEments represented in matrix form using haro-
CJI=OOOUS a:xn:'din.ate5 is described .
A discussion of existing techniques for reptesent.iN} p:>mts, lines, curves, and surfN"PS within a digital CXITpUter, as
'Mll.l as c:utpUter sofO.'are procedures for rmnipulatinq and displaying mcputer output in graphical form,
is then presented in the following 0\aptc.rs.
!'lathematical tcchniques for pt:oducing ~ ard perspe:ct.:ive views
are given , along with generalized tec::hn.iques for rotation, translation, and
scall.ng of qearotric figures.
aro
CUrve definition procedures for both explicit
palCiretric representations are presented for both
and three-
ditrcnsiooal. curves. C\.U:\Ie definition techniques include the use of conic sec:t.ials, ci.rcul.Ar arc interpolation, C\bic spl..ines , parabolic bl ending , Bezier
curves and curves based en a-splines. An int.roduct.i.a'\
to the mathematics of
surface description is inc:ltxied.
catplter algorlthns for uost of the fundarlental e.l.arents in an interactive graphics package are given in an appendix as BASIC* l..anguaqe subprograms .
ever, these algorithns deliberately stop actually display tre results.
srort
of the coding necessary to
U'\fortunate.ly there are no standard l.anqua
carmands or subroutines available for graphic display.
discussioo of
grapuc
HeM-
Although sate prel.imi.nary
primitives and graphic elem:mts is given in .Appendix A,
each user will in general find it r..eoessary to work within the oonfines of the
cx:nputer system and gra{il.ics devices available to him or her.
'1h:l furx1aJmnt.al ideas in this book have been used as the foln}ation for an introductory oourse in
c:utp.~ter
or sclcntific fields at the undergraduate l evel.
It
J.S
nann<:r at both I.D1iversit..ies and schools of technol ogy.
~mjoring
in technical
suitabl e for use in this
It is also suiUible as
a supplene1taxy text in rro:re advanced c:utputer programning courses or es a SH{l>letre:1taxy text in
sore
advanced mathesratics
courses.
Further, it can
profitably used by ind.iviCbals engaged in professiooal programning.
be
Finally,
tho docurcnted c::onputer ptograms srould be of use to m1puter users interested in
developing mcputer gra(Xlics capability.
The authors gratefully aclo'lc:Mledge the enoourageucnt and support of the
Urited States Naval Academy.
'llle acadatu.c envi..roon:ent provided by the aaninis-
Lratial, the f<-~t."Ulty, and espcc:ially the midshipnen was conducive to the de~lop
of the rmterial in this J:xxX.
*Msrc i.s a regiStered trade
nurk of
nu:U.OUt.h COllege
PREFACE YN
No book is ever written without the assistance of a great Here
\ole
\lo10Uld like to acknowledge a few of them.
~mny
people.
First, Steve COOns \lh::l reviewed
the entire m:umscript and made rrany valuable suggestioos, Rich .Reisenfeld \lh::l
reviewed the material on B-spli.ne curves and surfaces, Professor Pierre Be2ier \lh::l reviewed the 11\llterial on Bezier curves and surfaces and Ivan Sutherland ....no
ional recont.roction techniques discussed
provided the iupetus for the in Olapter 3.
Special
aclcncwled~t
CAD Group at ambridge University.
is due past and present rre•bers of the
Spec:Hically, \oOrlc Cklne with Jti>in Forrest,
Olarles Lang, and Tcny Nutlx:Jume provided greater insight into the subject of
CCJ1l",rt:er graphics .
Finally, to I.Duie Knapp who provided an original FORmAN
program for B-spline curves . '!he authors would also l..iJ
at the Evans and Sutherland CQipUt.e.r Coqx:>ration. a uthored the ~t fran ....ru.c:h many of the
j
Specifically, Jim can an who
deAs on representing , preparing1
presenting and interacting with pictures is based.
Special thanks are also due
Lee Bil101o1 "*'o prepared all of the line drawings.
t-t.ach of the art ...orlc: for Chapter l has been provided t.hrough the good
offices of •:ui.ous c:otputer graphics equiprent nanufacturers .
Specific acknor.ol-
led
Evans and SUtherland Caq;luter Corporatim
Fig. 1-S
Adage Inc.
Fig. 1-7
.Adage Inc.
Fig. 1-8
Vector General, Inc.
Fig. 1-11 Xynetics 1 Inc. Fig. 1-12 CALO:l-1P, California eoopurer Prochx:ts, Inc. Fig. 1-15 Gould , Inc. Fig. 1-16 Tektronix, Inc . Fig. 1-17 Evans and Sutherland catFuter Cox:pxaUon Fig . 1-18 CAiaM.', california c:atputer Prodttts, Inc .
David F. JO]erS .1. Alan Adams
•
CHAPTER l INTRODUCTION TO COMPUTER GRAPHIC TECHNOLOGY
Since o::rrputer grafhi c:s is a relatively new technology, 1.t is neoassa.ry
to clarify the current ternu.rology.
rather l OOEiely in this field .
A nmber of t.e.rTTs and defl.nitioos are used
In particular, carpucer aided design (CJ\D) ,
interactive graphics (IG) , m1puter graflh.ics (ex;) ,
aro
conputer aided manufac-
turing (01'.) are frequently used interch~ly or in such a rrlU'lner that
considerable confusion exists as to the precise rrean.ir¥]. is the nost general.
Of these tcrrrs CAD CAD rra.y be defined as any use of the caTpUtor to aid in
tr.e design of an individual part, a S\.i::)system, or a total system. 'Ihe use CJocs not have to involve gra._oh.ic:s .
'1he design process rroy be at tr.e systen
cxmoept lcw.l or at tho detal.l part design level.
It nay also involve an in-
terface with CA.'i. ~ ai
is the use of n CX>IIPJter to &d in the
manufacture or production of a part exclusive of the design pnx:::ess .
A direct
interfaoo between the results of a CAD application and the neoessacy part pr~xi'lmling
using su:h l anguages as APT (Autooatic Progt'W11l..>d Tools) and lNIAPI'
(United' s APT) , t.ht" direction of a IMChine tool using a haroNircd or softwired
(miniconputer) controller to
roae eata
-f'ro•·• a OUI'd"lCd pa._?Cr tap: and CJC!llerate
the necessary ~ to oonLrol a l"aChi.r.e tool, or the direct control of a
oochine tool USir¥J a minicarprt:.er nay be involved. Cbrputc.r graphics is the use of a ca•puter to define , st:.aro, nunipulatc,
i:."ltcrrogate, ar.a PLesent pictorial output.
ation.
'lhls is essentially a passive oper-
a:rputer prepares and presents stored infonration to an observer
Th'
in the fo:::r.. or pictures . bo.inj presented.
'lhe
'lhe cbsetver has no direct conLrol over the picture
application~
be as sirrple as the presentation of the
qrll{:h of a silrple C1lnCtion us~ a hicjl-speed line printer or a te.let.~
·
t.emi.nal to as m•plex as the simulation of the autaMtic reentry and
landU'lq of a space capsule.
Interacti"\:! qrapucs also uses the
ial rote..-ial.
llc:Uevar, in interactive grapti.cs the observer can influence the
picture as it is being presented; i.e., t:ht:! observer interacts with t:hc picture m real t:.in!.
To see the inportance of the Yea!
~
restriction, consider the
· onal picture a:xtposed of 1000 lines
pJ"Cblcm of rotating a cotplex
at a reason.tlble rotation rate, say, 15°/second.
As we shall see subsequently ,
the 1000 lines of the picture are JTOSt oonveniently represented by a 1000 x 4 nutrix of hooogeneous OX>rdinates of the end points of the lines , n::~St
tion is
a 4 x
~
aro
the rota-
oonveniently acca,plisbed by nultiply:ing this 1000 x 4 matrix by
tran.sfornation rratrix.
Accntplish.ing the required natrix nW.tiplication
roquin.... 16 , 000 aultiplicatialS , 12,000 additions , and 1000 divisions .
I f this
m'ltrix rrultipUcation is acx:n1plished in soft..lare, the tirrc is significant.
To
soc t.ltis, consider that a typical miniconputer using a haxd.-ta.re floating-p::lint pt~~sor
requires 6 microseconds to nultiply 0.'0 nurbers , 4 microseconds to
add O..'O fl\rlbers , and 8 rnicrosecorxls to divide t\\'0 nurbers.
'Ihus the matrix
nultlpl.ication requucs 0 .15 seconds. Sin~
mtputer cti.splays that al lo;,· dynamic notion require that the picture
be redram (refreshed) at least 30
~
each second in order to awid flicker,
i c 1s cbvious that: the picture canmL change snoothly.
Even if it is assured
that the picture is recalculated {\4?d!lted) only 15 times eadl seoond, i.e., ever)'
Hare.
tlcqrec , it is still not [X.>SSible to aco:JJtplish a sncoth rotation in soft'Ihus this is
to l.ntcractivcly
T"DN
no lollC}er interactive graphics .
To regain the ability
present the picture several th.i.ngs can be oone. Cl ever pro-
rJrarwli..ng can n:duc:c the t.irre to accntplish the required matrix multiplication . lfodever, 0 pOlnt will be reached wnere this is no longer possible .
plexlty o! t:hA.; picture
m'ly not
I~
~
accept.ab lP .
be reduced.
'llla
In this case, the resulting picture
Finally, the matrix multiplication can be acxntpli.shcd
by using a Spcclal-pl.ll"pC)Se digital hare);.;are rrut.riJ< nultipller.
r-osL prouU:sing CIR'road\.
'Ihi.s is the
It can easily handle the prcblcm outlined l!b:>ve.
lnt:h this tcrmioology in mind the renainder of this chapter will give an OVCIViL~
of cx:r·put.er gratillcs
displhJS
~va.i 1<-bl,..
W
~h....
am
discuss and c lassify the types of gr~ic
nt:Cl!SSary ronsiderations for devel~t of a soft-
S'fllt..an for the f~tal drcr,...l.T\g, device-control, and dato-handl:ing
aspects of c:atputer grapucs is given in J\ppe1rli.x B.
1-1
~GRAPHI CS
CNERVIEW
!J:
catputer
gr~ics
as defined
abo~
can be a very
o:xt~lex
arvi ell verse Sl.b-
I t cncx:~~paSses fields of study as diverse as electronic and rroch.lnical. desi gn of the corqx>nents used in catpUter gr
display lists and tree structures for preparing and presenting pictures to an
ooserve.r using a c:atputer gra[:irics system.
1\
di..scu<;sion of these aspects of
interactive c:atputer g:rar.hlcs is given in the bock by 1-l ) .
Ne~
and Sproul (ICf.
llcre ~ will atteupt to present only those aspects of the subject of
interest fran a user' s point of view .
From this point of via", CXlClputer
graphics can be divided into the fol lowing areas: Representing pictures to be presented
Prepari.nq pictures for presentation Pl:escnting previously prepared pictures
Interacting wi.th the picture Here the word "picture" is used in its broadest sense to rrcan arrt collection of lines , points, text, etc., to 1><:! displayed an a graJ=hics device.
A picture
TMY be as s.inple as a single line or curve , or it may be a fully sca.lcd and anrotatcd ~ or a c:atplex representation of an aircraft , shl:_:~, or auta•cbile.
1-2
la>REsa-rr1 ~ P1<:n~s
To BE PREsENTED
F'\lndaJrentally the pictures represented in cnTpUt:er cp:aphics can be considered as a collection of lines , points, and textual material.
1\
line can be
represented by the ooordinatcs of i ts end points c, , y 1 ,z1 > and
(~,y ,z 2 ),
2
a
point by a s.i.ngle-ooordi.nat:e triplet (xl'y , z ) , and textual material by col1 1 l.ect.ions of lines or points . 'Llle representation of textual material i s by far the rrost c:atplcx, involving in many cases cw:ved lines or dot matrices .
IICMever, unless Ule user is
concerned with pattern ['C(X)(Jnition, the design of graphic tmrd.,'are, or unusual character sets , he or she need not be concerned with these details, sinoe alIro6t all grapuc &vices have built-in ..harc},:are" of software c:ha.r.!K:ter gen-
erators.
'Ihe representation of curved lines is usually l!o:::ouplishcd by approxi-
rating them by short straight-line segrrcnt.s .
plished using hard.-lare curve <]CJlerator:s .
1-3
PREPARING
PtcrtR:s FOR PRESENTATION
Iio..Jever, this is sarctiJros acxntr
P1ctll.rCS
ult.i.r.ately oonsl.St of points.
are qererally stored
The ooordinatcs o! these poirlts
a h.le (anay) prior to beJ.ng used to present the plc-
1n
'nlis file (array) is called a data base.
t:ure.
Very mtplex pictures require
very m::plex data bases wtudl require a oocrplex Pn'XJrM\ to access them.
'l1lese
contain powts, slbst.ructures, and olher ronqraphic data. T"e des i
which the user can reddil} invent sinple data base structures \\lhich can be a~sed.
easily
'ttlcre are
Pomts are the b:s.sic building bloa'<.s of a graphic data base.
th.."""O basic rroth:xls or instructions for
trea~
a point as a graphic <_7El00etric
nove the be5n, pen, cursor, plottinc] head (hereafter CAl led the cursor)
entit'l:
to t:he point, dra.<.> a line to that point, or draw a dot at thal point.
m::ntally t:he.rt=- are
c..'O
wtJYS to specify the position of a p:>int.:
relative (incra·ent.al) ooordi.nates .
F\lnda-
absolute or
In relative or incranental ooordinates the
position of a point is defined by giving the displaoerne.nt of the p:>int with respect
to the pre\"ious point.
The specif1.cation of the position of a p:>int in either absolute or relative 'lhis can lead to difficulties 1.f a mrputer
ooord.inates requires a niJ!I't.)er.
Generally a fUll 0011puter lf.'Ord 1.s used
"''it.'l a l.irnlted lf.'Ord lengt.."'l is used.
to specify o cxx:mii.natc position.
ifl.ed by a full cr:r.puter word \
l.S
'lhe largest. integer rnmber that can be spec2n-l - 1, where n is lhe nutber of bits in the
For thl. 16-bit minia:rtp.lter frequmtly used to
displays, t.his
1S
32767.
5\.{l{X)rt
CQlPUter graphic
For many awlications this is ac::oE4"'table .
~var,
diffl.culties arc cnoount:ered ~.~.ne.n larger integer ru::bers than can be specified
are required.
At.
first
'-'C
might expect to 0\."ercare this difficult:r by using
relat.ivc ooord.i.Mtes to specify a nurrber sud'l as 60 , 000, i.e . , using an absolute
ooor:dinatc spcciflcation to position the cursor to (30000 , 30000) and then a relative cxxm:Unat.e specification of (30000, 30000) to position the beam to the final desired point of (60000, 60000) .
1-t::Jwever, this will rot \o.'Ork , since an
at!:.crtpt to aOC\I':Ul..ate relative position specifications beyond the nuxi1Tun rcp-
re:scnta>lc
val~ wi 11
result in the
ard erroneous nagnitu1c.
y1eld the
fhen::n:na
Thew~ hcrog€~us
generatia~
of a mlt'ber of ~ite sign
For cathode ray t\De (CRI') displays tins w11l generally
cal.l.OO wraparound.
out of this
di~ is to
use ho•oge.~l:S ooordinates.
'1hQ
o:x>xdinatcs introduces sare a&ii tional cuplc.xit}·, ~ loss
speed, and ~I'! loss in resolution.
tb~'ever, ~
~ighed by the advallt~ of bci.'19 able
use of lil
dlsadvant.aqes are far out-
to represent .l.ar9c wteger nwbcrs \·d.th
a cucputer of limited wonl size.
for this reason
.:lS
\o.
l ater, hom::lgene.ous coordinate representat:.i.oos are generally used in t:his bock . In harogcncous roord.inates an n di.rrensional space is rcpre.•;cnt:ed by n + 1
dim.:msions , i.e . , throo-d.inensional data where the position of a JX)int is given
by the triplet (x, y , z) is represented by four ax:n:dinat.es (hx,hy ,hz,h) , h is an
ami tra.ry
n\Jlber.
l f each of the coordinate positions represented in l ess than 32767 , then h "-ould be n-ade represented directly .
I(,
o:JUa1
c1
16-bit
were
c:arpu~r
to 1 and the CXX)rtti.nate positions
hcJ..rever, one of the Eucledean or ordinary CXX)rdi.nates
is larger than 32767, say , )( ::: 60000 , then the bea::tres apparent.
~re
poNer
or
haxogunuous cxxmlinat:es
In this case '''e can let h = 1/2, and the CXX)rdinatcs of the
point are then def ined as ( 30000 , l/2y , l/2z 1 1/2) , all acceptabl a nUtbers for a 16-bit CXIIlpllter .
Ho..-evet"1 sooe resolution is lost sinro x .. 60000 and x
59999 are both represented by the sama hooogeneous CXX)rdinate.
=
In fact resolu-
tion is l ost in all the ooordinat2s evan i f only one of them exceeds the 11\lX.i.nun expressable mmbor of a particular
1~
PRESEw.!~ f>REv iOUSLY PREPARED PICTURES ~lith
these CXJJtnents about data bases in mind it is
ne
to note that
the data base usad to prepare the picture for presentation is alnost never the sarre as the display file used to present Ute picture .
The data base represents
the total p1.ctura wile the display file represents only sare portion, vie.,r, or
s
The display file is created by transforming the data
The picture contained in the data base may be J:eSizcd, rotated, trans-
base.
lated, or part of it rorroved or vie..JOO fran a particular point to obtain
essary perspective before being displayed.
f"any
nec-
of these operations can be
aca:xrplished by using siJtpl c linear transformations which involve matrix trultip~cab,ons .
views.
As
1\rrong these are rotation , translation , scaling, and perspective
we shall sec later, hart:>gen(t>us ooordinatcs arc very oonveniont for
acxortplishing these transformations . As will be shown in det.ril in
to perform
arrr
0\apt:ers 2 arrl 3 , a 4 x 4 matrix can be used
of these individlal transfonrutions on points represented as a
matrix in harogcneous ooordinatcs.
\\hen a sequence of transformations is
dcs.i.:oc
points to ochieve the desired result.
If , ho.-Jever, the nurber of points is
slbstantial, this is i.neH icient and t.irc consU'lli.ng.
1\n alternate and rrore
desirable ret.OOd is to rrultiply the iOOividual rotrices reprcsantinq each ~
transfo11Mtion t:o<)et:har and then to finally rrul.tiply the matrix of
points by the resulting 4 x 4 transforl!ation matrix.
'Dri.s matrix operation is
It results in significant t.irre savings when performing
called cx:ncatenatioo.
C:OifX'UOO matrix operat.i.ons oo sets of data points.
Although in many grilliUcs applications t.'lle cotpletc data b.lse is displayed, frcqoontly only portions of the data base are to be displayed.
'lhi.s process of
d.ispl.ayll¥J ally part of the cotplete-picture data base lS called W1IlCic7wing . \\indo.;ing is not easy, particularly i f the picture data base has been transform-
Pertonnance of the winda.dng q:leration
ed as discussed above.
lJ'l
so.ft..rare gen-
erallY is sufflcicntly t.ine CXlOSuni.ng that dynamic real- t.irre intoracti\'C qtafhj.cs
is not possible.
.Again, sq:::histicat.ed graphics devices perform tlu.s funct.Loo
In ge.ncra.l there are two cypes of w~ - cllppinq
carded and not displayed; i.e. , they are not passed on to the displ ay devioo. In the scissoring t.edlnique, the display device has a larger fhysical drl!Winq space than is required . Only those lines or l.i.re segrrcnts within the specified
'".i.nclo..> are made are dra-m.
visible
e1.e1
though l.i.rles or line segmmts outside the w~
Clipping aCOJJ
than scissoring; e.q., clipping makes available a much larger drawirq area than
sd ssonnq . In scissoring, those lines or Se
a windcw is specified by values for the left, right, top ,
and bottx:lrrl ~ of a rectarqle.
Clipping is easiest i f the edges of the
rectangle are parallel to the ooordinate axes.
case, the rotation of the
w~
base 1..n the Of+OSite direction.
Fig. 1-1.
ll, ho..lever, this is rot the can be o:npensated for by rotating the data
'l'No-dirrensional ciiwirq is represented in
r..i.ms are reta.i.ood, deleted, or partially deleted , depond1ng on
whether they are mtpletely within or without:. the windo.q or partially within
or without the w.incX:w.
In three dinensions a wi.ndow consists of a frustum of
vision, as stn.m in Fig . 1-2.
.In Fig. 1-2 the near (hither) l::x::A..lOOary is at N,
the far (yon) boundary at F , and the sides at SL, SR, ST, and
sa.
JlS a final step in the picture presentatioo process it is ncoessacy to
a:>nwrt fran t:he ooordinat.es used in the picture data base called user coordin-
ates, to those used 1::1}• the displey device, called display ooord.irotcs .
In par-
ticuLu-, it is necessary to a::>n\.'ert ooordinate data which pass the winc:bd.ng procrss into display ooordi.nates such that the PJ.cture appears in socte specified
area on the displa-J, called a vi0-1p0rt .
'lbe viewport can be specified by giv-
ll¥3 its left, righl, top, an:l bottxxn edqes if dirren~nooal
~l.onal 0 1:"
U t:hree-
by also specifyinq a near (hither) and far (yon) boundat:y.
tn the
ltnc p.utl.tll~ '~tlhtn \\tndo"' DJrl from a b dtspla)'cd. pa fl rront b- c not dt~played c l tnc cnllrcl) \qlhlll
--
Wtndo '' cn l trt l1nc d •~P Ia ~ cd !II
J
L•nc:
paru;~ ll y
"'1th1n wwdow p~ll from b-e d t\ph•,,.d pall\ .t-h. c- d no t dt:.pla ycd
I tnt ent•rtly J OUISide O( w 111dow no t dt)p l aycd
~ di.Jrensional windo.o~ing (clipping) .
Figure 1-1
-I
Figure 1-2 ional frustun of vision w.indo.ol.
rcost general case oonversion to display exx>rdi.natcs within a specified t:h.medin'cnsional viewport~ a linear rrapping frau a si>Hu.d ed
[rustun
of
vision (wi.ndo.") to a six-sided vieNpOrt . An additional requ.Ue1ent for II'OSt pictures is the presentation of alpha-
nlm'Cric or character data .
There are in general two rroth::x3s of generating
characters - software and hard"are. If dlaracters am qenc.rated .1.n software using lines , they are treated in the smre manner as any other picture elCI'I\"'nt. In fnct this is rocessa.xy if they are to be clipped and then transforncd along with other picture e l aronts .
ha.rd,..rare character generator .
Ha.leve.r, many graphic devices hava sare k.ind of '-ben har:d.>are character generators are used , the
actual characters are generated just prior to beirxj drawn.
they are treated as only character oodes .
~
Wltil this point
Hara,•are character generation is less
flexible sinoe i t does not allow for clipping or infinite transfonration, e . q ., only li.Jru.ted rotations an:l sizes are generally possible, but it yields signifi-
cant efficiencies. \\hen a ha.z:d..rare character generator is used, the !'rogr& which drives the
qrapuc device nust first specify size , orientatioo, and the position the character or text strlll9 is to begin.
,mcz-c
'nle character c.oOOs specifying these
c:ha.rnct.&istics ;u-e then OOdod to the display file.
{k.lon being
character generator intexpreb the text string , looks up in
p~sscd
~
the
th<.:
neces-
sary .Lines to c:lra\..• the c:haractcr, and draws the characters on the displ ay de-
vice.
1-5
lrCTERACTWG HrrH THE PICTURE Inte.racti.ng with
t.l)c
picture requires sare type of interactive device to
CXllr.Uni.cate with the progra., while it is running.
In effect this
tix: program so that rc-1 or d.l.fferent infol:'T19tion can be provided.
devices have been used to accotplish this task . pe.n.G,
N\.ITCrous
The sirrnJ.est is , of course ,
the al[:haru.vreric keyb:lard such as is foun:! on the tel etype .
devices incll.JOO ll.ght
inte~
flore sq::hlsticated
jat sticl-.s , track balls , a rrouse, ftlnction swi tchc:s ,
control dials, and ana.lo-1 tablets .
We shall
exanu.nc each of these devices
I.:. ric fly .
A sjpple alpham..tmric keyboard as sham in Fig . l-3 can be a useful int:er-
actiw device. easily
Preci.sc al(ilabetic, ru.meri..cal, or ex>ntrol inforrration can be
~lied
to the proyram.
lbleVer, it is not capable of high rates of
interaction, e>
'lhc light pen
a:>ntai.ns a light-scn:ntivc rh;>toelectric oell and associated ci.tcuitry .
\>,'hen
I-OSitioned over a l.i.nc sc:qm:nL or other ll.ghted area of a CRr and activated, the posl. tion of
t:re pen is sensed
ard an inte~t is sent to the conputcr .
Gdl.eMatic of a typical light pen is sham in Fig. l-4 .
A
Figure 1-5 sharJs a licjlt
pen in usc for renu p1.c:::k.ing.
The
JCI'f
stick, rrousc, cllld track ball all operate on the
SCITe
principle.
By ITOIIirq a ex>ntrol, ~ional positiooal information is cx:rmunicatcd to
the catputer.
1\l..l thesr' deVl.oes are analog in
nature.
In particular, ro~t
of the a:>ntrol dlan<;Jcs tho setting of a sensitive potentiareter.
The result.i.ng
siCJOials are convortccl fran analog (voltages) to digital signals using an analog to dig.1tal (A/D) oonverter.
These digital signD.ls arc then interpreted by the
cn·puter as positiot"..a.l infoo:ation .
A joy stick is
stnm in Fig . 1-6. Jat
sti&s, rouses, t.rad: balls, azrl similar devices are useful for particular appll c:at.:lons.
IJ&C..'VC.r, they sOOuld not be used to provide very pr ecise posi tioo-
al information.
Control dials
a.c;
sho.m 111 Fig. 1-7 are essentially sensitive rotating poten-
ti(');Cters clnd
useful !'or activating rota im, t.ranslnt.ion, or soft..tarc
syst:.e~Tc.
7.0:11''
'l1ley are particular ly
fentures of ha.rc:Ware and
~hiiiiCI
hullnn 1 l1h cr npltc
hunulc I
I Ph ~>l••rnuluplu: r
•C/ .__., Pu l~c
Figure 1-3 Al{:tlanll!l'eric keyboard.
luhe ~hnl'lll!1
c trcull 1 > -
-
Figure 1-4 Schc:rnatic of a light pen.
Figure 1-5 Light pen in use for menu picking.
l et di~Piol) cnnr 111llc 1
Figure 1-7 COntrol
Figure 1-6 Joy stick.
dja]s.
Function switches, sJu.m in Fig . 1-8, are either toggle or push-button switches wtn>e fOSit.ion can be detennined by the
~cs
Li~ts
program.
1.ndicat.ing which S'"'itch or s.dtches are active are also usually provided .
The analog tablet is the ItDSt versatile and accurate devloe for cx::mrunlcat..irq {X)Sitional information to the mrputer.
Pxoperly used, the analog tablet
can perform all the functions of a light pen, joy stick , track ball, tion swi t.dles , or control dials .
IrOUSC , func-
Asscx:iated with the tablet is a pen which can
be roved over the surface a.rd whose position can be sensed.
It is also possible
to sense whether or not tho pen is in close proximity to the tablet surface.
PosiLional Wornation with
acx~uracies of
leLs, and Uiblct.s with accuracies of tablets arc sht::Ml in Fig. l-9.
:!:
:!:
0.01 inch are typical of many tab-
0 . 001 inch are available.
Typical analog
1'he position of the pen and its relative loca-
tion in t1'IL! picturo-di..splay area are tied to:)ether by rreans of cursor (a small visibl e syrri:nl) whose tTOtion on the pictl.ln! display area is in concert with that: of the pen m the tablet .
o"ur the 1 iqht pr·n.
'lhe analog tablet has two distinct advantages
IJanely, when the analog tablet is used to perfonn a
JX)inting function , the indication occurs in the data base and rot in the displ.dy file. 'n1us, the proqrdlmlin:] is sirrplified. Also, dr~ or sketching cr r-ointlng
~ing
.m ru'lillog tablet wtri.dl is on a horizontal surface is rorc
natural than perfonn.lng the san-e oper-
ation with a light pen in a vertical orientation. An analog tablet
Cml
be irrplcrrentod
in hard..are by using a variety of clcc-
trocnagnet.ic princi?lcs.
~of
these
are discussed in Ref. 1-1 in nnrc dctai 1. Except in unusual cnvl.roo!rents the user
need not be cau:rned with the precise
'' ' '' •• ' •••••••• •••'. •••. ' •• •• • • •
.........·~' ·-· ,....•.
'l)erating principle.
1-£
1l:SCR I PT I(XII
OF Sot"C
TypICAL
tAAPI·fl CS l£v1 CES 'ttlere are a large nll!bcr of different types of grCifhics devices available, far too many to describe them all hru:e.
'Therefore only a limited m..rrber of de-
vices representative of those available Figure 1- 8
E\Jncti.oo s.dtale:S .
will be c.cnsidered.
In particular,
three different t}?eS of CRT grapucs
devices - storage t\be, refresh, and raster scan; a pen and ink plotter; and a dot oot.ri.x plotter - will be described .
.Additi.ona1 devices and rore detail-
ed descriptions are provided in Ref. 1-1.
'llle three types of CRI' (cathode
~
tube) devices are direct vie,., storage
tl.be displays, refresh displays, and raster scan displays .
'1\le direct vi&l
storage tube display, also CAlled a bistable storage tube cll.st>lay, can be oonsidered as similar to an oscillosoope \17ith a very long persistence phospl'¥:>r. A storage tube display is shown in Fig. 1-9 . A line or character will remain visible (up to an rour) until erased by the generation of a specific electrical signaL 'llle erasing prooess mquires about 1/2 serond . Storage t\be displays have several advantages and disadvantages.
Sam of the advantages are:
the
display is flicker free, resolution of the display is gocd (typically 1024 x
1024 acklre.ssable raster points in an approx.irnate 8 x 8 inch square), and oost is
lao~.
FUrther, it is relatively easy arrl fast to obtain an acceptable hard
CXJPY of lhe picture , and conceptually they are sarewhat eas1.er to program and
rrore suited to tilre-shnring applicatialS than refresh or raster scan displays. 'ttle principill dis-'ldvantagc is that the screen cannot be selectively c.rascd; i.e., in order to dlange atrf el..E!rent of a picture the entire picture rrust be redrawn . Because of this the display of dynamic ITOtions is not possible. In
•
f'ig-..:.re l-9 Analog tablet. acldi tion, this C1aracteristic results in t."le interaction between
the
di.spl~ bci.ng'
scr.cwhat:
s~-cr
t.~e
user ar.3
t.flan with a refresh display.
A refresh CRr ~cs display is based on a telev>_sion-like cathode ray tltk.
Ho..oe\.oer, t.ie r.et.'10d of qenerati.ng the image is quite different.
'fule-
Vl.SlOn uses a raster scan tedmi~ (see below) to generate the reqw..red picture, ,,t\e.reas the traditional refres.'l CR!' gra!Xllcs display is of the call igraFhic or li.nr-draw.i.n:] type.
A refresh CRr graphics display requires
addition to the cat:hodc ray bbc itself : troller.
~
elerents in
a display buffer and a display a:m-
In order to understand the advantages and limitations of a refresh
display it is necessary to cxmceptually l..ll'lderst.md the puxpose of these devices. s~ the:
•
verJ rapidly, CQ'ISt.ruct.: t."'"
rate.
ptoS"t.hor used on the catl1ode rey tlbe of a refresh display fades .e., has a short persistenc2, it u; neoessa.ry to repaint or re-
enUre picture
rnny
tires each sccx:nd .
nus is
called the refresh
A refresh ra~ t."lat is too ].o..,. \..:ill result i...r1 a ~"nrCCOO called flicker .
'Ihi.s is SL'!Ular to the effect "ttich results fran runni..ng a rrovie projector too
minittun rcfrcs.'l rate of 30 tires per SC
slew.
A
'I!.e !unction of the display buffer is t:O store m
sequ"..l'lre
all of the inst.ruc-
tions neoc.ssacy to cirtlw the picture on the catliode ray t\be.
fu function of
the display controller is to access (cycle through) these instruct.ions at the
Imccdiately, a limitation of the refresh
refresh rate.
~lay
is obvious:
'lbe OOiplexity of the picture is l.i.mited by the size of the display buffer and
Ho..-ever, the short persistence of the
the speed of the chsplay oontroller.
ilnac]e CM be liSE"d to advantage to sho.-r dynanic rrotion.
In pcuticular, the pic-
ture can be \.{)dated every refresh or, say, every other refresh cycle if dolble
buffering is used.
fUrther , si..nm each elcrrcnt or instruction necessar; to
drcr,.. the mcplet.c picture exists in the display buffer, any individual ele!mnt
can be changed, deleted, or an adlitional e1crncnt £Kklcd: i.e., a selective erase feature can be iltplarented.
One additional disadvantage
or
a refresh CRI'
graphic device is the relative difficulty of abta.i.nirxj a hard oopy of the pic-
ture.
1\lthough refresh CRJ' graphic displnys are generally rrore e>epcnsive than
storage ttiJe displays , the above characteristics make them the Uisplay of choice when dynamic rrotion in real t.irre or very rapid interaction with the display is
required.
A raster scan CRT grafh,ics display uses a starxlard television monitor for In the raster scan display the picture is cnrposed of a
tile display c:or.sole.
series of dots.
These dots are traced out using a dual raster scan t:.cchniqoo,
i.e., as a series of horlwntal lines . used to reduc:e flicker .
'l\o.1o rasters as sho.m in Fig. 1-10 are
'lhe basic electrical signal used to drive the display
cxnsole is an analog signal wl'Y:>se nodulation represents the intensity of the individual dots which cntpot:;e the picture.
In using a raster scan display con-
sole it is first ~sary to convert the line and d\aracter infotmatlon to a
form <:x:ccpatible with the raster presentation. wrsion.
'Ihis pn:leeSs is called scan
ccn-
once it is converted the infox:mation mJSt be stored such that it can
be ae«:Ssed in a reasonable manner.
ques this is
~9
rrore feasible .
display, the odvantagcs and
With the advances in data storage tedmi-
In considering a raster scan CRT graphic
disadvanta~
are similar to those for line-drawing
displays , with sate additional cx:nsiilerations .
Narrely, they are generally sore-
what sl011er, the selective erase feature is rrore difficult to inpl.erent, and they 11\:lY be directly
interfa~
Digital incre:rental pen
bed and drum.
plotter.
to cloGed-circuit television systans.
am ink plotters are of two general
Figure 1- ll sho..ls a flat-bed plotter and
F~g.
types - flat-
1-12 sho,..rs a drun
l'bst diun and flat-bed plotters qJe.rat.e in an incrorcntol rrode: i.e.,
the plotting tool, which need rot be a pen, noves across the plotting surface
in a senPs of small steps , typically 0.001 to 0 . 01 inch. her of directions in which the tool can nove is limited,
directions stn.n in Fig. 1-13. a series of small steps .
Frequently the run-
say, to the eight
'lhis results in a curved line appearing to be
In a nat-bed plotter the table
ll1
generally station-
ary aro
the ·..ni. ting head rroves in o.u
di.rrtmsions over tho surface of the table . A d..>"\r.\
f-t n t ra~ lff '-t:conll ra ~ tt:r
technique to achicvo bo."CHl1.!:-cnsional notioo.
--
_.... _,_.
--- - --
ing tool.
-~
le.ngth.dsc notion is obtained by roll.i..nq
,
,;::;;JI-...c..-- -
-
-
-
the paf>"'r
-::-~
and forth
uroer
the rrork-
high-quality hard ~ o f qrat:hical output.
O::crpared to CRT
they are qw. tc
sl~ .
grapucs deVl.ces COnsequently t.'"eey
are not generally used for .rea.l- tiJre i.ntl!ractiw gratilics .
--
~~ ...-.:. ~' -----~-- L _____.-'" - '\. ...... -~-<..: --:11.:-..T>
-__ -
a particular aplllcat.ion, a flat-bed plotter can be utilized as
oot
a coubination
digitizer and plotter and an interactive
Figure 1-10 Paster scan technique. ':he elect.rost.1tic
no--ever, where
large dra.rings are noill\llly required for
-' -- --- ----------' ----
__,-
~ck
Digital i.ncrem:mtal plotters can prov:ide
--
---------
. r=::
c:..
Here the marld.nq tool rroves bac:Y.
and forth across the paper while the
-__,....--::: =
plotter uses a sarewh8t different
mtputer grllfhic system S'n::essfully cb.-elopcd (see Pef . 1-2) •
matrix printer/plotter operates by depositing part-
icles of a toner onto small electrostatically charged areas of a speoal paper. F1gure 1-14 sho-"9 the qenernl
~which
lS esrployed .
In detail, a specially
coated paper which will hold an electrostatic charge is passed owr a writing head -.,tu.ch oont:cu.ns n ro..- of SJMll
per inch
iU'1:'
ial paper.
typical.
~-riting
nibs or styb..
Fr'Oia 70 to 200 styli
The styli deposit an electrostatic charge onto the spec-
Since tho alectrostatic c:ha..rges a..re therrselves rot visible, the
charged P
particles. ~
'lhc particles are attracted to tr.e electrostatically dlarqed areas ,
then visible.
The paper is then dried and presented to tOO user.
Very
high speeds can tx> d:>tainOO, typically fran 500 to 1000 lines per minuto . 'lhe elccLrosl..c:iUc cbt rnatri.'< printer/plotter is a raster scan device; i.e. , 1t. pres~ts
mformation one line at a t.irre.
Bec.!suse 1t is a raster scan devire
it. requires .:1 scl>stantial crount of mcputcr storage to construct a CXJC•pletc picturo. 'lhis p lu.,. the fact that the ~vice is useable for only pass1.ve gr~ ics
'll'T'
the principle disadvantages.
A further cti..ecedvantagoe
accuracy ar.d rcsolut ~ m, typically ~ 0 . 01 l.lldl.
1..9
relatively lew
'1l\c principal advanta<}es are
th.. very high speo..-.d wi th "''hich drawing!> can be prtdloed and an excellent reli-
ooility rcoord.
Figure 1-lS sho.ls an electrost..'ltic dot rrat.rix printer/plotter
Figure 1-ll flat-bed p1otter.
Figure 1- u
Drum plotter.
• , Pll pe 1
drne
I one r upp iJCiltor
Rcill e lee t rode
.._ ....
OJ
-I
,
Pa Jl e r $Uppl)i
I
Fi'JUCe 1-13 Direc:t..i..ats for incre
Figum 1-14 Conceptl:.a1 description of electrostatic dot rntrlx printer/plotter.
and typical output.
1-7
CLASS l FICATIOI~ ()- f1wt.i i CS l£vlCES 'lhcre u.rc a n\.UTbcr of rrethods of cl assifying couputcr graphics devims.
Each rrethoJ yields ~via.:.s.
~It'"!
sate
insight into the sa1etirres confusi.ng array of possible
will dic;cuss several different
t~Ethods .
f'irst let's consider the difference beb.'een a passive and an active grafhics d>vi~.
A IldSSive qr~cs devi~ s.inply draws pictures lrder O:XIpulcr control;
i.e., it
alla..~S
are
c1
the o:x•puter to Cl001'1lln.icate graphically Wl.th the user .
t.elctypa , a high-speed 1..i.ne printer, and an clcctroGtatic
EXillTplcs
oot matrix
printer, pen and ink plotters, and stor~ tlbe cathcxie ray tlbe (CXJ') and refresh CRI' devices . Exar.'plcs of sate of these deVices curl the typical pictures that th~ rniljht qenr;:rab: an. sho..-n in fig. 1- 10, 1-U, and 1- 15 t.h.ro\ql 1-20. '!be
re~r
might wonder aboot ronsidcrinq thP tel etype and the high-speed line
printer as gr.J(..hics devices . l~ver. the'; have been used to drcr,... sirple qrafhs or plots for a nurber of years .
-
Figure 1-15 E.l.ectrostatic dot trat:.rix printer/plotter and typical ootput.
•
I
~
~
,p -
rill
J. • I ~~iii:'!
~
..: Ill'! :;;;,
1.11 :00
Figure 1-16 Storage tube CRT gra(i\i cs device am typi cal output.
Figure 1-17
~ CRI'
graphics device and typical ootput.
=--"'\~
~r ·
t
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c: ..
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=~
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-·...., -- -,.,..._,.,.. --- --·- :.-.; •
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:.
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---: . -• -
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~_.,__
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----
f'=oii_,___
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......
-
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.
INrR:IXX:l'ICN TO C'Of-IPI.1I'ER GPJ\PI JICS 'l'mNXDGY
llo ............ • ••• I •
• •
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... ···-· ·······
•
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Figure 1-19 High-speed printer and typical graphic output .
• • •
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.' '
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... ... .... -· "
•
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I
Figure 1-20 Teletype and typical grapuc output .
I f points are plotted close enough t:.ogether , they will awear to the eye to be a solid line.
All the storage tli>e CRl' and rrost refresh CRI' gnph.ics devia:!S
are line-dra.ring devices. refresh CRr grapucs
All the pen plotters are
devi~,
·
devices.
5atc
particularly raster scan (televl.Sion-lil'..e) devl.ces,
can be oonsidered as point-plotting gra(ilics devims .
'l'eletypes, hi
line printers, and electrostatic dot matrix printers are classed as point'lhe uti 1 i ty of a device can
pl otting devices .
f~y
of its resolution; e .g . 1 a teletype has a resolution of
aro :
be CX)OSl..deted in tet::ms
! 1/20
i..ndl horizontally
l/1 2 inch vertically, ~reas an electrostatic cbt matrix device
a resolution of
:!
may have
0. 01 inch.
Still another rret:hod of classifyin] graphics devices ~ dctenni.ning whether a device can aooept true three--di.nensiooal data or whether threc-
d.Urensianal data nust first be converted to n.u-d:irrensicnal data by the appli-
cation of sare projective transformation and presented as C.u-dirrensional data. In essence this rrethocl requires detex:mi..ning whether a gra._chics device has 0..'0
or three registers to hold ooordinate data.
In the case of a
al
device the third or z ooordinat.e is usually used to control the wtensity of
a CRl' beam.
'"tis feature is caned intensi ty m:xlulation or gray scaling.
is used to give the illusi on
or
It
depth to a picture.
Each of the classification rret:hods will f-requently plAce a particular
gra{ilics device in a di fferent categoxy.
Ho«Ner1 each tmthod roes yield sorre
insight into the characteristics of the vari ous graphics devices .
Appendices A and C contain the architecture o f a software sche:fre based on the concepts presented in this and in subsequent chapters.
description
or position
Of
vectors , lines , cu.rves , and surfaces
C'Olli'SC ,
aro
the
the transfor-
mation of these gearetric entit ies can be accarplished mathenaticaJ.ly independent of any display technique or display software.
'lhe remainder of this book
is conoerned with these mat:hefratical , devioe-i.ndependtmt techniqoos .
REFERENCES 1-1 Newman, ''~· t-1.. ,
am Sproull,
R. , Princi pl es o£ Interactive O:ltputer
Graphics , r-ldinm..-Hill Book Catpany 1 New 'tori<, 1973. 1-2 Bezier, P . E., "E:xarple of an Existing Systerrs in the t-bt.Or Industry: '1he Uri.sur! Systan"
1971.
P:roc. lby . Soc. (I.orxbn) , \Ql A321, pp. 207-218 ,
CHAPTER 2 •
POINTS AND LINES
2-1
ltrrROIU:TION ~·1e
p.lter
begln our study of t.l-te furxla.'ielltals of the matharatics underlying ca•r-
grapti.cs by oonsidering the .representation and transformation of points
and llJleS .
Pol.Jlts am the lines wbidl join them
Me
used to represent oojects
or to display infot:rnation graphically on devices, such as discussed in Chapter 1.
'1he ability to transfonu these points and lines is basic to CXX1pUte.r gr a-
rtUcs.
\-,'hen visualizing an d:>ject
it II1!!IY be desirable to scale , rotate , t r ans-
late, distort, or develop a perspective view of the object . All of these transformations can be ac:carplished using the mathematical techniques discussed in thi!i and the next dlapter.
'2-2
ltP~scrrrATJ CX~
OF Ponrrs
!\ point can be represented in two dirrensions by its coordinates.
These
bJO valu:s can be spoc.ified as the el.en:!nts of a o~tn.~, b.oo-c:ol~..~~m nllltri.x
(x y).
In tJ'\rnP dirrensions a one by three rclt.rix [x y z)
lS
used .
point can ~ x~resented by a two-row, onc-cohrn matrix
ly,
c1.
cl
·ruions t:>r by
rotrlc.x:s like
[r]
r~J
in three di.Jrensions .
Alternate-
[¥ J in
bNC
ION rMtrices like (x,y] or oolurm
arc frequently called vectors .
A series of points, ~ of
f'OINI'S N:O LINES
25
\o.hlch is a position vector relative to a local ooordlnate systGn, may be stored in a ttluputer as a matrix of ru.mbers .
'Ire position of these points can
be controlled by manipulating the rrotri.x \"trich defines the points.
Lines can
be dr.r..-n beo.oeen the points using afP:roprlate cxxtputer ha.rd../aro or software to
generate lines , curves , or pictures as output.
2-3
TRANSFf:»4ATICl'lS f\.'40 1'\.\TRICES 'lhc c.lerrcnts that make tt> a matrix can represent various quanutics , such
as a mmber store, a nab.'Orl<, or the coefficients of a set of e:JUations .
'111e
rules of matrix algebra define allaVclble ooe:rations on these rntrioes (cf Appen-
dix B) .
t1an}• physical prd.:)lerns lead to a rrotrix fonrulation .
m::ry be fonrulated as:
Nr = B.
Hero the prd>lGn
Given the matrices A and B flnd the solution matrix, i.e.,
In this case the solution is T ,. A- la, where A-1 is the inverse of the
square matrix A (see R:!f . 2-1) •
a matrix is to treat the '1'-matrix itself as an operator. Here rotrix trultiplication is ~ to perform a georetrical transformation on l\n alternate usc of
a set of points represented by the position vectors contained in nutri.x 7\.
matrims
A and T
'nle
are assurrcd krn-'1"1 arrl it is required to det:.e.rrnine the elerents
of the matrix B.
Th.is interpretation of a mat.ri.x nultiplication as a
~trical
qJerator is the foundation of the ll'athernatical transforrrations useful in cx:mputer gra!itics.
2-4
TRANSFO~'ATI~ (J= PoiNTS COnsider the results of the matrix nultlplicat.i.on of a nutrix {x y) con-
taining the ooordinates of a point P and a qeneral 2 x 2 transforttation matrix:
(x y]
[:
:]
[(ax + cy)
os
(bx + dy})
=
[x*
y*1
(2-l)
'lbis mathematical rotation mans that the initial CXX>rcli.nates x and y are trans-
fenc-ed to x* and y*, where x* .. (ax + cy) and y *
= (bx + dy) •
We
are interested
in investigating the inplicat.J.ons of considering x* and y* as the tra.nsfornoo
CXX>rdinates .
\'le begin by investigating several Sfeeial
Consider the case where a
=d =1
arrl c = b = 0 .
~,ses.
'lhe t.ransformltion Nt.rix
then reduces to the identity matri.x and (x
y)
=
l~ ~]
"'
[x
y]
= [x •
y* )
and oo change in the ooordi.nates of the point I' occurs . tJext
consider d : 1 , b
=c
= O, i.e.,
(2-2)
26 t-fi\l'HEW\.TIO\L EIDIM'S FOR CQ-!'Ul'ER GRAPHICS
(X
produo~
'!his
r :1q. 2-la.
[~ ~ J •
y)
(ax
y*)
y) = (x•
a scale change since x'" =- ax.
(2-3)
'Ihis transformation is sho.om in
llenoe this natrix nultiplication has the effect of stretching the ~on .
original ooordinatc in the
r-o,., consider
b • c
=- o, i.e.,
f~ ~J
(x y]
•
= (ax
dy]
= [X*
y*]
(2-4)
'lllis yields o stret:dting in both the x- and y-ooordinates , as s ho.m in Fig.
If a ;. d, then the stret.chinqs are rot equal.
2-lb .
en.Uw~rent
If a = d > 1, then a pure
o r scal.i.nc:] of the c:oordinates of P occurs .. If 0 < a = d < 1, then
a m tprcssion o f the ooordi.nates of P will occur. If a ar¥.\/or d are negative, then reflections occur.
b - c - 0, d .. 1 and a (x
-1.
s
'Lhen
[-~ ~] =
y]
(-x
Y1 =
[x*
ard a reflection about the y-axis has occurred .
t.ion is sho..m in Fig. 2-lc . about the x-axis occurs. origin will occur.
To see this, consider
~
If b = c
If b = c =
o,
(2-5)
y*]
The effect of this transfonna-
0 , a::: 1, and d = -1, then a reflection a = d < 0, then a reflect.l.on about the
'Ihis is slnm in Fig . 2- ld, with a = -1, d
= - 1.
Note that
reflection, stretching , and sc:al.ing of the ooordinates involve only the dia
o f the transfoDMtion rratrix .
No..' <::a)Sider the case where a [x
[~ ~]
y)
= d = 1,
= [x
(bx
arxl c - 0 .
+ y) 1 =
[X*
'Ihus
(2- 6)
y*]
and lhe x-ooordi.nate of the point P is unchanged, while y* deperds linearly on
the origuvll coordinates. Similarly \o.iten a -. d
= 1,
to the y-c:xxu:d.t.nate , as
This effect is c11 led shear and is stnm in F1g . 2-le. b = o, the transformation prodtJCCS shear proportional
sho.rn in Fig. 2-lf .
Thus , we sec that the off-diagonal
tcnns produoo a shearing effect on the ooordinates of the point P. Bofore m •pletirq our discussion or the transfoll'Mtion of points, consider the effect lD
Of
the qcreral 2
X
2 transformation qiven by Eq . (2-1) when af{>lied
the oriqin, i.e.,
(x y)
:J [: :]
= [(ax+ cy)
(bx + dy)]
[: or for the o rigin, [0
IIP['P '1\"e
OJ
= [0
0 ) = [X*
= (x*
y*)
(2-1)
y*}
see that thP o rigin is invariant uroer a general 2
X
2 transfonmtion.
'ltn s is a liJTU t.ation whidl will be overc:nre by thP. use of ha•ogenrous coordinates .
POINt'S AND I..JNES 27
) ·.)
•r
,.}
[~ ~]
=
p• I
>·~
•
,, ,. -.
I
I
I
•
I'
' ,.
•
,.
[~ ~]
T=
I
•
- --
)
T= r-10 -1OJ
.
~ ~
,,,.T= ·" ,.
!-
r
, ••
I'
,
•
'·~.
L
....
\
.
[~ ~] h~
)
.
".
)
, . .T [~ ~]
n
,
(\'•\
I'
)
h\
',.
•
"' ".
•
~ "'
f
Figure 2-1
2-5
•
c
\. \
,,
I'
I.
\ , A•
~
,,
~ [-~ ~]
T
I
TRANSFORI'V\TIOO
OF
Transfonnation of points .
STRAIGKT LINES
A straight line can be tlcfined by bov rosilion v.....ctors \olhi.d1 specify the CXJOCcli.ndtes of its end (X>ints. uq these b.."t> points can be ~
'nle position and orientation of the line joirr
c:::hatY.Jed
by operatul<} on t:hcsc two pJSition vectors .
actual q>eration of dro:JWing a line bet\-'{!(:11 two points will depend on the Hen. we cx:m sider only the mdUtunalical operations on the
output ck.'Vi.ce used.
er¥:1 posi lioo vectors. A straight l ine bebw'cen b>o points A and drd;IJ'l in Fig. 2-2 .
(2
u in a
~ional
'lhc posit ion vectors of r.ou1ts 1\ and D
3) rest=eelively.
CJ.re
(0
plarE is
11
aro
No.I cons1der the trrlllStoDT\llion malrix T =
[~ ~]
(2-7)
....tuch
.,.,'C
~.Sing
mJ.tri.x ITllltipliC
~w
recognize fran our previous discussion llS producing a shearing effect.
Lransfomed vectors N· cmd
AT~
to 11
a•
CJl.Ven by
[~ ~]
(J
1) • A*
(2-8)
8'!' :
(2
(2-9)
3)
'Ihus, the rcs\.O.ltinq el.e"ents in A• a..-e x* = 3 ani y* f'l&l [X>int
~·
Similarly 1 B* is a
f'bre mtpaccl.y the line AB
specified by x* ... 11 and y• • 7.
rcprcse."'tcd by the 2 x 2 ~TUtrix L = (~
= 1.
l·atrix nul.tipllcation t.~cn yields
LT ""fo 1] '"1 2] = r3 1] .. L* l2
3 l3 1
rror:t be
(2- 10)
u1 1
"''here the mtp:ments of L* represent the transfoll""Ed position 'v'ect:ors A" and
B*.
The transfot:r:\ltion of A to A* and B to B* i s sho.m• in Fig . 2-2.
'Ihis shearing effect
tial axes are x,y and the transfonred axes axe x*, y *.
has increased the length of the line
2-6
am
'Ihe ini-
changed its or ientat ion .
MIIYOINT TAA~SF00"1f\TION Figure 2-2 sho..'s that the 2 x 2 transfOllMtion rratrix tral'.sforms the
straight line y = }' a
{3/4) X
-
X
+ 1, beO.-een points A and B, .into another straight l.ine
5/4, bet.-.-een A* and B* .
In £act a 2 X 2 natrix transfOJl!'S atrj
straight line i."lto a secord straight line.
Points on the seo:m;i l..lne have a
one to one correstx>l"lder::oo with points en the first line.
this t.o be trua for the c!'ld p-rints of the line.
o:::nsider
B.
t.~
Letting A
\'ie have al..ready s.l\a..'n
To further ccnfinn this we
J
transfolT.Iation of the midpoint of the straight line beboleen A and
= [xl
Y1l, B = (x 2 y 2 ) 1 and T • [:
{X)ints sir.ult:aneously
and transforr.U.ng both
gives
[:~ ~~1 r:
:J
=
r:~: :~
(2-11)
Hence, the transfomed end {X)ints are
A*
= (axl + en
B* • {ax2 + c.y2
bxl + dyl) bx2
= (xl•
+ dy2) =
Y1*l
Y2 * )
'Ire cidpoint of the initial line AS is
]
(2-U)
'Ihe t.rar.sfo rr:'dticn o f this midpoint is
] [: :]
=
axl + ax2 + c.yl + c.y2 2 bx1 + bx2 +
•
2
ay1 +
(2-13)
l<>INI'S lWD LINI:S 29
'lhl..s transfomation places the rnidpo111t of AS on the midpoint ot i\*B11 since
the
rniqx>~.nt
of A*B* is giwn by
ax1 + cyl
x• =
+ ax2
~
cy2
2
bxl
+ dyl
+ dy2
t- bx2
r - ----...---2
=----..,.,----2 bx1 + bx2 + dyl + dy2 2
------=----...;;...
For the spec1.al case sho.m in Fi9 . 2-2, the [1
(2-14)
axl + ax2 + en+ cy2
uli~t
of line 1\B is H1
=
'1hls transforms to
2] .
[1
2]
G~ J :
[7
\,hlch gives the mic\X>int on line A*B*.
(2-15)
4)
'lhis process can l::e
rep"'.lt.ed
for any
subset of the initial l i.ne, and it is clear that all points on the initial line Lransform to points oo the second line .
Thus the transformalion providas a one
to one correspondence bet..;een points on the original line and points on the
transfor:rred line.
Fbr o:::rcputcr graphics applications, this nmns that any
straight line can be transfo.rtred to an.t end points and then
2-7
redrawi.n9
nEM
position by s.inply transfonning its
the line betlo.-een the transfomed end points.
PARAU.£L LiNES
a 2 x 2 rratrix is used to transfonn a pair of parallel lines, t.~ result is <1 seoond pair of parallel lines . To see this, oonsider a line beb."OOn A = [xl y 1 ) and B .; lx2 yz} and a line parallel to AD beo.Aoon E and f' . To shari that these l..incs and any transfomation of them are parallel, we exa:-.U.ne \•lhen
'1he slope of both AB and EF is
the slopes of AB, EF, A*B*, and E*F*.
Y2- Yl
(2-16)
mt • X2 - Xl
Transfo.cning AB usi.ng a general 2 x 2 t.ransfonnation yields t.he end points of A*B11 :
xl [><2
Y1
Y2
J
[a b] = c
d
rnx1 + cyl
bXJ. + d.Y11 = [>
lax2 + cy2 bx2 + b.Y2
~
yt] Yi
=
[A"]
(2-17)
B1t
'the slope of A*D* is then
(bx2 + dy2 ) - (bx1 + dy 1) m.. = ........---..,..---,--.;.;_-~ · ··~ (ax2 + cy2) - (axl + cy1)
:::0
b(x2 - x1) + d(y2 - Yl) _...,..---:---:-----:aCx2 - xl) + c(y2 - Yl)
or
b+d m 2-a+c
Y2- Yl x2 - >
b + dnl ~
a+ on
1
(2-18)
12
•8
I(I
T== [ll -32]
8
10
T:
[~ ~] • l ( 1/2 J/] lA l •
6 ) .\ .
4 ...
·u
,
·2 - 4-
,
0 0
· ll
-
~·
0',\
~ H13
b
•
IU
•
•
12
'·"
2-2
Figure 2- 3 Transformation of intersect.ing lines .
Figure Transformation of straight llne:s.
Since the slope~ is indc~t of x1 , x 2 , y 1 , and y , and since "1• a, b , c, 2 and d are the sarm for EF and AS, it follo.,'S that ~ for both E•F* and A*B* are equ.al.. 'll\us, parallel lines remain parallel after t.ransfonration . This ~ that parallelogra:rs t.ransfonn into other parallclogtans as a result of operation by a 2 x 2 t...ransfonnctt..wn matrix. 'Ihese siltpl e results begin to sho.o.' the po..-er of using r.atri.x rrul.tiplication to produce g:rlltiric:al effects.
2-8
INTERSECTING LINES '1\-.lo dashed 111tersect.ing lines AS and EF are s.ha.m in Fig . 2-3.
'Ihe
~int
4/5 and y = 1/S. We no" rrult.iply the rratric:es oont.al.ning the end points of the two lines AS and EF by the transfonnation matrix
of intersection is at x
[:
~] .
'll\us ,
- 1 2
3 2
1
2
3
-2
1
-3
- 1
_!! 2
12
romrs
-1 -1] [1 [ ~ 3
3
2] -
1 -3
A'llD LINES
31
-2~ 1 3
1
This gives the solid lines A*B* and E*F* sho..m in Fig. 2-3.
'lhe transfamed
point of int:ersect.ion is
[~ ~)
[l
1)
'lhat is, the point of intersection of the initial pair of lines transfotli'G to the point of intersection of the t.rclnSforned pair or lines .
Close examination
of the above transformation of intersecting straight lines will sha.l that it has involved a rotation, a reflection, and a scali.nq of lhe original pair.
However, the total effect of a 2 x 2 matrix transformation l.S easier to see, considering the effects of rotation, r eflection, and scaling separately . In order to illustrate these .itdividual effects, we shall oonsider a sinple plane figure - narrcly, a triangle.
2-9
fbTATION Consider the pl ane triangle ABC s.ha.m in Fig. 2-4a .
'Ihis triangle is ro-
tated through 90° about the or igin in a counterclockwise sense by operating on
ead\ vertex with the trnnsforTMtion [ 0 1]. -1 0
If we use a 3 x 2 matrix conta:i.n-
ing the x- a00 y-coordinates Of the vertices 1 then
3
1 [01]'"'
4 1 -1 0 2 1
-1 4 -1 2
'Ih.is produces the triangle A*B*C*. ed by using
r-1
(2-19)
13
A 180° rotation llbout lhc origin is abu.i n-
0] and a 270° rotation about the origin by using
0 -1 Note that neither scal.in;1
lo
-1].
1
0
nor reflecticn has occurred in these cxanples.
2-10 REFLECTION ~\!lereas
a pu.re
~onal
rotation in the xy-plane occurs about an
ax:is norm:ll. to the xy-p1ane, a reflection is a 180° rotation <~bout nn axis in the xy-plane.
'IWo
reflections of the tri.anqle
reflection about lhc line y
=x
[EF
occurs by usinq [~
'lhe trans foetal, new vertices are qiven by
am shcMn in Fig. 2-
~] •
A
•
•
• •
(}'
I•
a
!:>
s ,-• ro
•
~
I
3
~6
2
J.• .ref~on a::xr~ y =
1
l
oJ
-
•
....' e-
0
-
3
-
,(.
2-20)
6.
:..s cbt.a:. nee
fr.::clfl _,0, . .. (;
a-'"C
s-h~.
by •
~.q
sca 1 ~ r.g
a l- 1 7
3
.e
2
0
-~J
--
a.:.r d' sc.:.ss!.on of
•
8
-l
-
7
3
6
2]
~-...e r:...~.sfot~ar::oo cf
is cxr.trolled ~· t.'le. rogr.itu:ie of ~"'.e
cf t."le ratri.Y..
If the
t'iltr-Y.
-4
t;.O
r2 0 .:.S ..:sed as .0
r=oi:ts, W'£: can see
te~
Zi!".
or.
mat
~..e: p~-.:tri ~.al
0""~-.ttor cr: the ..~_ioes
2j
o f a ~~"l'il~, a .. 2-t.i:'cs• e.-: .arge: -er.t. occ;.n:s abo"~ the oric;.r. . If the r..ac;nitn1es nre ur~l, a d.lSt.OrtJ..on oo:.-.!..""5 . ~..t:S= eff~...s are sha--n i:-. fig . 2-'>. Tria."¥]!~ 1-& if: t=c;J"Sfc:::;:-.-,d
at 2 Q'j ; 0 :2
~'l:.s a sCil.l.i.N; OCC\!I"S , a.'"ld
POD\'TS A"''D t..niES C£f' is transforrrod by
33
{~ ~] which yields
a distortion chr to the unequal sc.osle factors . It is n::w clear how plane surfaces, defined by ~oes joined by straight lines, can be mmipulated in a variety ~ of ways . By performing a matrix operation on the position vectors which define the \.~oes , the shape and position of the surfa~ can be controlled. Ho...-e\...:rr, a desired orientation my require no;re than one transformation . Sinre mtri.x
}
.~ I•
• I•
•
1\ t-
~ ;'
(\
~ ~
r
r
-
nultiplication is noncntuutatiw, the order of the transfoxrnations is inport-
'
•
ant when performing cntbined q:Jeratior.s .
,
'I
-
It
c·
~~~·
'"'
I)
- {
,
-
-'
">; "8 ,_ ~
--
\
-
•• \ \
h
,.
Figure 2-5 L'niform and nonuniform scaling or distortion.
In order to illustrate the effect
of ronc:amult.at.lve mat:rix nult.lplication,
consider the operations of rotation and reflection on the
(x y] .
If a 90° rotation is follo.,'ed
~·
verti~
of a triangle
reflection about x- o, these two
consecutive t:ransforr.ations cfive [x
0 1] "'
y] [
[-y
x)
-1 0 and then (-y
en the other
X)
[-~ ~]
•
(- x
y)
hard, if refl ection is follor..ted by rotation, a different result
given by y] [
(X
-~ ~1
=
[-x y]
and
[-x
y) [
0 11 • -1 0
(-y
-xl
is d:>t:aizled . So far we
ha\."C
concentrated on the behavior of points and lines to deter-
mine tho effect of sillple rratri.x transfonrations. correctly considered to operate on
C:Vt.l'll
However, the rratri.x
point in tr.e plane.
lS
As has been shown,
the only poir.t that remains invariant in a 2 x 2 matrix transfonnatjon is the origin.
All other points within the pl ane of the coordinate system are trans-
ferred.
~
pla~
transfoll'l\ltion nay be inte.rpreted as a stretching of the initial
and coordinate system into a new sh.apQ.
M:>re forna.lly, we say that the
transfonMHon causes n J'lk'!IPPi.nc; fran ore plane into a second plane.
Exanples
of this rnawin:J are sm.,., in the next section.
2-13
TRANS FO~\ATIOO
OF A ~IT
O::nsidcr a square-
SooAAE
~l:k
consisting of unit squares in the xy-plane.
four position vectors o f a unit square with one CX>J.'Tl& at the origin of the
coordi rote system arc 0 0,
o rigin of the coordinates - A
1
0
unit point on the x-ax:is - B
l
1
outer oorner -
0
1
unit point on the y-ax:is - D
c
'Ihis unit squru:e is shom in Fig. 2-6a.
:J
transformation [:
::
: ~e results
Application of a general 2 x 2 matrix
to the unit squaxe yields
~ ~ 1[: ~ ~J
:J- ~ la: c
~
(2-21)
:::
b : d:
First ootice fran Eq. (2- 21) that the origin is not affected by the transformation, i.e., A • A* o f this transfonnation are sl'la.m in Fig. 2-6b.
= (0 OJ. fUrther, ootioe that the coordinates of B* are a;JUil.l to the first row in the genera l transformation matrix, and the coordinates of D* are t:q1..li!l to the scoond row in the general transfo.rmation matrix.
dinates of B* and D* (the transfomed unit points fl
'nlus , onoe the coor-
01 and [0 1) respectively)
are J
Since the sides
of the unit square are o riginally parallel and sinoe we have previous l y shotm that parallel l ioos transfoDn into parallel lines, the transfor.med figure is
a parall elogram. The effect of the t..enrs , a , b, c, and d in the 2 x 2 rrotrix can be identified Sep
'1he t:.ez:ns b and c cause a
~ in they- and
,IP
shea.rin:J
(cf Sec. 2-4) of the initial
x-directions respectively, as can be seen in Fig. 2-6.
terms a asd d act as scal e f actors as noted earlier .
Thus, the general
2 x 2 rotrix produces a ooub ination of shearing and sca.lintJ. It is a l so possibl e to easily det:errrU.nc the area o f the parallelogram J\*ti• C*O• shown in F'i.g . 2-6 .
'Ibe area within the parallelogram can be calculated
POINI'S PND L.INES
35
)
.. l •
d
I
tJ'
~-
u
-
I
a
-x
b
Before
After
Figure 2-6 General t.rar.sfomatian of unit square.
as follc:MS :
~
•
(a
t-
c) (b + d) -j{ab) - i(cb) -¥(b + b + d) -
~(c
+ a + c)
which yiel ds (2-22)
It can be shown that the area of any t.ransfoJJTed of the transformatioo natrix det:e.J:rninant
initial square
~
aro
parallelogram~
is a function
is related to the area of the
by the sinple relationship (2-23)
In fact , since the area of a general figure is the
area of any transfomcd
figure~ is
SUTI
of urut squares, the
related to the area of the initial fi«]llle
~by (2-24)
'!his is a useful technique for
2-14
AAB ITRARV
2 X 2 l'bTATI~
de~
the areas of arl:atrary shapes.
r•'ATRJX
'lhe general 2 x 2 rrat.rix which results in pure rotation of a figure about
~~ICAL £IEJ.fm'S
36
FnR CCl.rorER GAAPHICS the origm can be obtained by consider-
ing the rotation o f the unit square about
the Origl.n . 'lhis is sl'loNn in Fig. 2-7 for positive rotation in tl'.e c:ounter:clockwise direction . Figure 2-7 stXJWS that p:>int B, [1 01 , is transfomed to the po-1.1\t B*, with x* = (l)c:osa andy* = {l)sme and that p:>int o, [0 ll, is
))
transfonred to D*, with x• and y* = (l)cose .
= -(l)sin6
Recalhng our previous
cbservatjoo that for an amitraxy 2
X
2
transfol:I'Mtion rratrix the cxordinates of B* nust equal the first
ro-o• in the trans-
fot:mation matrix and the coordinates of
.. _..,.
D* form the second
;
roN,
the general 2 x 2
rotation rratrix is
I - • - _.._______...0....__ ~B X 0
)...
I
(2-25)
c:os9 sinA] [:-sin p cos 6
For the special case of 90°, this reduces
to the result t!Sed earlier in Sec. 2- 9, i.e.,
Figure 2-7 lbtat.J.on of unit square.
[_~ ~] In effect F.q . (2-25) is a mrbination of scaling and shearing ~ch leaves the
final dirrensions unc:h.anged. pure rotation through an
2-15
'lherefore, this transforrrat.ion rr.atrix produces a
amitraey angle e.
Two-DJtelSI
AND
Up until the present t:.im! we have
~EOUS CooRDI NATES . rot discussed the translation of points
and llnes or the figures or shapes resulting from COJtbinations of points and li.Jl,e,s .
E\.rndM"entally this is because within the fran-e ,.ork of a general 2 x 2
l1\3t.rlx it is rot possible to introduce the constants of translation.
Foll.ow.l..ng
Fouest (Pcf. 2-2) we rote that this difficulty can be overc:x:JTC by introducing
a thlrd corponent to the p>int vectors {x y] and [x* y *), naking them [X
y
1) and [x*
matrix, say,
Y"
1) .
'l'he transfonnat.ion I'T'atrix then becares a 3 x 2
'Ihis is necessary since the nurber of ooh.rms in the !X)int nntrix nust equal the llUiber of
in the transformation matrix for the operation of matrix
ro.tS
I!Ul.tipllcat ion to be defined. [x
y
1)
'Ihus, 0 ' = [x + n
1
y + n] "" (x• y•]
(2-26)
0 1
m nJ where W"e note that the constants m, n cause a translation of x" and y• relative to x and y. tblever, a 3 x 2 rratrix is not square and thus does not have an 'Ihis difficulty can be
inverse (cf 1lppCNiix B) .
ovenx:t~1c
by utilizing a full
3 x 3 matrix, say,
1
0 0
0 1 0 m n as the t.ransfonmtion matrix .
Notice that the third exxtp:)nent of the position
vectors i s not affected by the addition of the third col~..~~m to the transfo:tm:ttion matrix. Using this matrix in SJ. (2- 26) gives (x* y" 1] for the transfonred vector .
'lhe adCi +-..ion of the third elarent to the position vector a00 the third
co1mn to the transformation matrix a.l.l.oNs us to perfonn a trar.slation of the position vector. Alternately W"e may consider the third elerre.nt as an addit.J..onal coordinate of the positioo ·vector. For notatiooal pw:poses we assuro that when the fX)Siti.on vector (x
y
11 is operated on by a general 3 x 3 natrix it beoorres
the p:>Sition vector [X Y H) .
y'* 1} .
that (X Y Hl • [x •
Previously the transforrratioo was perfo01ed suc::n If we consider the t.ransfooration to be taking
pl ace in thrce-dimmsional space , this confines our trar.sfor.retions to the plane
H ,. l.
If, t-o,..ever,
W"e
assure that the third colum of the general 3 x 3 trans-
fomation matrix T is rather than
r0 0
ll matrix rrultiplication yields [x y
l )T = (X
Y
H) , where H
1- 1.
'lhe general plane in 'Whidl the transforrred position vector rt::M lies is in th;t:'e(~.immsional
what happens in
space .
However, at the present tiJre we are not interested in . ional space.
Hence we d>tain x• andy* by proj ect-
ing ftun the general plane back onto the plane B = 1, using a pencil o f rays
thrcu.Jh the origin. 'Ihis effect is shewn in Fig. 2-8. Csing si.rnilar trumgles \o.'e see that H/X = 1/x* and H/Y ,.. 1/y* . With three ca1pooents it follo.-."S that: X y (2-27) (x*
•"*] = [:r H -H 1]
\ree-Cli.rre''lSional vector representation of a two-di.Jrensiooal positioo
H, sc nc rat plan CHII
Figure 2-8 Geatetric representation of hoiT'Ogeneous c::x:ord.ir.ates •
vector, or in general the representation of an n-a:a,pJnent vector by an n + l cx:arp:>nent vector, is called J:orogeneouc oool'dinat
like an additional c::x:ordinate .
'1he extra ccttpJnent in the vector acts
'111e ordi.nacy or "physl.cal" cx:x:>rdi.nates are re-
lated to the hcrrogeneoos ooordinates by X •
~
and
y =
~
(2- 28)
'Ihere is no unique haroge.ne::>us c::x:otdinate representation of a point in tw::>-dilrensl.Onal spaoe . For ex.anple, the lnrogeneous c::x:ordinates (12 8 4], [6
4
2], and (3
2
1]
all represent the ordinary point [3
of calculation we dloose [x y
For ease
1) to represent a rontransfonred powt in bolO
dirensional houoycneous OXJrdinates . r~
2] .
'then the transtormation*
y*J - rx yJ
*Note that here we have put the result of the tr&'\Sfotrnation oo the left-hand side of the equation rather than on the right-hand side as was done previously: i.e . , we write rratrix equations as B = AT rather than as 1'\£ = B. Fl:an hereon we w1.ll adopt this c:onwntion for ease in translat.i.ng results into cmputer algori tlurB.
romrs
J\ND I....INES 39
in regular coordinates is given by
= [x
(X Y Hl
in hatogeneous coordinateS. that X
= x*,
Y
= y«,
1]
y
b
0
carxying out the above two t.ra.n::>fotmaLions sllOoiS 'lhat is, since the addi banal conrdi.nate II is
and H .,. l.
unity, the transfonrod hatogeneous coordinates are the sasro as the transforrred In general, H 1- 1, and the transferr-ed ortli.nary coordi-
ordinary coordinates.
nates are d>t.ainOO by noxnallzin<} the transfomed hatogeneous coordinatr$, e.g., x* ..
~
and
y* -
~
(2-29)
In a gearet.rical sense, all transfor:mations of x andy fall within the plane H=l after normalization of the transfonred hatogeneous coordinat.cs.
'n\c advantage of i.nt.rcrlucir¥j honogeneous coordinates occurs in Ute qene.ral 3 x 3 trans formation rMtr.ix a b
p
c d q m n s where space is new available to include other traJ"I.sfoxnations such as translation.
n-.e
scaling and shearing effects of the IMtri.x terns a, b, c , and d have bc-€0
sho.m previously.
To sb:w the effect of the t:him colunn in the 3 x 3 transformatioo matrix,
consider the follOioling operation: [X
Y H}
= [x
11
y
1
0
p
0
1
q
0
0 1
Here X • x, Y • y , and li • px
= Lx
y
(px
+ qy + 1))
(2-34)
+ qy + l. 'Ihe coordinate H ~ch defines the
plane containing the t:ra.nsfonred [X>ints expressed in harogcncous coordinates 00;1
has the equation or a plane in thlre.<:Hii.tn:msional space.
'lhi.s transforma-
tion is sho.m in Fig. 2-9, where the line AB in the two-dimensional xy-plane is projected to the line CD in the
ional plane pX + cfi - II + 1 = 0.
In Fig. 2-9 the values p "" q = 1 are used .
!l:onrallz.i.n<;J to obtain the two transfoll'OOd ordi.nacy coordinates,
t;le
,.'rite
X X x*-n=pX+qY+l y
y* ; if
=
y pY. + qY + l
Letting p • q • l.O and USl.f¥1 [X>ints A, [1 gives
3) and B, (4
1} sho.m in Fig. 2-9
'
.)
4
>"
2-
(
.
,
-
-'-7'~o·- . -
(
•
'
A
. ..
II I
D
Figure 2-9
Transfoma+ion in harogeneoos coordinates .
for the transfoil'I'Ution and projection of A to C* , and
4 2 :x* •l+- 4+l =J
1 y*=6
and
for the transfomation and projection of B to D* . for C* and D* sN:Ml in Fig . 2- 9 are (1/5
3/ 5
'!be harogeneals coordinates
l) aJ".d [2/3
1/6
1) respectively .
'Jlle effece of rottralid.ng is to project the th.ree-di.Jrensional line
a two-cllirensional line C*D* in the plane H
= 1.0.
rn back onto
As shown in Fig . 2-9 , the
center of projection is the origin . 'Ihe C}eneral 3 x 3 t:ransforrration matri.'< far O.'()-(llirensiona.l honogeneous mordinates
Cdl\
be subdivided into four parts: ra b 1 p d
C
I 1
q
-m ---n ' s 1
l..s we have seen a , b , c, and d produce scaling, shearing, and rotation; m and
n pttxluce translation; and p and q prcxluoo a projection .
'lb shcMr this , consider the transformation
elcrrcnt s , produces overall scaling . (X
Y
H) "" [x
y
'Ihe f ourth part,
1)
1
0
0 1
lo o
01 • (x y 0 sJ
s]
(2-30)
POimS NID LINCS Here , X
= x,
Y :: y , and H = s .
41
'1his gives X
~-9 'lhe effective transformation is [x y the p:liSition vector.
1] •
y/s l}, a unifo tm scaling of
(x/s
If s < 1, then an enla..r:gen:mt occurs, and J.f s > 1, a re-
ductioo in scale occurs.
2-16
Po iNTS
AT
INFINllY
The use of harogeneous (X)Qrd.inates gives a oonvenient and efficient ..,ay
of rrapping a set of (X)ints fl:un one coordinate system into a oorresponding set in an alternate exx>rdinat:e system.
usually the infinite ranqe o f one ooordinate
system will rrap into a finite range within the second coordinate system.
1\l.so,
lines which are parallel in one system will usually not be parallel in the ot:l1cr .
Hoo.oever, geatetrical prcperties such as p:>ints of intersection can be evaluated
in either (X)Qrdinate system as s.ho.m bela¥" . 'Ihe O..Q lines given by
x+y
•1
2:x+3y=O
have a p:>int of intersection at x .. 3/5, y .,. 2/5 . ed in terms of the horToge.neous ooordi.nates.
'!his solution can be cbtain-
By rewriting the equations as
x + y - 1 • 0; 2x - 3y = 0 it is possible to represent the bolo equations by the
matrix equation [x
y
1)
•
1
2
;
[0
0)
1 -3
l-1
0
Ho..oever, for a matrix to have an inverse it must be square . '!1)e follc:Ming representation of the sane set of equations can be used to rreet this rcquirenent; (X y 1) r 1 2 0 e [0 0 1] (2-31) 1 -3 0
l-1
1
0
The inverse of this square matrix M is
t-fl
,,/~
j ol
1
1
3
2
55 LS ~tiplying
5
0 1
.J
both sides of Eq. (2-31) by
identity TMtrix yields
11C1 and
noting that
l·r·C1 = I,
the
y
[x
l 11 • 'S" (0
0
3
2
1
-1
3
2
ll
'nlus, x "" 3/5 and y = 2/5 es :req..:ired . Conslrlc- tloiO parallel lines defined by
x+v=l ~
x+y=O Proc::ee!ing as above leads to tt.e r.atrix EqUdtion (x
y
1) ~ 1
1
01
•
(0
0
1)
1 1 0 ..-1 fb.'C\.'C.r, C\te.:'l
an
0
•
lj
though a square matriX exists in this equation, it does not haw
since t..o ro..:s are identical .
~rse,
sible t..trich does have (x
y
Another al t:.emate eguation is pos-
a., invertable mn:riY.. 'Ihis is 1] j l
1
11
=
[0
0
gLvcn by (cf
~.
2-3) (2-32)
x)
1 1 oI -1 0 o.. The :i.m:erse of the 3 x 3 rratrix in Ec;. (2- 32) is
r~-1
=
r~ ~ -~l Ll
-1
!-hlt!.p1y.L'¥3 both sides of Eq. (2-32) by this inverse gives lx
1] ~ (0
y
0
x)
ro
1
0
1
1
-1
o. .
I
l1 ~
-x 0]
-11 - (x
1
resulting hatO<)el'leous CXlOrdinates [1 -1
x[l
-1
01
0) nust represent the "point of
intersectJ.on" for the n-o pa.rallellines , i.e., a point at infiruty.
tr.e t::...'O-d.urensicmal hatOgeneous vector (a b on the line ey - bx = 0.
(2- 33)
In ge."leral,
0) represents the point at infinity
'Iho fuct that a vector with the harogeneous carponent 11 equal to zero OOe5
indeed represent a point at infinity can be illustrated by the limiting proc:ess ConsJlder the line Y* = (J/4)x• and tr.e point (X
Sh.'Cn be1cw. calli r.<:J that
3] .
Re-
a unique represe.r.tation of a position vector docs not exist in
b:::m.::xJcneous cx:ordinates , tr.e point [4 co:>rd.i.nat.cs
Y) • [4
in all
t.~e ways
31 can be represer.ted in hcr.cgeneocs
sl'lcwn in Table 2-l.
See ~f . 2-3 .
Note t:Mt
lll
Tal;le 2- 1 as tH>, the ratio of yi/x' ren:rlrs at 3/11 as is rEqUired by the gove...>ning cqt2tion.
F\lrther rote that
SIICO"!SSiW
paj rs Of [x•
!all on the line y • - (J/4)x•, beoat·e closer to i.nfiru.cy. as 11-.o , tl1e p:>int at inhnity (x* "r
11
= (m
(I)
y*), all o f .._'hid\
'ihus , in the linit
1) is g~ven by (X 1
H) •
POINIS .aND LlNF..S
43
Table 2-l Harogeneous Coordinates for the Point (4 H
x*
1
4
l/2 1/3 l/4
X
y
3
4
3
8
6
4
3
u
9
4
3
16
u
4
3
40
30
4
3
400
300
4
3
3)
• • •
l/10 •
• •
1/100 • •
•
[4
3
0) in hCiroqeneous ooordinates.
It follows that the position vector
(1
0
OJ represents a point at infinity on the x*-axis and [0
0) represents
1
a point at infinity on the y-axis. In this rranner horrogeneous ooordi..nAtes give a convenient representation of points at infinity in addition to the capability for ~ali zed transformations .
2-17 T\'K>-DweNstONAL Previously
fbTATION
ABolJT AA ARBITRARY Axts
have considered rotations as
~
about the origin .
Han:>geneous cx::ordinates provide a JreChanism for aCCX)Itplish.ing rot:ations about
points other than the origin.
In general a rotation about an amitrary point
can be acc:atplished l:7j first translating the center of rotation
the origin, performing the required rotation, and then translating the result back to the to
'ltlus, rotation of the position vector (x y
original center of rotation.
1)
about the point m, n through an arbitrary angle can be ao::ntplished l:7j (x
y
1]
1
0 ""'1'1\
0
0
1
0
-n
1
rc:ose -sine
l
0
sine cos a
0
01
~J
r1
0
0
b
1
0
n
l
•
(X
'i
H)
(2-34)
By
canyinJ
out the n.-o interior rratrix products we can write
(X Y H) = [x y
For e:-carrple,
1) f cos6
sinO
0, (2- 35)
-sine
cosl3
o
+ n(si.ne)
-'"1ll (c:xl50 - 1)
-rn(sin8) - n(cose - 1>
the center of an cbject is at I 4
~
rotate t.'ic cbject 90° counterclocJG.tise about
3 ) and
lJ
it is desired to ~tion
its o.m omtral axis .
by the matrix
0 1 0 -1 0 0 0 0 1j causes a rotatiGn about t:.hc origin, rot the ooject axis . ~~
procedure
l.S
As sho.om
above , the
to fust translate the OOJect so that the desired center
of rot.iltion is at the origin by using the t.rnnslntion n'iltrix
r1
o
0
0
1
0
-4 -3
1
then apply tr.e rotation mat.rix, and fil'lally translate the results of the rot.:r-
tian bac:X to t.'ic original center by rreans of the reverse translation rratrix. 'Ihc: entire q:Jera tic.n [X
Y
H]
y
[x
1)
1
0
0
1
L-4
-3
o1 rl 0 01 0 -1 0 0 0 1 0 0..,
0
1
1,;
0
0
l4 3
1J
(2- 36)
1J
can be cn•bined into or.e rratrix cpo:>...ration by conc:atenat.J.ng the t.ransforrnatic.n Ntrices, i.e. , y
[X
I
'11-.enl x•
•
H] = [x y 1)
0
1
ol
-1
0
0
7
-1
(2- 37)
1J
X/II andy* = Y/H .
'lWcrdiirensional rotations about each cuds of an orthogonal nxis system are represented in Fig. 2- 10.
A cntputer al90rithm for performing a t:wo-dirrensional
rotation about an arbitrary point is given in Appendix
c.
a.long with other
algori thn'6 for ~ic:nal translation, scal j ng 1 and reflection .
REFERENCES 2-1
f'ox1
L.
An Introduc:t:.ion to N\marical Li.near Algd>ra, OXford lhriversity
PresS 1 1.nndon 1 19 64 •
2- 2 Forrest, ;,.
P. .
1
"Co-ordinates, Transfoxmations , ard Visualization Techni-
que.•," CAD Grol..p
~t N:>.
23, Cafl'brid:;e University, June 1969.
POINTS At;o (,fNE:S
2-3
Ahuja, D. C. , and Coons, s . A., "Gecm::ltry tor Const.ruction
45
Displa~·, ..
IB15yst. J., vol. 7 , oos . 3and 4, pp.l88-205, 1968.
,.
•
l
,.
1
>" }
0
0
0
1
\ \
• CClS)' SlJl)
\
l
0 -silry oosy \ \
.->"
l
'{.
""I I
'
->
a
,.
1
oosa
0 -sl.nS
0
1
,..
"
\
\
0
sinS 0 cos
,.
-' -
\
a
\ \ \
\
..
.,.,.
~
I
j·
I
b
OOSil
sim 0
. -smn coso 0 0
0
]
>"\
}'
\
\ \
\ \
...
,..J
..>.. ... ... •
' c
Figure 2-10 ROtation .
CHAPTER 3 THREE DIMENSIONAL TRANSFORl\fATIONS AND PROJECTIONS
3-1 INTROOXT I00 The ability to represent or display a t:hree-d.ir'Cnsional object is fundalrent:a::l to the understanding of the shape of that ooject . tO
Fu.rt:herlrore, the ability
rotate, translate, and project views of that object is also,
f~W to
the underst.an1.i.r¥J of its shape.
nus
picking up a relat.h.-ely c:ancJlex unfamll.iar object.
oolds
1.n
mmy cases,
.1s easily c:lErnlstrated by
Imrediately one rotates it,
it at ann• s length, stands back fran .it, etc ., in order to obtain an under-
starding of its sh8pe.
To
oo
this with a mcputer we rrust extend our previous
t:wo-d..ironsia-.al analyses to three dim.msions . Based on our previous experience we imrali.atcly introduce N::rrogeneous exx>rd.i.nates. Hence a point in ~ diJrens1onal space (x y z) is represented by a four-dimensional pos1.tion vector
lx y z to
1)
or
IX Y z H).
Again the transfonration f:n::n N::rrogcneoos coordinates
ordinary coordinates is given 17.1
[X Y Z H) • (x y z l)T X Y Z
lx• 'Y* z• ll • Iii ii ill
(3-1 )
1ooo1'1ero T is sore t.rans!or.rut1oo catrix. 'I'hc c_,'Cnarali7.&l 4 x 4 transfoil"oaticn Ntrix for three--dirnlo_nsicnal haroc}etlDOUS
coordinates is
T•
ab cp
(3-2)
defq
hi j r
tmns The 4 x 4 t.ransfonnat.ion matrix in Eq. (3-2) can be partit.loned into four
sepsrate sectiCX'lS: 3
J
X
3
X
1
-------1 X 3 I 1 X1 'llle 3 x 3 ootrix produces a linear transfomation* in the tonn of scaling, shear-
The 1 x 3 row matrix produces translation, and the 3 x 1
ing, arrl rotation.
oolunn matrix prcxluces perspective transformation.
duces overall scaling.
ThL final single e 1mcnt pro-
'Ihc total transfonmtion obtained by op&Bt..i.ng on a posi-
tion vector with the 4 x 4 matrix arrl normali2ir¥3 the transfomed vector i.s called a bilinear transfomation. nus gives a cutbinatioo of slwari71!,, weal
scaliff(l, l'Otal.i.on, rej'lection, translation, per'BiX"Ct.iVl, and WnY111 cwatitlfl.
3-2 Tffi.EE-D WENS IONAL
ScALI t-.6
The d.i.agooal terms o f the general 4 x 4 transfonrauon produce local ard
overall scaling .
Consider
(x y z 1]
wch
a 0 0 0
0 e 0 0
0 0 = [ax ey j z 1] : [x* y* 0 0 j 0 0 1
sln-ls the local scaling effect.
scaled as a unit. cube.
2*
1]
(3-3)
Figure 3-la shows a parallelepiped r e-
Overall scaling is obtained by using the fourth diagonal
elerent, i.e.,
(x y
2
11
1 0 0 0
= [x y
z s]
0 1 0 0 0 0 1 0 0 0 0 sj
= (x* y"*
z* 11 • (~ l.! 11 s s s
-
linear trcmsformation is ale \
1. •
0 ll I
•·
o1a
• c.lOl
t• ' f9
S l I
20:1 ,1 I l ).
•
o;oe
0 l DI
0 0 l 0
011
2001 •OOC1 : ) 0 l
1'
fi 0 0 0
~
r
I 0 0 I I 1 o1 1 r llClC 1 I 1 I I
A 0 0 0 1
8 00 1 : c o ac J
t
,. •
Figure '3-1
-
0
C . 1~
0
O.l l
0
o.•
I I
0
-.n
0
0
0
1
-o.~
-c.n
--c.a~
o.~
0 1
0
I
I
l :1'
O.f
I "'
c.•
r. 1 0 II J I 1 0 1 I
'
&.!.
0.15
l.~
I l l t• I t•
0 l I II 1 J ' 1 l 1
0.1S
0.1~
0.¥.
r•
·'~
1.1~
l,
•
D 0 1 1 :
s·
I
-c.,.
1., t' ,1!;
'lh1s effect is shct.m in Fig. 3-lb.
:Incidentally, the same effect can be obtained
by means of equal local scal.ings. be
In thls case the transfornatHX1 matrix "'Qlld
!9
0 0 0
0
!s
0
0
0
0
!s
0
0 0
3-3
0 1
T~E-DIMENSI
erAlized 4 x 4 transfonration matrix produce shear in three dimmsialS, e.g., (x y z 11
1 b c 0 = (x + yd + hz
bx + y + iz
ex + fy + z
1)
d 1 f 0 hi 1 0
0 0 0 1 A siJtt:>l e t:hJ::ee-dirner1Siooal. shear on a unit cube is stxlwn in Fig. 3-lc.
3-4
T~EE-DI~I
cx:up:lC'lent
matrix produced a
lt:lwever, i f the deteminant (cf Appen:lix B)
CXlibinatioo o f Si2ling and shear.
of tbe 3 x 3 oc::mpcnent matrix is +1, then it produces a pure rotation about the origin.
Before considering the general case of
about an arbitrary axis,
we
examine
ional rotation
several special cases.
x-ax.is, the x-d.irrensians oo not cl'la.NJe. 'lbls, the transformation matrix will have zeros in the first row and first co1l.llm, except for unity oo the main diagonal. The other tenn9 are determined by considering In a rotation about the
the rotation of a unit cube in the same m:mner as discussed in Sec. 2-13.
leads to the t:ransfontatioo matrix for a rotatioo of angle
e about
the
'lhis
x-axis
given by T=
1 0
0
cose sine
o -sine 0
0
0
cos6
0
0
(3-6)
0 0
1
Rotatia\ is assunel to be positive in a right-han:i sense as one looks fr:au the
origin outward a.1.oog the axis of rotation.
aro.xt the x- axis.
Figure 3-2a smws a -90° rotation
final result.
In order to sln.l this, o:ns:ider a rotatioo abot..lt the x-axis fol-
lowed ~ an equal rotatioo about the y-axis.
e
=~I
Using F.qs. (3-6) and (3-7) with
we have T =
1
0
cos¢ 0
0
0
0 cos6 sine o o -sine cosS 0 0
-
0
0
cosa . 2e Sl.n
1
-sin~
1
0
sin¢
0
0
0
0 cos¢
0
-sin6
0
cos6sine 2 cos8sine -sine cos e 0
0
~J
0
cose
0
('3-9)
01
0
0
1
0
On the other hand, the reverse operation, i .e . , a rotatioo about the y-axis
e=
~1
0 sine cos9
0 0
0
1
f01J.o..Jed by an equal rotation about the X-axi.S with
T = cos¢ 0 sin¢
0 •
o -sin~ 0 1
o
0 0 cos¢ 0
0 2
o
1 0
case
0
-sine
0
1
0
e
-oost sin6
case
sin
o
cose
sin&
0
0
0
2 sin6 -cos6sine cos e
0
yields I
(3-10)
0
01 o
Caq:larison of the right-hand sides of Eqs. (3-9} and (3-10)
~
that they are
not the sc:ure. The fact that three-d.iJrensional rotations are oonoc:mrutative nust be kept in mind when tJDre than one rotation is to be made. 'Ihe result of transformatioo. of the upper left object in Figure 3-3 consisting of
o-o 90° rotations using
.sho.m as the dashed figures .
the rratrix prcrluct given in Eqs. (3-9) is
hllen the opposite ordcu of rota.tioo. as specified
by Eq. (3-10) is perfumed , the solid figures shaHn in Fig. 3-3 graphlcal ly
dem:nstrate that different results are obtained by changing the order of rotation . f'requently it is desirable to rotate an abJect about one of the cartesian
coordinate axes . y-, and z-axes.
3-5
Algorithms given in Appendix C allow rotatioo. al:nlt the x-,
REfLECTIOO IN Tt«E DWENSIOOS Sc:m:! orientatioo.s o (
a three diJren.s.i.oo.al object require reflections.
three ctitrensions the sizrplest reflectia\S occur through a plane.
In
For a rigid-
.
)
~
1
0
0
0
0
0
1
0
0 -1
0
0
0
l
0
0
..
../ 0 0
~
0
~
0
,
0
1
0
0
0
0
0
1
0
0 -1
0
0
0
0
1
0
../
) )
•'
/
-r---71
//
/
/ I
f. ...... /
/
/
I
J--~--j
/
// z• , I//'--..'( / ~ ....... I / / ' ....V.,.. /
•'
/
Figure 3-3
0
0 -1 0
0
1
0
0
1
0
0
0
0
0
0
1
/ /
-x;•
------~
I .,.-" I } - - - ~ ~-- ..) ,~ /
J&.-.-~--1-
I
___ _v /
.,.
/;/'
/
7
8
a·
0/ l .l "
0 -l 1 -1 0 -2 1 -2 0 -1
1
A
1
B
1
c
1
D
1
E
2
F
2 1 -1 1
G
2 0 -2 1 2 1 -2 1
I
1 1 1
0 0 1 0 0 0 0
1
-
0 0 0 0 -1 0 0 1
r1 0 1 1 A* 111 1 B* l 0 2 l C* 1 1 2 1
D*
2 0 1 1 E* 2 1 1 1 F* 2 0 2 1 G* 2 1 2 1 I*
1
· ensiooa.l ref1ectioo through the xy-plane. Figure 3-4 'Ihl::ee-d.i.rr body re.f1ectioo the determinant of the reflectioo transformatioo will be -1.0. In a reflectioo through the xy-plane, only the z~.rdi.nate values of the
cbject ' s positioo vectors will be cl>.anqed.
sign.
In fact they will be reversed in
Thus, the transfonratioo ratrix for a reflectioo through the xy-plane is
T
=
1 0 0 0
0 1 0 0
0 0 -1 0
0 0 0 0
(3-11)
54
l'~CAL ~
FOR a:t-tPlTI'ER GRAPHICS
The reflection of a uru.t cube through t.~ xy-p1anc is shc:7..m in Figure 3-4 .
For
a reflection through the yz-plane,
T
-l 0 0 0
=
(3-12)
0 1 0 0
0 0 1 0 •
0 0 0 1 and for a roflcct.i
T= 1 0 0 0 0 -1 0
0
0 0 1
0
0 An
(3- 13)
0 1
0
•
alqoritln which INill perfom t:hree-ctirnensional reflecticns is given in
1\ppendix
c.
Reflections al;o.lt ot:OOr planes can
oo
obta.ined by a oanbination o f
rotation and reflection.
3-6
TRANSLATIOO IN TJ-REE DWENSICJIJS Three-dimensional translation is obtained by writing
[X Y Z H) • [x y z 1] r1 0 0 0 0 1 0 0
(3-14)
0 0 1 0 t mn 1
'
•
t'lhen expanded this yields [X Y z H)
= I (x +
t)
(y
+ m) (z + n) 11
(3- 15)
It follo,.s that
x"' • ~ • x+ ~ H
(3- 16)
y* = 'i = Y+m H
z"' - !H = z + n An a1gorithn for translations in three dimonsions appears in
3-7
l\ppe1¥ii.x
c.
T~-REE-D U-1ENSIOML RoTATIOOS ABOOT ~ AABITRARY Ax iS The rrcthod of O.t>-dirnensional plane rotation D'to.lt an arbitrary axis was
discussed m
axis in
sec.
2-7.
The generalization of that probl em is rotation about any
t.hr~sional
space.
Again , the procedure is to translate tte ob-
ject and the desired axis of rotatioo so that the rotation is made about an axis passing l.hrouqh the origin of the ooordinate system. th~im:!ns1onal
The rrethcxl invol ves a
translat1oo, a rotatioo a'to.lt the origin, and a translation
bdck to ~ init.LAl positioo.
1f the axis about "'hlch rotation is desi.rEd
passes throu.g h the }X>int A • It m n 1] , then the form of the transformat.ioo rratrix is
(X Y Z H)
= (x
y z 1]
1
0
0 0
r1o o o
0
1
0
0
0 1 0 0
0
0
1
0
-11\ -n
1
-t
R
(3-17)
0 0 1 0 t mn 1
where the eleSJents in the 4x4 rotation matrix R are in geroral given by 2 n1 + (R)
Q
(1 -
n1 ~(1
n1 ~ (l
'1 2l008 D
-
oosO) • n 3sJ..ne
- cos9) - n sJ.r,(, n-/ + (1-
n1 ~(l -
3
n/)~
~n 3 (1
-
008) -
~n3 (1-
0066) + "2s.J..nk n n (1- cosO) - "ls.inO n 1 2 3
2
~
0
(3-18)
C060) + n 1 sintJ 0
(1 - n
2 )0C1Sll 3
o l
0
0
0
"2~1n
a derivation for dete.rming the elments of the rotat.l.oo matrix R
.1.S
given in tho
next section. Fqu.ations '3-17) and (3-18) can be nserl as a basis for c::arputer software algoritbns .
SUch an algorit:l'lm is particularly useful
for c::a1puter anirrations.
'Ibis transformat.ial , as well as others discussed in this dlapter, can also be
produced by use of hardware ~ts in a display device . An algoritbn which will produce a · jooal rotation al:xJut an arl>i-
axis in space based oo Eq . ( l-17) ~ given J.n Appenchx C. The alqont.hns for rotation about the in:lividual x-, y-, and z-axes previously discussed are special cases of the present algorithn. trary
3-8
Ei.&Etfl'S FOO n£ taERAL RoTATIOO f"AlRn<* We begin by considering Fig . J-5, whid1 sho,.;s an arbitrary axis of rotation
(Q'l) transl ated so that it passes through the origin.
We then rotate a }X>int oo
tl1e translated cbject (5\d\ as }X>int P) about the axis by an angle
JTOve the }X>int fran P to P* .
oo.
P•s
u. This will
In Fig . 3-5, PQ = P*Q, and both of these lines arc
cx::nst..ruct.ed to be parpendic:ular to PQ. To determine the elerrents in the matrix R, we rrust relate the transformed coordinates of P* to three variables: the coordinates of P, the rotatial angle () , and the • This dcridirect.ioo of the axis of rotatioo as specified by the unit. vector n. perpendicular to
J.S
vatioo i.s given below. ~ dircctioo of the axis of
three directJ.oo cosines,
rotatioo is nost easily expressed in terms of
"1 = cosa , "'2 = cose and
n
3
=cosy.
*1hl.s scctioo rmy be Oll'li.tt.ed withoutloss of mntinuity.
The angles a, B,
':he d.irection cosines are tOO t..ltree
a..--d y are 1der.t;.fied 1.n Fig. 3-5 .
of 1:.~ ~t ~_or wc':.Or
u
.. .. .. ,
..
For arry gc.r.eral "l!Ctor
0 = qlI •
q2j +
c;.j. tlu.s
tr.it
ca.lc-..:l.at:.ed by n • C/ C 1 where C is the absolute value of the vector i3 .. 2 + + s 2 1 ard I 1 j, and k are u:ut veco-..ors l-"'1 the x-, y-,
sl
/q1
g1ven by a.."ld
n.
~ts
%-di:'~..l.a'Ui
3
It folla..'S tr.at:
tesfE!Cti\lely.
0.1
"" n
COS'l •
;'
ql
2
... q2
2
~
q3
2
1
q2
oos3 -
--:::::::::==::::::;==::=: = "2 / 2 • 0-1
+
2
0.2
.,. q3
2
•
q3
COS'( •
':1le t:.it vector is
-
t""d':.r!.X
~
form t.l-Us
I• ql2
+ q2
tr.e.'l given by n ..
!.."'1
--;:::::;=====:;- • n3
15
other results fra:1
+
q3
2
-
"2J.... + nl~ as n {"t "2 n3] .
"11
written
2
...,..
-
Z<
~....or
calc-J.lus1 tl'.e dot product. a."ld t."le cross
•'
/
'
/ • _ :;/ •
I p"
-J •
t)l'?-=:;:_-= -:--- - 1
-
•
Figure 3- 5
n
•
, .. •
·-
.\
produ:::t, are needed to relate P* to P.
1h! vector dot prtxlu:::t is 91ven by
P . ~ .. IP nlcos~ • IPI~
.P =
..
where ¢ is the angle bet....-een the tw:> vectors, as shcl.m in Fig. 3-5.
since
1 n 2 n 3 ), the dot product can also lx! expressed in matrix
[x y z) and n •
(n
form as
•P
• =~
YJ7
+
• n
+
~
[x y z}
c
'1. "2 "3
~
vector cross product is give."\ by .,.
.....
1
n x P • det n
1
t"
+k
n
n
)
2
X y
=
3
Z
!cn2z- n3y) + j(n)x- nlz)
a
In IMtrix form the
cross product is +
+
0
n
-n3
0
-"2 '1.
"2
-"1
0
n x p = (x y z)
It 1s also true that where
+ is again
... 1n
.....
x P
.. ,.... n
-
P sino) =
3
.~
P sin~
the angle between the b.'O vectors.
'lbe foll.o.oling three vector equalities are obtained fran Fig. 3-5:
0 + ci• • p•
(3-19) (3-20) (3-21)
.......
4-
....
....
Fran Sl· (3-21), oP = (P - Q} .
the ma.gnitldc of vector +
OS • IOPicos9.
Thus
QS,
which is in the direction of +
CS •
•
(P -
..
Since the magnitudes of QP and (F)* are equal,
Q)cose .
QP,
is given by
Using this result and
m 1bining
Sis. (3- 19) and (3-20) gives
P.
= Q+
+
sP*
'l1le nagn i tude of SP111 is equal to SP* -
'nle ch.rection of
s'P•
is
parallel to the vector ~ x
P.
loP,sine
>=
IP - olsine to the plane F'Ctl and is therefore
·
'Jbe unit vector m this direction is
.. ..
•u= nxP
... .. 1n x PI
-
...nxP.•
IP i sin~
- -
n• x P•
SP• -
p
P)sW
rut
loP!
=
zo
IP - • aro
-.
p - Q Sl.n J
Str.~
thus
en x P}s1.ne
~ ~
•
Eqt.:.11 t; cr. ( 3-21) can oow be written ...... ..... ... p• • Q + (P - Q)cos6 + (n x P)sine
.
Fl.9'..:.re 3-5
of rotation. t:Sl.."lg
..
~
•
'lb! mag:nita'!e of Q
o..
p• As r~,
•
!1-.at the vector C
the e<;Uality
l..S
J.S
(3- 22)
t
....
t.lte proJection o f the voctor P on t.lv! axis
:z: ..
-
t'· n = P cos; , a.'ld the direct..ion of
(3-22) beocJres
{P·n)n(l - cose) +
a
~tion
this
.
-
Peose
+
...
Q
is n.
•
(n X P)sine
gives the transfor ned pomt 1.n terms of the iru.tial
we
point, the angle of rotatioo , a."ld the d.i..rectioo o f the axis o f rotation. :.trite this result m rot.n.x form. ...
ro.~
.
l!smg the Ntrl.X ex-press~oo give."\ earlier for the dot pro:3uct (P· n ) ard for the cross product x P), the equat.iC!'l for p• beo::m...>S • ~ · • (x y z) n 1 !~1 ~ n3J (l - COS7) • (x y z ~ t lx y zl 0 n 3 -n 2 Sln (3- 23)
(n
n
-r.
2 . r.l ~
l
0
3
"2 -'1
n
1
0
em ~ writte."'l in a."'l equi\'81ent form usi.'\9 hatt:a;ie.neous coordil'...ates :
P '"' (X )'
~
l)
r n1
2
r1 n 2 nl':l 0 Cl -
r:l:-"2 ~
2
"J "2"3
t
n1
0
0
COS• J •
r 1 0 0 0 C08
0 10 0
n2"3 0 2 n3 0
+ " "
~ -n 2 0
SUl
()- 24)
-n3 0
0 0 1 0
u •~
LO 0 0 l J
1'11P. three t.erTT'S within the braces give- the required rotat.ioo rratrix (R] .
L
0
0
l
0
It 1s seen th.::lt to pro:lt.X:'C .a general rotat.lcra by use of F.q. (3-17) , the (x y z ) O'.:lOrdinates of t.'ll2 JX)int P, t!le directioo oosiJ'les of t.lx! u.xis of rotatioo
ln 1 As
~
n l, and the angle of rotat:a dbout the
a spcc1hc exa:rple consider rotat.J.oc al:ntt
an::l n
3
• 1.0.
ax1s
t.~
are neOOod to defl.ne (RJ.
z-
'~hen
n = 0, 1
For this spt..>cial c1se t.~ rotaticra r .:ttrix R b..~
~ =
O,
•
THREE-OIMNSIQW.. "l'AANSFC>Rt-17\TIOOS AND PI'D.': 'cr'l'IQS
R=
r COSH
59
sine 0 0
• cos., 0 0 -sm
0
0
1 0
0
0
0 l
'lhis rMtrix a
3-~
f or rotdtioo about the z-axis in
the xy- plane.
3-9 AFF1NE
PMJ PeRsPECT 1VE
GEa-t: 1RY
The idea of a vcct..or Lo describe a point. was .initially oonc:civcd frQI'" qoormtrical c:ooccpts.
Gearet.ric theorG'l\S have been developed for boili perspect.1.vc 'Ihe theorems of affine geaoot.ry arc identical to t.ta;c for
and affine g
Euclidean geonetry .
In affinQ gecmatcy parallelism is an .inp:>rtant. concept, and
the ratios beboleell parallel lines is an inte<]ral p.trt of the geatetry.
Euclidean
ga:t~Ctry
has been taught in
scrools
Si.nec:!
for many years , draw.Lng and sketch-
ing techniques based upon this gearetry have bec:are standard rrethods for graph.i-
cal ccmnunication . 1\n affine t.ransfor.rration is a canb.ination of linear transfornations folla..'ed
by translation .
For an affine transfonration, the last. oolum in the gcllcral
4 X 4 transformation IMtrl..X must be
ro .
OthenoJJ.SC 1 llS sh(l...n
in Sec. 3-11 belo.t,
0
0
1 the t.ransforrred torogcneous coordinate H is not unity and there is not a one to
one oo.rrespondence bet\.-een the affine transfol:'lfatioo and the 4 x 4 matrix opera-
tor.
Affi."'le t.ransfornations fonn a useful subset. of bilineAr transfornaLions,
si.rce the product of n.o affine transfo:rm3tions is also affine .
This alla..-s ooe
to perform a g
coordl.nate H. Alt:hot.Jgh perspective views are often used by artists and architocts because
they y1.eld rroro relaistic pictures, t.hey are seldan used in t..cchn1cal work becaU9C
of the difficulty of C'a'\Struction.
HoNcver, when hcrlogencous CXX>rdinates
are used to define an object ~'ithin a mrputer, c.1.the.r affJ..ne or perspective t.ransfOJ:T"Ut.Lcns can be obtaint::rl with equal ecu;e . In perspective gearctry no bolO lines arc parallel.
ln fact the perspecbve plane can be oonstdered "lS a hemi-
spher1cal surface and t:ha
pcrspcctt\"C
t:lon fran cne t hree BE
'"·
\>J'Cl the elerrenls
• 1n
a
11
1'
tran.sfornatioo considered .:lS a transforrra-
r.,... ~
spa....tJ.
the lasl oolum of the general 4 x 4 transfotmltioo
11\;ltrix arc not zero, a perspective transfOLtMtion is fotned.
A perspective
trcmsfoCTitltiOO is frequently associated with a projoct.ioo ooto a plane, su::h as
z = c , fran a lcx:al center of proJectiOO. 'nle catb:ination of a perspective 1 transfotl'"\ltiOO with a projectiw transformatioo is called a perspective projectioo. O.Q
A p:.rsp:cllve pxojectioo represents a transforrratioo fran three space to I f the center of projection is located at infinity, then the perspec-
space.
tive proJectiOO
.LS
Ccllled
an axooc:rr-etric projection.
Ot:heiwisc, the form depends
upon the locatioo of the projectioo FOint.
3-10
AxoMMT'RIC PRoJECTIOOS An axonanetric projectioo is produced by an affi!le transforrration which has
a zero value for its determinant.
'lhere are several different types of a>a::lOCI'OOt-
ric projections which find applicatioo
descriptiw gearotry o.nd engineering
I"Mt:.herratically form an axonaootric projection, the 4 x 4 transfo.rmatioo matrix is used to pxcdu::e an affine transforrratioo on a set of p:>ints . 'n1e p:>mts are then projected onto a plane frun a center of projection at infinity. 'l1le canton axonarotric projecticns are classified in Table 3-l. draftl.ng.
..
l.l'\
To
'nsble 3-1
Ax.onc::r.etric Proj ecti
The trans£orrratioo matrix causes pure rotatioo. 'Ihus, the coordl.nate axes
Trilrotric
remrln orthogonal
~'hen
projected
Oiztetric
Two
Isaretric
All three axes are equally foreshortened when projected
of the three axes are equally foreshortened when projected
J\Xl:lnc::lrotric projectioos
~
shown in Fig. 3- 2 and 3-3.
An axonaret.ric
projection fran tlu:ee-c:l.inenional space onto a plane z • n can be obta.i ned by the followll"'9 operatioo:
0 0 01 ""
(X Y Z H) = {x y z 1) 1
(x y n 1)
(3-25)
0 1 0 0 0 0 0 OJ 0 0 n 1
'Ihcn, [X* y* z• 1)
a
(X y n 1)
Notio:! that. this transfoD1'1<'1tiOO ~ts a translatiOn in the z~on by the amJU'lt n CJ1vP..n by the transfomatioo
l
'l'HRai-OINENSlOW. TRANSFOffofATIOOS AND PR:mX:TIOOS
T' =
61
l 0 0 0
0 l 0 0 0 0 1 0 0 0 n 1
follo.,'9d by a projectioo
ftu!l
now
infinity ooto the
~
.. 0 plane given
by the
transfm:matioo
T" = l 0 0 0 0 1 0 0 0 0 0 0
0 0 0 1 'l!le concatenation of T' and T" 1 i.e. , T" T' 1 will yield the transfoz:matjoo matrix
given in F.q. (3-25).
'lhe effect of tl1e translation is to nove the z .., 0 plane
to sate other position
in the object.
Proje::tioo onto the z .. 0 plane then oor-
responds to projectioo onto the z ,., n plane.
Axonc::Xretric projection transfotmations onto the appropriate zero plane always contain a ooll.ltn of zeros, \olhi.ch oorresponds to the plane of projection. Such project.i.cns are frequently called orthografhic projectioos. ncnly used in rrechanical drawing.
= R.
the x
Exarcples of an axcn:ttet.ric projectioo onto
and y • m planes are given by
T' •
They are can-
1 0 0 0 and T" =
r1 o o o
0 0 0 0
0 l 0 0
0 0 1 0
0 0 0 0 L.O m 0 l
t 0 0 l
As noted in Table 3-1 the transfomation natrix for a trilretrix a>col'lCI'I'etric
'Ibis type of transformatioo \.as studied in
projection causes pure rotation. Sec. 3- 5.
tia1s.
Aux.il.iary views
u.c;ed
in engineering drawings are tr.itnetric projec-
An exanple of a trilretric projection of the unit cube onto the z • 0
plane after a 90° rotation al:x:>ut the x-axis is shown in Flg. 3-6. In order to develop the oonditicns for diJretric and isaretric projections,
recal 1 that caubined rotatioos folla.;ed by pr-ojection fran infinity f orm the
basis for generating all
~tric
In particular,
projectioos.
WI:!
a rotatioo about the y-ax.is, follONed by a rotation about the x-a.xis.
Eqs. (3-7) and (3-6) the matrix project which gives this is
(X Y Z HI • [x y z l)
cos~
0
-sin~
0
o
1
o
o
sin'P 0 cos¢ 0 0 cax:atenatioo yields
0
0
0
1
0
0 0 o oose sine o 0 -sinO cosO 0
0
0
0
0
consider Fran
>· ,
L" c·
A".\"
~·
B" D"t -- - --e i " J"
A B
c D
E F
G
I
0 0 0 0 1 1 1 l1
ll
Figure 3-6
(X Y Z H]
ll ,. .
ro I
0 0 1 rl 0 0 0 = 0 0 01 1 0 0 0 0 0 0 0 1 a• 1 0 1 0 -1 0 0 o 1 o 1 c• 1 l 1 lo 0 0 1 0 1 0 1 o• 0 0 1 1 0 0 1 E* 0 1 1 1 1 0 1 F'* 1 0 0 1 ~ 1 0 1 1 0 1 I• 111
= [x
~c
y z 1}
Projections .
coso sm¢sin6 -si.n9cose 0 cos9 sine
oo]
(3-26)
~J
0 By use of this transformation matrix, the unit vector on the x-axis, (l 0 0 11 , 0
transfo~
0
to [X Y
z H)
= [COSQ sin¢sinS
-sin~cose 1)
and
x*
= cos¢,
y*
= sln¢sin6,
z • • -sin¢cos9
In projecting ont:D the z• • o plane, z* is effectively neglected as previously
slnm by fl:i. (3-25) with n = 0 and Eq. (3-28) . tion, the mitut1 z- axJ.S unit vector / x*
2
+ y• 2
OCM
a result of this transform!lhas a magn~tude of
= ks2~
As
+ (sin¢sin6) 2
Slrri.lar1y, the unit vector on the y-ans, (0 1 0 1], transforms to [X Y z HJ = (0 cose sine 1) and
x* = 0,
y* = cose ,
'Ibe maqn1.tudc of the transfoi'T"Cd ~s lD'lit
cos
z• =
sin~
vectOr
&= cose
is
T'OII
'lb create a dilretric projectioo, the I!B
vectors is shortened by an equal mount. y-axi.s wctors yields
El:]uating thL magnitude of the x- and
2 2 oos2¢ + sin ~sin e
2 Usi.n.J the identities cos ¢
\-lhen the value of sine is
:
sin2~
cos2e 2
and cos & = l - sm2 e, it follo..'S that . 2,. sin2e SJ.I\ " = (3-27) . 2 1 - sm e
=1
-
crosen,
and
is calculated by the use o[
~
D:J. (3-27),
the matrix in F.q. {3-26) produces a d.imetric projection.
now consider a specific e.xanple far a di.rretric projectioo. Qle rrotln:l of choosing sine is to shorten the z-axis by a fixed mrount. Consider the unit. vector on the z-axis, [0 0 1 1) . 'Ibis transforms to [X Y z H} = (sincj. -cos~sine cos¢cose 1) \lie
x* = sin¢, Now
we shorten
y*
.= -cos~sine
the magnitude of this transfomed unit vector to a value of l/2
by setting
or ~~2~ + {1 5~· 'f'
-
. 2 ) . 2&
Sl.Il ~ SU\
""
1 'f
Introducing Eq. (3-27) loods to
4 8sin e
-
2e + 1
9sin
=o
'l1lis equatioo has roots of sin2e • • 1/B and sin2 t • _.. 1 . 0. \'hen sine 11/8 or e = 20 . 705°, then sin¢ ::: 1(1/8)/(1- l/8) or Q .: 22.208°. 'l1le terms 1.0 tlle transfornatioo matrix in F.q . (3-26) are thus determi.ned.
Using the values for a dilretric projectioo, the t:ra.nsforrMtion rretrix {a:J. 3-26) is ¢ven by
(Z Y Z H) = (x y z 1)
0.925820
0.133631 -Q.353553
0
0
0.935414
0 . 353553
0
0.377964 -0 . 327321
0.866025
0
0
0
1
l
0
A clinet..ric projectioo is sln-m in f'iq. 3-7. Perhaps rrore cam at is the isoretric projectioo .
'1'0 form an isaretric pro-
jectioo, all three transfomed axes are equally shortened. l::oth
'l1lis requires that
~
•
X
y
Z
!. 0 0 1
l 1 0 1
1 !2
1 1
1 01 1 0 0 0 1 0 l 0 1 <
0!2 1
;
1
0 0 1 1 •
I 5 .,..
~
•
>" O.S25820 0.133631 -~.353553 0 0.9354H 0.353553 0 0 0.377S64 - 0. 327321 0.866025 0 1 0 0 0
I! )"
I, -
0!1
l
~
I0 -
•
... 0 I
,y·-.--==--___.. ' -4
OS
·0
s
D netr1c
-0 !
•
-
I <
OS
OS -
4
•
The first oondition a9"in gives
2
sin ¢ =
. 2,.. Slll .,.
. 2-
sm 1- sin2 ~
2 ;; 1 2sin ~--'-~i'--
::
1 - sin2A
2 It follows that sin e • 1/3, or sine • 11/3 and 2 sin ¢
e
t- ..
35 .26439°.
Then
r¥h = !
and
oos 2 ~
= 1 - sin ~ = ~ or ~ = 45° The angle lolhich the projected x-axis makes with the horizc:nt.al is given
2
by t.ano
= sin¢sine cos~
For sin6 =
/i73, sin2~
2 • 1/2, and cos ¢. .. l/2, this angle is Q
1 = tan- (-;) = 30°
This result is well knc:J..m to those wtx> have prcdnced an isaretric projectioo on a drawing toard.
A plastic right. triangle with incluOOd angles of 30° and 60°
is a cx:xutonly used tool for this purpose.
USing the angular values for an isoretrlc projection the transform:ttion IMtri.x in Eq . ( 3-26) OOoc:mes
(X Y Z H) • (x y z 1)
0. 707107
0 .408248 -0.577353
0
0
0 . 816597
0.577345
0
0. 707107 -0 .408249
0 . 577353
0
0
0
1
0
An exanp1e of an isaretric projectioo is also sh:Mn in Fig. 3-7.
Cblique projections such as cavalier and cabi."''et do not preserve ort:hcx:Jonality of the coordinate system.
Table 3-2 .
The characteristics of these O.;o are shown in
Since they are not frequently used in technical '-()rk they axe not
di.scussed f\lr"ther. A si.JTple orthograiiUc projection onto a zero plane perpendl.cular to any
of the three ortrogonal axes .results when the corresponding co1llm in the 4 x 4 transfo.mation natrix contains all zeros. ~ =
For exanp1e, a proJection onto the
0 plal1e will resull when the follOWl.llg JMtnx T =
1 0 0 0
0 1 0 0 0 0 0 0 0 0 0 l
~s
used . (3-29)
66
MT>.'Il01l\TICAL £IDn.1I'S FOR
Table 3-2 Cblique PrOject..ials Oeser iptioo
Pro)ec+..l.O'l
'1\.Q axes appear perpendlcu.lar and are not
cavalier
foresrort.enerl; the third axis is inclined with respect to the horizontal and is not fore.sOOrtened A special case of the cavalier projectioo
in which the third axis i s foreshortened by a factor of 1/2
AI: algo.nthn for sirrple projections is given in .AppeD::lix A
c..
c.
trirretric projectioo can be abtai1led by using the three-dim.:msional
rotation alg:>ritlrs presented previously followed by the sinpl e projectioo lllg:>rithm given al:x>ve . An alg:>ritl'ITI which inplerrents the specific diJrotric and isaretric transf or-
TI'ittion rratrioes discussed above [i.e. , Eq. (3- 26) with P • 20 . 70481P ,
: = 22.20765°
and ~ • 35.26439, ~ = ~so respectively} and assuming projection on the z • 0 plane is give.'\ in Appendix c.
3-ll PERsPECTIVE
TRANs~rta-ts
As previously nentioned, nonzero elements in the first three
ro.~S
of the
last co1um o f the 4 x 4 transformatioo rratrix yield a perspective transforma-
tion.
A
perspective projection is obtained by ooncatenatioo of a perspective
transformat.l.on
foll~-ed
by a pro)ectial onto sare two-dirren.sic:nal
"vie.r.ri.n9"
plane. A perspective projectioo onto the z = 0 plane is given by tho transf o rrration (X Y
z
H] = [x y
z
1]
1 0 0 0
= (x y 0
(rz +
1))
(3-29)
0 1 0 0 0 0 0 r
0 0 0 1 'Ibe ordinary transfol'li'Od coordinates arc
X
t.l)en
X
x*=H=rz+l Y"
'lhc C]"...oretrical
eff~
( 3-30)
= yH = rz ___.Y~-,-.,... + l
of this transfonnation can b.! indicated with the aid
'm:REE-DlNF.NSiaw... 'rRA.~FUif.WI'IGNS A.~ Pinli:L"l'I
67
I
The nontransfor:trod p:>int
of Fig. 3-8 .
P is transform..'>d to P* by the aoove operatial,
WJ.t:h
r • 1/k.
In Fig. 3-8
the center of projectia'l is located
at [0 0 - k ) dnd Lhe plane of projectial is z "" 0.
Smoe the above ope.r-
atial prodtrcs no translatial, the origin is unchanged and U1C xy- and x*y*-coordinatcs share thO sane
origin.
By using the
similar t.ri-
angles shc7...n in Pig. 3-8 it foll(1.o1S that
x•
k •
X
{z + k)
•
or
xe
X
z +1
1<
Considermg the yz-plane in a similar
rrrumer gives
r OJCCIII'n Figure 3-8 Perspective projectial of a point. (
'F'
y
2
z .. 1
(3-31)
C'crllcr ul
k
'Ibese are tho transform.."
r • l/k.
This cperatJ.a'l produoes a perspective projectl.al on tl'lC'
fran a center of projoctial located at
z
=-k.
z ..
0 plane
In a perspective transfotmatial
the transfonrod space is rot Euclidean, since orthoqalaliLy of lhc a>-.cs is not
preserved.
!l.ft'\en k approaches J.nfinity, the ~tric Lransforrralion discussed
above is cbt:ained . Equations (3-30) and (3-31) sho.ol that when z
rx.e to this iclentJ. ty transforn'at.ion, changed by a perspective projection .
= 0,
then x•
= x,
andy* --' y.
p:>ints in the viewing pl ane z "' 0 are not
'Ihis is ooly true,
rc.•~vcr. when
the horro-
gencous ooord.inatc H is unity after the transfol:malicn has been a,wlied to a p:>int [x y z l j .
Since aff.inc transfo:rmal.ians retain a value of nnity for II,
a perspective transfornatial 11'\lY be preceded by an arbitrary sequcncc of affine transfoi"'T''ltions.
Thus , to obtain perspective views fran an arbit.rary view point,
affine t.ransformst.i.als are first. used to establish a ooordi.natc system with the
z- aJU.S alwt.s alto the viewing plane z • 0 of the current coordlllAte syst.ern.
In prac-
tice , it is often l'lCO'Ss.cny to apply further affine transfornutl.ons to scale and positton the proJected new (~f . 3-3).
fb..-eve.r 1 this
Above we have considered projectioo onto tl¥3 2 • 0 plane .
ray not a!.wars be desiiable .
Q1e
reason is that projectioo onto the
2 •
0 plane
loses the 2~rchnated infanration which can be useful in controll.ing the bnghtness o f displayed points oo a plane (Pef. J-4 ). To include the 2-coordinate infoOM.tion the perspective transfomation (X Y z H) E {x y z 1) 1 0 0 0
Q
lx y z (rz + 1)]
0 1 0 0 0 0 1 r
00 0 1
rx * Y* 2 * 1)
-
{
-
rrey be used and the full
X y Z 1] rz+lrz+lrz+l
ional results obta.ined.
·
Here we note that the origin of the coordinate
unch.ancJed.
(3- 33)
Notice that the point At infinity
a'\
system ro
0 0 1] ls again
ro
the Z-axi.s l
0 1 0] 1
'('0;1
uansfoms to the noninfinite p-.>int [X Y z R] = [0 0 1 r) w.ith the ordinary o:x:m:U.nates [x* y* z* lJ • [0 0 lfr l], i.e . , a finite point m the z*-axis . nus rreans that the entire positive half of the dcrrain within a finite danain (0
~
parallel to the z-axi.s will
z•
~
~
z
~
'"') is projected
Further, lines which were originally app>ar to pass throught the point [0 0 l/r 11 .
rv::J;~
1/ r ) .
This point is frequently cal l ed the
a
%I
co
vanishing point of the perspective transfomation.
\\hen the results o f
the transformaticn are projected onto l n11ial It n c:
,..
a plane, a perspective projection as rre."''tiooed arove is obtained .
Assuni.ng
that the z = 0 plane is the projectioo
plane, the center of projectioo is A
•
located on the sarre axis as the vanishing p:>int at an equal distance on the
opposite side of the zero plane; i.e ., i f the vanishing point is located at
z • l/r oo the z-axi.s, the center of projection is located at z :: - 1/r oo the z-axis.
'lhi.s is
s.~n
m Fig.
3-9.
~
·t , Cen 1cr
t )f
p• QJC:CI•on
Figure 3-9
P.roJectiat of a line parallel to the z-axis.
Similarly the perspective trans-
fooration
~I'IOOS
(X Y Z H)
z
(x y z 11
1\ND POOJfel'IQIS
- [x y z (qy + l)J
10 0 0
69
(3-34)
O l Oq 0 0 1 0
0 0 .0 l with ordinary coord.in.ates
(x• y• z* 11
= [qy+1qy+lqy+l x Y z
11
(3-35)
will result in a vanish.i.ng point located oo the y-axis at [0 l/q o 11, perspective transfonnatioo
rx Y Z H) = [x y z 1]
1 0 0p 0 1 0 0 0 0 1 0
(3-36)
0 0 0 1
=
(X
y
+ 1))
Z (px
with ordinatey coord:i.nates
rx*
y* z• 1) -- Ipx+lpx+lpx+l x Y z 11
(3-37)
will result in a vanishin:] point located oo the x - axis at (1/p 0 0 1).
'1lle
transfoonatioos given in E>qs. (r32), (3-34), and (r36) are called single-point or parallel perspccti ve transforrraticns . Fig. 3-10a.
'!bey result in views such as sho.m in
Fic;ure 3-loa shows a single-point perspective of a unit
pxojected onto the z • 0 plane.
C\t)e
'nle vanishing point ~ on the z-axis at z • -10.
A rore natural single-point pe.rSfECtive can be obtained by first centering the
face OCFG with respect to the z-axis . the unit cube by
tion.
X ""
That .is, translate the eight corners of
-0 . 5 and y = -0 . 5 before awlying the perspective opera-
'11-.e resulting view is sham in Fig. 3-lOb.
perspective i s the type that artists often use.
'1bis centered, single-point It can, hc:7,..1ever , give a dis-
torted vi&t of the depth of the object i f the vanishing point is noved too far
away fran the center of the picture. This is what causes the "unnatural• view in Fig. 3- loa . Lf b;o terms in the 4 x 1 colum nBtrix are nonzero, then a two-point or angular perspective is obtain~ .
'Ihe o.o-p:>int perspective transfotm!ltion
[X Y z 1) = (x y z 1]
1 0 0 p
a
[x y z (px + qy + 1)]
(3-38)
~+1
(3-39)
0 1 0 q 0 0 1 0 0 0 0 1 with ordl.naty coordinates [x*
y* z* 11 "" [px +
results in two vanishing points : the y-axis at (0 1/q 0 1).
~+
1 px +
px +
~
+ 11
one on the x-axis at [1/p 0 0 1) arxl ooe on
This is shc7.n'l in Fig. 3-lla, where the results are
t lo \1 PI on ) • 111.1\
>"
)'
I
l
I
·.o·
.11 }
10
G" •
-ToVP1 on
A·B l< l \
IIJ. 10 (
(j
E ll
A"
Po5it.iat
Vectors
.....
s·
~ tl.VI? and Projecticn
,..
E"F" Folo.iuon
Nolll\1 II zed l'criiu oo Vectors o f
Vectors o! o f o ng- Tr.lnsfOYt:D- 'rransforrred 1na.l Clile t ion Mat.."'U! Cube
F A
B
c D
E F G
I
0 0 0 0 1 1 1 1
0 0 1 1 0 0 1 1
0 1 1 1 1 1 01 0 1 1 1 1 1 0 1
1 0 0 0
0 1 0 0
0 0 0 0
0.1 "" 0 0 0 1 0.1 0 0 0 1 0 0 1 0 1.1 l 0 1 0 1.1 1001.1 1 0 0 1.1 1 1 0 1. 2 1 l 0 1. 2
Trons fo l'l"f'd C\.bP
A•
0 0 0 0 0.91 0.91 0.83 0.83
a•
c• o• t• p•
c• r•
0 0 0.91 0.91 0 0 0.83 0.83
0 0 0 0 0
,
l .. 1 s• 1 c1 o• l e-
0 l f"'
0 1 c• 0 1 t•
).
c·.o· A
--
G".l"
-·
X
~
l
A.
'-
I
n·
-
•
I ·.I •
Pcrspec-
Positlon Vectors
of orig1na1 CUbe
0 8 0 c 0 D 0 E 1 F 1
11.
0 0 1 1 0 0
0 1 1 0 0 1
t1ve Md Projoctlat Tr.vuofor'IT'dt.iat TransfoDMtion ,.latrJ.X !'latrix
1 1 1 0 1 0 1 -o.5 1 1
G
1 1 1 1
I
1 1 0 l
0 1 0 4>.5
Pos.1.tion \'ectors
lbtm1lized Pos1.tion
o f Tr&ns!or:rood
voctort. o r TransfO,!t'IOd CUbo
CIJbe
0 1 0 0 0. 1 = -o.s -o.s -o. 5 -{). 5 0 0 0 1 0 0.1 -o. s 0.5 1 0 0 0 0 0 -0. 5 0. 5 0 l 0001 o.s -o.s o.5 -o.s 0. 5 0.5 0.5 0. 5 0
1
A• =
1
a•
0 0 0 0 0 0 0
1.1 C" 1.1 D" 1.1 E* 1.1 f"' 1.2 c•
0
1.2
r•
Fiqure 3-ll 'IWo-p:>.int pe.rspectivcs.
-o.s
-0.5
0 -o.s .-o.s 0 -0.455 0.455 0 -o.4S5 0.455 0 0.455 -0.455 0 0.455 -0.455 0 0.417 0. 417 0 0.417 0.417 0
1 A• 1.
~
1
c•
1
c-
1 E"' 1~ 1 G1 r•
proJc<:Ud onto the z • 0 pl.ane.
In Fig. 3- ll.a, t !\e vanishing p::>ints aro at the
x-a.x.is at x - 10 aro on the y- axis at y • 10.
If ""-e first center the z- axis
with r~ to the front face as s.~n in Fig. 3-l lb, a suitable t\olo-point 'tt'.is is du: to the fac t that the obJect
p:::rspective is still rot obtair.ed.
sh::lu.ld fl.rst be rotated aboot the y- ax.l.S in order to attai n an esthetically pl&'!SJ..."lg t:w:>-pol.Tit per~ve .
'!his is discussed later .
I!: three te.rrS in the 4 x 1 coh..-:n fl"atriY. are nonzero, then a t:hree-pomt
or obL.que ~...ive is obtained .
z 11
(X y
= (x
yz
The throe-point pe.rspective transfo rmation
11r1 0 0 p, ~ [x y z
(px
+ qy + rz + 1)
(3-40)
0 1 0 q
0
ol
rJ
•
~0 0 0 1~ w~th
o:rd.ina.r:'J a:x>rdinates
(x• y • z- 1) • [px + qyx+ rz + l px + qy + rz + 1 px + qyz+ rz + 1 1) results L"l t.i-.ree vanishing pomts :
one at the x-axis at (1/p 0 0 1 ), one on
the y-axis at [0 1/q 0 1), and one at the z- axis at (O 0 1/r 1).
perspec"'_ive vie-N of a unit cube is
(3-41)
sl'~•n l.TI
Fl.g . 3-12 .
A three-point
'Ihe r esults are proJected
In Fig . 3-12 the throe vanishing points are located on
onto the z = 0 plane .
the x-axis at x = 10, on the y-axis a t y • 10 , and on the z-a.x.is at z "' - 10.
0:
0 c 0 D 0 £ 1 F 1 9
0 0 1 1 0 0
PoSJ.t.ia'l Vector s o f Transf oLHed
vectors ot
Vectors of Ori9inal C\.lbo A
~tm'll ized
PosJ.t:.1 on
Po5itioo Trans formauon :-auix
0 1 1 1 1 0 !. 1 0 0 1 0 0 1 11
0
0 0.1
Trnnsfocrcd
-
0.1 0 0 -0.1 0 0 1
l
Cl:be 0 0 0 1 A• "' 0 0 o o o 0. 9 a• 0 0
Cl.lbe
0
o 1 o 1 c• o 1 o 1.1 o• l 0 0 1. 1 E• 1001
F"'
o 1.1 c•
G
1 1 1 1
1 1
I
1 1 0 1
l 1 0 1. 2 P ·
)
.
Dr-----.'
0 1 A•
0 0 0 l 0.91 0 0 0.91 0 0 0 1 0 0. 91 0. 91 0 0. 83 0 .83 0
u,. l
•
o• ~
p
G• 1•
c ·~----G"
a1
,., "'
,, "
>-10
T u VP2 on ~-.u.•~ .H 11. IC'
A .- --+--~-
E
c-
Tc. \ Pion)·•nn
Tn \ Pl on , •• ," ·IU
8"
)"
o··- --c-I "'-." -- -
lll I •
1 1 1 l 1 1 1
~
.,
I
Figure 3-12 Three-point perspcctJ.ves .
•
TEOtH
3-12
h1len a perspective viLv of an cbject is created, the horizontal reference
line is norrrally at eye level, as shown in Fig. 3-13.
Vanishing p:>ints arc then
p:>ints oo the horizon line (eye level) at which parallel lines converge. 3-14a shows heM the vanishing p:>ints are forned.
In genenU. , different sets o f
parallel lines will have different vanishing points. Fig . 3- 14b.
Figure
This
l.S
illustrated w
Planes of an ooject which are tilted in relation to the ground
plane have vanishing points which can fall above or below the horizon eye level.
'ltlese are often called trace points and are sluwn in Fig. 3-14c . \\hen viewing a CRl' display the positi.oo of the viCWJX>int or eye is normdly
fixed .
'Ihus , rather than rroving the viewpoint to ootain an acceptable perspec-
tive view, it is comon to ITW'Iipulate the position ard orientation of the OO)ect displayed on the CRT.
This will normally require ooe or rrare rotatials and
translatioo.s of the object. As we have seen, i f the viewing point is on a line normli to the center of
a wti.t cube face, then a single-point perspective "looks rl
In this view
all reoed.i.ng ~!.anes con'll'er9e to one vanishing point oo the horizontal eye-level line, and all perpendiculars are drawn as true vertl.cdls. cube
'n1e front and back
faces are parallel to the picture plan and do not converge.
Papel
VP
- - - - - - - - - -Hor11on 1- yc lc\C I
I
St.liHIO po1nl
Figure 3-13
Perspective t.ransfoll!Ution to paper.
H r r
-
-
\P
\P
-- -
( ,.
-
\p
'
-
•
<::-::> -- - --- ~
-- --
~
-
~
:cr •;;In.o..:s ::ltlra:lel L.-.es . •
Hot ron
\ P
--·
TP .
, _.-
---
n u
--- -- -- ------
--~---------
--
- - - -..:::--..,.. \P -:::- -
--------
\-E will roN rotate the urut cube about the y-ax.is and translate l.t to the
point [0 m n].
'l11e resulting figure
~;.'ill
then be viewed fran a P'Sition k on
the z-axis and finally projected onto the z = 0 pl.aoo.
Uswg Eqs. (3-7), (3-11),
and (3- 29) with r = 1/k yields
-sre
cos~;
o -sine o
0
1
0
0
sine
0
cos6
0
0
m
n
1 0 0 01 0 1 0 0
-
,.cose 0 0
0 0 0 1
k
1
Is:~
1 0
L0
mO n + 1
0 0
0 0 0 1
0 cosO
k
k
Since projection 01to the plane z = 0 was desinld, the third oolum of t..'Lis
ll'atrix is all zeros.
NOtice that
t\Oo
of the t hree terms in the fourth col 1.1m
which oontrol perspective in the matrix are nonzero.
'Ihus a t'..\'rpoint perspec-
tive view will result. '!be eight vertiocs of the unit cube, expressed in honogeneous coordinates,
can be represented as an 8 x 4 matrix .
r-hltiplication of t..l)is rratrix with the
above transf0%l'll4tioo rMtrix produces the required transformation on oach point.
'Illis
l.S
given by • -suw .. ~
0 0 0 11 fros6 0 0 0 0 1 1 1 0 0 0 1 1 1 • Sl.n 0 0 0 1 0 1
1 0 0 1
0
1 0 1 1
1 1 l 1 1 1 0 lj
mO
k 0
-
m
0
!!.+1
m
0
cosO + !!. + 1 k k
sin'"
1+m0
coso + !!. + l
0
1 + m 0
!!.+1 k
0
.
S.ln;.
cos
k
~+
1J
k
+1 k
m
0 -sinG + n
cose +sine
m
0 -sine + oosO + n + 1
k
k:
k
mo -sr-e + J<0
1 + m0
oose
e-
k
cose
oose + sinS 1 +
For
k
-sre f +
K'
..
~+1
+ l
60° 1 m :- 12, n : 1, an0 k = 1 , this matrix Of transfomcd, horrogcncous
ooordl.nates is
ro
-2
o2
0.86 -2 0 2 . 5 0 . 86 -1 0 2.5 0 -1 0 2 o.s -2 o 1.14 1. 36 - 2 0 1. 64 l.36 -1 0 1.64 0 . 5 -1 0 1.14
I
l
'
!ben, since
[! H !B
[X"' •'* 1] I
ll
the! proJected pcn.nts x•, y• are given by
0 0. 344 0.344 0
-1.0 -0.8 -0.4 -0.5
0 .44
-1.75
•
0. 83 -1.22 0. 83 -{).61 0.44 -{).88 'Ire transfotned cube is shown in Fiq. 3-15 . the parallel sides of the cube.
0
...
•
Cl:lserve what has happened to
Lines which were initially parallel to the x-
intersect at ooe of the two vanishing points. 'lbe intersections of the initial parallel lines occured at infinity. '!hey now occur
and z-axes respectively
nc:1n'
at finite values along the x•-axis.
<:nly lines iniHally parallel to the y-axis
remain parallel after the t:ransfor:natioo.
Following sim.U.ar logic, a three-point perspective can be created and
vie..-ed fran a position k on the z-axis by first performing rotatioo.s about two y
oose
,..
0
o - sine -sine k 1
0
sine 0 oose 0
m
n
0
cose k !!+1 k
Vfl l
VP1 _.-~I ....--
//
_, /" " I I -'/ I
/(
I I
I I
I
I
e ..
60°
m • -2 n • 1
k. 1
Figure 3- 15
Perspective transfonnatioo .
For exanpl e , one might rotate 6 degrees about they-axis and
different axes.
then ~ degrees about the x-axis before applying the perspect i ve transfooretim .
A cooplete set of transfo:anatia'\5 on a unit cube to obtain a three-point perspective is then 1 000 0 1 00 00 1 0
.,...,
I
0 ~
0
lan.l
o-....c 0 0 0
0
0
0
I
c
0
..,.. .,..
•Ulll
0
0
.... .....
L
0 0
3-13
-
all.I!
-
0
.,..
0
0 000 I [ 1 DOC! I
-1
....~ .. w
1 0 0 0 • J'? oI 00
• rrnr
~-·
l
0 r•~
c ~._..-
·11"1
..,... • ( l&1rl' - """'0 II w
0
..
I • ~Ulll • C.lrl'i •
f!!:!'"l
PoiNTS AT INFINITY To sho,..t rrore clearly
\~t
has happened to points at infinity, consider
the following exanple.
'Ihree po1nts are given by 11 0 0 OJ , (0 1 0 OJ , and [0 0 1 0].
These
wct:ors represent points at infinity on the x-, y- , and z- axes respectively.
!New, performing the boo-tx>int pe.rspective transformation, given by, '"1
o o o· ·o.s
0 1 0 0 0 0 1
o.
o
o -o. s6·
0
1
0
0
0
1
0
0
0.86
0
0
0.5
0.86
0
0
0. 5
0
Thus, the point at x
=co
-2 0 2
g
oo,
o o -o.86 I
.
transfoil'RS to x* • 0.5/-0.86 • -0 . 58 andy* • 0 .
This is the vanishing point VPl in Fig. 3- 15. to y* •
·o.s
'lbe point at y ..
oo
transforms
ari1 the point z •"" transfoxms to x* • 0 . 86/0 . 5 • 1.7 andy* ... 0.
This is the vanishing point VP2 in Fig. 3-15 .
'1he result is that points at
infinity in me coordinate system have been t.ransfox:nel to finite FOints in a second system by use of harogeneous ooon:hnates .
ic of harOgenCOUS ooordinatcs , we again consider the transfometim of a l.1ne parallel to the z-axis onto the x*z*-plane.
The line AB is shown in Fig. 3-9 .
Applying a perspective trans-
forrratim rratrix to a general FOint m the line gives
t-~TIID·~TICAL EI~
78
FOR OJMPOI'ER GRi\PHICS
(X Y Z H)= (x y
Z
1] rl 0 0 0 =
y
[X
Z
(f + 1))
z
1]
1~
AB to the line A*B*
0 1 0 0 0 0 l
!..
k
0 0 0 1 (x+
[
This operation
(\o.~th
shown in f'ig. 3-9 .
at (0 0 k).
~hen
y
X
y- z* 11 =
c; +
1>
~+
1)
(~ + l)
k = l/4) transforms the initial
A*B* lies oo a line which passes through a vanishing point the Z-<:OC>rdinate is neglected, the line A'*B* is projected onto
the pl.:me :z• • 0, as shc:Ml by line A' B' .
The oot result is a projection from
the line AB onto the plane z* = 0, fran a center of projection at (0 0 -k).
Notice the symret.ry of the vanishing point and the center o f projection. An algorithm which will genexate a general perspective vie..~ is given in l\ppendix
3-14
c.
RECONSTRU:T I(J.j OF T~-DIMENSIOOAL INFORI'IATIOO 'l11e reoonstructioo of a thrce-dinlmsional ooject or positioo in sp:x:e is
a
ca11:al
problem.
It occurs c:ontirn.ously in utilizmg rrechanical drawings which
are orthographic projections.
The rrethod of reconstructing a thl::ee-diltC'ISional
object or positioo frau o.o or nore views (orthographic projectioos) given on a n'!Ciw\ical drS'Ning is well kno;.m.
,...
!Jo..-ever, the technique of reconstructing a
three-d:inl:msional position vector fran two perspective projectioos, for exanple,
tw:> photographs is not as well koown.
Of course if the trethod
l..5
va 1 id for
perspective projcctioos, then it is also valid for the sin'pler orthographic
projectioos, and in fact for all the projections srentioned above. ~
will shc7,.. belON, i f certain other Womab..oo
~s
FUrther, as
available , then no direct
kno,..lcdge is required about the transfor:JMtion . 'Recall that the general perspective transfomation can be represented as
a
gr>Jlet"al
4 x 4 nutrix.
Thus (x
y z l)T '
T' =
T' u
e
(X Y Z H)
(3-42)
~re
T'tz T ' 13 T'14
T'21 T' 22 T'23 T'24 T' 31 T' 32 T' 33 T' 34 T'41 T'42T'43T'44 l1lO rosults can be pro)ectcd ooto a t::wo-diJrensiona1 plane, say, z "" 0 by using
e":l• (.3-28) I i.e . I T" ::. 1 0 0 0 0 1 0 0 0 0 0 0
(3-28}
0 0 0 1 OJnc:atenatioo of the 0.0 matrices yields
T = T,.T'
=
Til T 0 T 14 12
(3-43)
T2l T22 0 T24 T31 T32 0 T34 T41 T42 O T44 It is useful to write the transformatioo, say, [x y z 1)
Til TU 0 T
14
::: (X 'i 0 1]
(.3-44)
T21 T22 O T24 TJl T32 O T34
T41 T42 O T44 = [x* y*
0 1)
(3-45)
Note that x* and y" are the coordinates in the perspective projeet.ioo onto the
z • 0 plane. the x
= 0 or y
'nle same procedure could be accutplished using proJecticns erato
: 0 planes .
Writi.ng oul F.q. (3-45) yields
T x + T2ly + T31z + T41 = Hx* 11
{3-46a) (3-46b)
T x + T y + T z + T44 34 24 14
~
(3-46c)
H
using H from fl:J. (J-46c) and substituting yields
(Tll - T x*)x + (T2l - T x*)y + (TJl - T34x•)z + (T41 - T44 x*) 14
24
(Tl2 - Tl4y*) x + (T22 - T24y")y + (T32 - T34y*)z
=0
(3-47a)
+ (T42 - T44y*) • 0 (3-47b)
As suggested by sutherland (Ref . 3-9) this pair of equatioos can be considered
in three d.ifferent ways . b.Q
First assure T and x , y, z are knc7.-m .
cquatims in the t:w:> unJ
'lhcn t..hero are
'lllus, they may be used to solve
directly for the ooordinates of the perspective projectioo.
This is the approach
taken in all the previous discussions in this chapter .
Alternately T, x*, y* can be as.suted J
In this case
equatioos in the three unJmown space cxx:>ntinates x, y, z. ticns cannot be solved .
\ole
have two
'!he system of equa-
fioNcvcr, i f b.u perspective proJecticns say t:w:> photo-
graphics are available , then F.q . (3-47) may be wn.tten for both proJectims.
ru.s y1clds
~~1
(T~
-
~4x«1)y + (~1
-
~4x*1)z
+
(~1
-
(-rfl - ~4y.1)x + (T~
-
~4~y + (~2
-
~l4y*1)~
+
(~2
- rr!4y*l) = 0
1-?11 -
~4x*2)x
+
~~1
-
~4X*2)y
+
- or;4x*2}z +
(~1
-
~4x«2)
= 0
(~2
~4y*2)X
+
(~22
-
'S~.,2)y
+ (T;2 -
+ <-r!2 -
~4y*2)
a 0
-
-
~4x*l)x
+
(~1
~4y*2)Z
~4x*l) = 0
wher& tt.o superscripts l and 2 illdicate the first and second perspective proJectiOO view'S.
Note that the transfor.mations
These oqucttions rr.ay be rewritten .in matrix
~ and ~ need
not be the sarre.
form as •
[AI (X] .,. (B]
(3-48)
~.ere
[A] ""
~ll
- rri4X*l
~1
~4X* ~2
-t-12 -
1
-
-
T~4x«l T~l 1
~4Y* ~2
2
2
~11 - ~4x* ~21 - ~4x* ~1 2
2
~u - ~4y* ~22
-
~4y* ~2
-
~4x*1
-
~4y.l ~4x«
-
2
~4y*
-
2
y Z )
(X) T :
[X
(B] T =
l~4x« 1
-
~l ~4y*l
-
~2 ~4x« 2
-
~l ~4Y*2
-
~2]
Equatloo (3-48) represents four equations :in t.00 three unknown space o:x>.rdinates x, y, z.
The problen is over speci.fied and thus can be solved cnly in sane mean
or best fit sense.* '1hls is an extrerre1y threc-diJrcnsional positional data.
E:x..arrp1e 3-2:
pcJWerful
technique for obta.ini.ng
'lliREE-DIMENSICNAL
Asslll'C that the rreasured position of a point in one perspective projectioo is (0.3763 -0 . 8258 0 1] and in a second perspective projection is [0.6548 0 0.2886 1].
The first perspective proJectioo transform1tloo is knc:7.m
to be the result of a 30° rotatioo about the y-ax.is follo.--ed by a 60° rotation *l\ rean squares tit can be
a:ltPJt.ed by solving the eqootioo
(A)T[A} (X) ~ (A)T(B)
~re
(f\) T (A) is square and of the co.r:rect d.irrensian for (B) . In particular I>!J=[(AJT[AJJ-l[AJT[s]. Note that i f no solutioo result.s, then the iJrposed
conditl.oos •'lre redl.l"ldant and no uni.que solutloo which qives a "least" error concli tion f"..xJ.Sts.
arout
the x-axis plus a translatioo of 2 units in the negative y-diJ:ectioo.
'Ihe point of projection is at z .,. 1, and the result is projected onto the z •
o
plane. 'l1le second perspective projectioo is the result of a 30° rotat.ioo about
the x- and y-axes .
The point of projection is at y ... 1, and the result ~
projected onto the y =- 0 plane.
~
=
o.s
~
o o -0. 867
and
il
are t!'l.ls
~ ~ o.867 o -o.s
and
0 0 0. 867 0 0 0.5 0 -2 0 2 0
o
0.25
0 0. 433 0.867 0.433 0 0.75 -0.5 0 0 0 1
1
The A-flBtrix is then
[A] •
0. 826252 -0 . 715969 0. 867 -0 . 5
0
0.67885 1 0.4129 -0.317712 0.7604 0. 182784 -0 . 8943
'
and
solution yieldc:
[B)T e (0 . 7526 0. 3484 0.6548 0.2886) X ... [0 . 5 0. 5 0. 5] , i.e. , the center of a unit cube.
An al gorithm which utilizes this technique is given in Apperrlix C. As a third way of CXX'lsidenng Eq .
(3-47)
\~
rote that if the location of
several points which awear in the perspective projcctioo are J<.no,.,n in object
space and in the perspective projection, then it is possilile to determine the transfonnation elerrents, i.e. , the T .• ' s. ~J
'Illese transfoD'I'atioo elerrents can
subsequenUy be tt
technique descritEd alxlve.
~
points using the
To soe this ~ again examine Eq. (3-46), which nay
be rewritten as
- T34zx* - T44x* = 0
(3-49a)
T12x + T2~ + T32 2 + T42 - Tl4xy* - T24yy* - T34~ + T44Y* • O
(3-49b)
Tllx + T2ly + T3lz + T41 - Tl4xx- -
T24yx~
Asslll\ing that x• and y* as well as x, y, z are kno.-'1"1, the Eqs . (3-49a,b)
represent bolo equations in the 12 unknown transfornetioo elmcnts Tij.
Apply-
ing these equations to six noncoplanar kruwn locat..ions in d:>Ject space and in
the pers?£tive projectial yields a system of U equations '1hese equations can be solved for the T .. •s. ~J
transfo.x:natian.
~
U \11"\l<:Jn.m.s.
Thus we have determined the
Notice that in this case ro prior Jan..r1edqe of the transfor-
rration is required.
I f , for ex.arcple, the pers?£tioe projections are photo-
gra{tl.s, neilher the locatioo nor the orientation of the carrera is required. In
matrix form the system of U equatioos can be written as
-x1xl * yl 0 -yl~--
x1
0
0
xl -x1Y1* 0
~
0
0
x2
yl -ylyl*
zl 0 -zlxl* 1 0 0
-zly1*
21
1
* Tu
y • 1
• 0
(3- 50)
Tl2
y2 0 -y2X2* z2 0 -z2"2* l 0 X2• Tl4 I 0 y2 -y~2· 0 z2 -zi'2* 0 1 y2 • T21 -x~ • 2 -xyc3* y3 0 -y3X3* z3 0 -z 3x3• l 0 X3* T22 - xy 3* 0 y3 -y3y3* 0 z 3 -zl'3• 0 1 y 3 * T24 -x4x4• y4 0 -y4x4* z4 0 -z4x4,. 1 0 X4* T31 - x4Y4• 0 y4 -y4Y/ 0 z4 -z4y/ 0 1 y 4 * T32 -x2~*
x3 0 0
x3
)(4
0
0
)(4
)(~
0
0
xs ->
-x5x5* Ys 0 0
-ys~·
zs
-z5x5*
0
Ys -ysys* 0
zs - zsYs"
x6 -x6y6* 0
y6 -y6y6*
0
26
1
o ·x5•
T34
0 1 y * T41 5
-z6x6*
1
0
)(6
T42
-z6y6•
0
1
y6
T44
)(6 0 -x6x6• y6 0 -y6x6* z6 0 0
0
Xl
where the subscnpts correspc:nd to points WJ.th fgeneous, they will contain an m1n.tra.ry scale factor . T 1"1\:ly, for ex.arrp1e, be defined as unity and the resulting lransfo.rma44 tl.oo norruUzcd. ·nus reduces the rcq..Uroncnt to 11 equations or 5 1/2 points . ~
If the transfornatioo
lS
nornalized,
t.~
the right-hand suie arrl the
Exanple 3-3:
the last COlUITI'l in (A I
us Jratr ix equatioo
lS
I is 11'0\ied
to
solved.
EI»ari'S fOR REXXhS'I'ROCTIQl
AS a specific ex.anple, cx::nsider the unit cube with the six Jcno..m corner pomts
1.n
the physical plane q1.ven by
IPl =
ro 0 01 0 0 1
lo 1 1 10 1 0
l 0 0
l
1 0 l.J
thc corresponilinq points in the transfonrod vifM are q1.vcn by
(P*] •
fXiUation (3-48) then becares 0 0 0 0 0 0
0
-1
o.J44 0.344 o 0.44 0. 83
-o.8 -0.4 -o.5 -1.75 -1.22
0 0 0
1 0 0
Tu - 0
0 1 0
T12
0 0
0
0 0
0
0 0
0 0
0
0
0
l
0 0
l
0 1
Tl4
0 0 0
0 0 0
0
1
0 1 0
T2l
0 0 0
1
0 0 0
0 l
0. 4
0
1
0 0
0
1
0
0
0
0 0
0 0 0
0
1
0.5
0 0 0
0
0 -0. 344 1
0
0
0 -o . 344 1 0 0.344 T22 0. 4
0
0 1 -0.4 1
0 0
T24 T31
-o.s
T32
0 0 .44
T34
0 1
1 0 -o. 44 0 0 0
0
0
0
0 0 0
0
0 0
0 1 -1.75
T41
1 0 -o. 83 0 0 0
1
0 -o . 83
1
0 . 83
T42
0 1 1.22 0 0 0
0 1 1.22
0 1 -1. 22
T44
1
1.75
1
0
SOlut.ia'l yields [T) =
0. 867 0
-o.s 0
1 0
0. 5 0
0.867 0 -2
2
An algorithn which utilizes this technique is qiven in ~ C.
3-15
STEREOGRAPHIC ~CTJOO Q1ce
an object has bePn transfoxrred to a desired positioo, many techniques
can be used to actually display the abject. Tec::hniques such as shading, coloring , etc. , are frcquP_ntly device-Oependent and are .beyald the scope of this book. Others, such as hicklen-l..ine reaoval and cliArlng can be acmtplished in either software or ha.rdnare. Arother useful display technique is the use of stereo pairs to create the illusi.oo of depth.
Often stereogxaphic projection
can be used for this purpose rather than hidden-line rerroval. .
Graphical output
of cnqineering struct:ures such as bridges and towers lend therreel ves nicely to ste.r:eographic projcctloo .
Also, the representation of three-dirrensi.onal
arra.nqerents such as piping systens, a.rchitect:ural designs, rredical >t:-ray
studies , the chcrni.cal structure of organic nolecules, and solid crystal oonfiguratio,s can be irrproved by the use of st:aJ::eographic projection . Jn a stereoytafhic projectioo, a separate perspective view JTUSt be created for each eye.
'
Then a stereo viewer such as sho..ln in Fig. 3-16 can be used to
view the apparent single thr
·
object.
If desiied, a
stereo~
can be designed and attached directly to the cntputer controlled output device.
For a hunan with average eyesight, the strongest stereo effect occurs at a d.ist.ance of about 50 en in fralt of the eyes.
Thus, for an eye separaticn of
Ph..). /}
I
I It ll.
Plane
or
P rOJCC: I ion
k
Figure 3-16
stereo~ -
Figure 3-17
stereograpuc perspectives.
5. 0
Cll,
-1
the stereo angle is c • tan
For a ste.roo vie-acr ~
(5/50) = 5 . 71° .
tr.e
let d be the scaled scparatioo distance betwetm
eyes (cf Fig. 3-17).
If
~ viewer has a focal length of k units, then the value of d .1s fixed by tl~
requ.i.reJtent tan-l (d/k) = 5 . 7l0
correct stereo angle c .
'lhus, d = k/10 i s required to 11'\;'lintam the
•
To obtain this scaled eye-separation distance fr
single view o f the obj ect, a -+d/2 = +k/20 horizontal translatioo is ~rforrred before creating the left-eye perpsective view and a
-<1/2
= - k/20
oorizontal
transl.atioo is perfarrrcd baforo creating the right-eye perspective vie.... Before generating a stereogra{tlic projectioo it
LS
oonvenient to center
the object with respect to the z-ax:LS and then rotate i t about tile y- axis in
preparat.lon for a tw:>-point perspective, or about y and x in preparation for a
three- point perspective . One final pl;'E!Ca.utioo is advisable . ~tatioo rmy cause sate of the points to be behind the p:>int of observatioo, z c k. 'lllcse over tbe shoulder points cause a distortioo of perspective views.
To prevent this a
final transl ation in the negative z-cl.irectia1 can be rradc to ensure that all p:>ints on the object lie on o r behind the x*y*-plane, e . g ., -z* !._ 0.
preliminary operations result in
re.~
'Ibese
positioo vectors \vhi.ch descr ibe the t.rans-
forned initial object. To create a stereo pah, the rr.at.rix contai.n.ing the new posi tion vectors l.S
t.ransfonred by use o f the two matrices Left eye
Right eye
1
0
0
0
1
0
0
0
l
0 0
0
1
0 0
0
0
0
0
-~
k
k
0
0
0
20
0
0
-20
1
0
'Ihis creates two separate pers pective vie-'S , one for the left eye and o ne f or
At this int:e..orediate step the left-eye view is to tho r i ght o f
the right eye.
the ori gin, and the right-eye v i ew is to the left, as 50CMn in Fig . 3-18. Le fl
I
_ .. ----
-
c ~ c: \le~
,.. --~
c~·~.L~----- o - - - - - --ci•l F
l.tl l C:}( v
Figure 3-18
Intermediate Stereo Position.
-·
'Ihc last ~rat.ia'l is to sep.-Lrate the two pcrspcctivc vie..rs by the p:rqer di~
aloog the x-axis in order to place each view directly in front of the
proper e:;ye.
As
stnm m
F~g.
3-18 , (X*)R
l.S
the average
val~
between the
largest and S'T\l.Jlest value of x*R and (x")L is the averag-: value be~ the
largest ard smallest value of x*r.·
t'e let. D = (X'*)L- (X*)R.
mJSt be calc.llatcd by searching the no:rr.ulizcd
~-val~s
The value of D
of each perspective
• VJ.C:.l.
A final hori?.ontal transl.atioo in the x-directioo of L .. D + (E - 0)/2
for.
nght-eyc view and - L for the left-eye view places the stereo pair in
t.">)c
prqx>r positioo.
'Ille value of E is the actual eye separatioo distance for the
hll'\an observer,
"~u,·c'.i i~
1!6
:wale c f thr output device .
for cxanple, i f
the final points for the right-eye view fall bet:..-een 2 !. >
ootwcen -4
~
xt_ ~ -2 ,
then one might c:h;:x)se
to plot these two views on a standard sheet of paper, using a 10-unit scale
bet;...-een
-s
l"n'lSurcd
x* $.. 5.
Then, the actual 5 an distanoe
relativo to this scale o;..uuld be about E
bet\.~
= 2 . 5.
Q'le
hunan eyes or two iterations
r.ay be useful to determine the best value of E to use. An algor1tbn to generate a stereo pair for an ob)C'Ct wtu.ch has been p:l!Si-
tiooed and rotatG1 to form the desired view and parspective is given in Afp>-Niix C.
I t uses the left and right ste:reo rratri.x pair presented above to
Vigure 3-19
Sinple stereo vi<.......
•
create the t:\10 perspectives and a final translation matrix to generate the required eye separatioo distance. An
exarrple of a stereo view of a siJTple
generat ed using the STERFX> subprogram of Apperxhx
ional wire frcwne
c
is sho.m
1.n
rig.
3-19.
The
positioo vectors describing the wire fra:re are 0 0 0
1 0 0
1 0 1 111 0 1 1
0 1 2 0 0 2 -1 0 2
-1 0 0 -1 1 0 0 1 0 Prior to cal ling the S'l'ERID subprogram, this daUl was first translated by usi.ng Sli:lprogrcm n =
-o.s.
J[YJ'RII.~
(Appendix
c,
Sec. c-10), with £ • -0.5, m = -1.5, and
'lhen a 20° roUltion about the y-ax.is was per!orned in preparation
for a b.o-point perspective. u9:!rl for this operation.
The stbprogram 3DYOOI' (1\ppcnd.ix C, Sec. C- 7) was
At this p:>i.nt a catpJter search sho..>ed that the
maxinun z-value was 1.9226 .
Then 3m'RA.~ was used
a<3<1in
n = - 1.9226 to place the object behind the x*y*-plane.
with 1
m = 0 and
At this p:>int STEREX>
(Appendix C, Sec . C-17 ) was call ed, using k • 4 and E .. 2.
the stereo pair slxJwn in rig . 3-19.
= 0,
'lhis resulted in
However, a final overall scaling
~-as
a pplied to prcxluce Fig . 3- 19 .in order to fit the output to a particular storuge t ube cro'.
REFERENCES 3- 1
lti:lerts, L. G. , "IJarogeneous Matrix Representation and t-'. arup.Uation of
N-D.iroonsi onal COnstructs, "
r:o:::urertt t1S 1405 , Lincoln L1.boratory, M.I.T. ,
carrin"idge, l1a.Ssachusetts, May 1965. 3-2
Ahuja, o. V. , and Coons, s. A., ..Gecrtetry for COnst.ruct.ion and Display," I.Et-1 Syst. J ., vol. 7 , nos . 3 and 4, pp . 188-205, 1968.
3- 3
\-bodsf otd , P . A. , "'lbe Design and lnplerentation of the GIN:> 3D Gra{iri.cs
Soft:
\bxlsford , P. A., "GIN)-{)eSign and Inplesrentation , " CM Group Doc. 27, U1L, carrin"idgc University, 1969 .
88
~11\TICAL EIJ.l.!EM'S
FtlR COMPl1I'ER
GRAPHICS
3-5 Cocns , S.A., ''Transfor.nation and 1-'atrices," unpublished notes , of
:-~clugan
Univers~ty
Short Course in CbrTpuler Aided Design , 1971.
J-6
Forrest, A. R. , "co-ordinates, Transfor:TI'Olt.l.Ons, and ViSUil.lizatioo niques," CAD Group n:x:. 23, Catnbn.dge t'nivcrsity, June 1969.
3-7
forrest , A. R., "c,c,.pumtional
Gc~t.ry,"
Proc: .
roy.
~
Sex:. (London) , vel .
A321, pp. 187-195, 1971. 3-8
Kubert , q., SZl\00, J. , an:i Giulieri , S. , "The Persp:!Ctiva Representaticn
of F\lnctions of '1\.u variables," J . Assoc. c::atplt. t-\!ch., 193-204, April 1968. 3-9
\1'01.
15, 2, pp.
S.!the.rland, I . E., "'lllree D.irrensional Data Inpul by Tablet," Proc. I EEE, vol. 62, 2, pp. 453-461, April 1974. •
CHAPTER 4 PLANE CURVES
4-1 INmOIXJCTI ()'~ Today a multitude of t....">Chniques are available for ltli1lllla11 y drawl.OCJ and designing curves.
'Ihe instrunent userl can be chosen fran
pencils, pens , brushes, knives, etc.
A m1pass,
stra~ght
~
wide vanety of
edge, French curve,
ships curves , splinBB , tmplates, cm:l nany other rrechanical devices , as 'h'el.l
as r.odels , col or dta.rt.s , and perspective grids , rnay be used to assist in the oonst:.rucLion of a c:urve.
A great
many different techniques and tools for cuxve
generation and display will also be needed \
sare
of the ideas that have proven useful for generating plana, o.o-<11mensional curves
ill'e
discussed in this chapter .
The previous b.o chapters have treated the transfo.rrt\ation of points.
A
curve may be presented by a rollection of points, provided they are closely spaced as the niC3i us of cw:vature decreases.
Hc::lwe\ler , there are several
reasons why a mat:rematical representation of a curve has an advantaqe over its representation by a grid of points . 1.
SOrre of these are
A !Mthatatical represen~ticn is precise, and the properties of the
curve such as slope and radius of curvature can be eas.Lly calculated . rra~tlcal
2.
A
3.
~s
representation can be mt,w=t.ly stored
1.n
a cooplter.
o! an object represented mathematically in a muputer can
be easily ptcduoed.
4.
\\hen
C'lli"w"eS
1.nterpllation schsre
are analytically defined over the region of interest , oo l..S
needed to cxrpJte in~te p::>i.nts.
'nle use of p::>l!'lt.s to represent curves has proven difficult
5.
\ooile."l it is
desirable to o::ntiru*"llY alter t:he shape of the curves to reet certa.L.n des1.gn
cr.1.ter.ta.
A rmthrn-atical representation of the curves has prove."\ t:llCh r.ore
cse.ful !or t.his PU-"1X>S8·
we
ro.~
consider various techniques for math£!!\atically
repre~ti.ng
tiNO-di.rre.n-
sio.na!. curves.
4-2
•
tb~PAAAVETRIC URvES
Either a p..lranetric or a nonpararretric foon can be used to represent a CU..""\;'e mat:hm.atically . A oonpararretric expressJ.on can be explicit or inplicit in form.
For a plane curve, the explicit, nonparamettic expression takes the
form y • f (x)
(4-l)
Il' this fonn there is only one y-value for each x-value.
For t:lus reaso:'l the
explicit fotl':' cannot represent closed or CJ.l.tiple-valucrl curves . 'lhis limitat.la'l can be overCXXll!' by usin<; an inplicit eJ
{4-2)
Hc7..Jever, a p::>i."\t on an in'plicit curve se;:p.eat rrust be calculated by fl.nd.ing the
.-..
correct root of an algebral.c equation.
'!he detem.ir.aticn of roots of inplicit
curves satctiJres requires le."l~· calculations .
Both eJ
'n'lus,
the choice of the coordinate systen can affect the ease of USl.nCJ the curves and calculating their properties .
In spite of tl-.ese li!r.itations, nonparametric
curves have been 115M with success (cf Refs . 4- 1 and 4- 2) .
Silrple s.:ca'ld- and
thlrd-degrec cqwtions provide a wide vnriety of curve forms. Fl:an a rotharatical p::>i.nt of view,
la'o..n set of data p::>i.nts
l.S
t:r.e problem of defining a curve frau a
cne of interp:>lation.
'lbe c:u.rw can be made to pass
tl'lrough all Jcnc:Mn t:0i:1ts by use of ?Jlyncnial i.nterp::>lation.
'lhis requires
~ter:'ilining the coefficients in the ll'Btheratical equaticn for a ?Jlynanial of
SOI'e ~ee.
~ actual
shape
be~-en t:.he data
p::>ints dept!nds
lplt'l
the order
o! th:: p::>lyroniaJ and the Ntheratical t:oundary cxmd1t1.ons "'hlch are applied.
:. the other hand, i f
~
data p::>ints are only approxit!ati.o:".s to sare
unknown "true" ~ues, then a curve is required which indicates tht: correct. trend of the datu.
L"l general tlU.s curve l'l'c'lY pass t:hrou:jh ally a few data
FOint:s an:!, in fact , may not pass t:hrough .lilY of the data FO.i.nt.s.
'Ihis
PU\t.:F. OlRVES 91
requirmcnt exists when
fran
<:1
curve is needed to fit (intc.rpolat.e) data obtained
exper~tal
rroasu:rarent.s or other observed data. est:imaled values over the r
This curve ht. glVes
\\hen the data points inherently conbin unc::ertainb,..s, the rrethod of least squaros a.Ff>r<»UJT'dtion is a cx:mronly used curve fitting (or c:llJ:'W fairing) technique. 'this technique prod\IO:lS a curve fit of pr~ fonn y c f' (x) ~ch
minimizes the sun of squared y-deviations bcb.t.>en the observed and
estimated values. {:tlysical
'!he
~n
pn;scrib...~
fonn
~
usually b.:tscd oo a Jcro,.rlcdgo of tho
whl.ch produced the data.
CCmronl.y prcscribe:J e>
technique are fOI-& functions , ~~tial functions, and polyrx:mials . l\ fX"""Cr function might be prescnbacl as y = axb and c'lJl eJqX>nential fonn as y ,. aabx, l\ general p:>lynam.al has
\\ilere a and b are constants.
c-r2 + •.• + crml,f. ReqardltSs of techm.que requires solv1.ng a
sel
the presc:riOOd
'1
the fonn y =
C\..II'V\!
-+
C'r +
fonn, the least squares
of sinultaneous, linear algtiliraic equations in
order to determine the I.ITlkzv..m constants in the prcscr1lxx! equation. Z.bst mrp.1ter facilities have library programs for least squ,...rc curvo fitting .
Reference 4-3 oontaUlS a "'ell docunent..ed BASIC alC]Orittrn which irrple-
rrents this technique. For t.ha r~ of this chapter~ consider plane curves which pass through the spccl..fied data points. A general secord-deqree, urpll.c1.t equatloo 2 2 ax + 2bxy + cy + 2cy + f
.lS
"''ritten as
=0
(4-3)
If the curve is defined rc.J.ativc to a lOC2l coordinate system through which the
curve passes, then f ... 0.
By c.Wfi.Jung
the oonstar.t coefficients a, b, c , d , e,
and f , several types of plane curves can re produced.
Hound'lry oonditions arc
used to establish a particular curve through specific pomts , as shown below.
rf we now choose c =- 1. 0 m t:h.o genPral equation, then ~t
bcb.leen
l:w:)
to
dcf inu a cu.rve
[X>ints , hve independent oond.itions rust bP specified
determine the values of the remaining five coefhcients.
to
one choice is to
specify the position of the 0..0 end points, the slope of the. curve
segrrEnt
at
each end point, and nn i.nt.e.rn:;odiate point t..hrough wtuch the curve ITUSL pass . If, instoad , \,~ specify b
=0
nnd c == l.O, then the rmalytlcal ~scription
of ti'C resulting curve is fi..xcrl by specify.iJlg only four additional oonditions, su:::h as the boo end points and the u..o end slop.>S. An even s.wpler curve can be defined t7j first sct.:ting a • 1.0, b • 0 , and c
1.0.
'Then t:.hc fonr of the
curve is given by x
2
+/
+ 2t1x + 2ey + f = 0
'1lle three. ex>nd.it1.0ns rcqu1rro to flx d, e , and f can be the
(4-4) 1:'-0
em
po1.nts and
e.1.thcr the slop.. at the bcqi.nning or the sJ.or;c at the end of the curve: ~~t .
1\oot.her chou:e is to specify the t...o end IX>ints and a third internal rx:>int through \o.hich the A ~ght
curve nust pass.
line is obtained by setting a • b = c "" 0 .
~ the
equatioo
is
dx+ey+f .. O
(4-5) •
or y •
-(~)X
-
~=
11'0(
where rn is the slope of the straight line.
+ b
(4-6)
'Ihe values of m and b are fixed by
spec1fyi.ng the t:'-0 end points of the line.
create an inflection point in the curve ~t , it is necessary to use a hiqher order curve such as a cubic. 'llris curve form·can be expressed by 'lb
3 a + bx + cc 2 + dx - y ... 0
(4-7)
Specifying the t:'-0 end rx>ints and the slopes at these two p.:>ints fixes the
0
values of the four coefficients. A difficulty
arises \,'hen using axis-dependent nonparanetric curves i f the
end rx>int of a curve has a verti.cal slope relative to the chosen ooordinate system. •
'lhi.s infinite slope cannot be used as a ni.JI'et'ica.l bourdary conditi on.
It is necessary to change the coordinate system or approx.iJnate an infinite slope with a l..arqe positive or negative value.
FUrther, when points on an axisdependent nonparanetric curve are calculated with equal i.ncrem::nts .in x, the positioo of the rx:>ints will not be distrib.rt:ed evenly
curve .
alCI'lg the length
of the
This can affect the quality and accuracy of the gra{irlcal outplt..
difficulties lead to an interest in para!'llrtri.c curves for
'1hese
gra~ .
q_3 PAAA"ETR I c WMS In paranetric form each cxxmlinate of a point oo a curve is represented
as a flDiCtion of one or nore paraneters. For a curve of one pararreter, the p.:>Sition vector for a p:>int on a curve is fixed by all! val ue of the pare.rceter . If the parameter is t and the curve is t:INo-diJTcnsialal, one writes X= f(t)
(4-8)
y .. g(t) The
positioo vector for a point oo the curve is then a single
ro.1
matrix given
by P(t) = [f(t) g(t) J (4-9) If required, the nonparasretric fmm can be obtained f:tun the pararretric form by eliminating t:.re paraooter fxan Fq. (4-8) to obtain one equatioo in terms of x
and y. In a
PlANE OJRIJES 93
representing closed curves and curves with multiple values at a given value of
an independent variable . 'nlc ~t vector at a point on a para~"retric curve l.S
given by
P' (t) =
\
Y-~
( ) L - -_ __
rx• y' 1 ::r
[f' (t) g' (t) 1
o
(4-10)
_
10
\
~ the ' denotes differentiatial with LOt---._
_.
0
I
respect: to the
parar~eter .
the curve, dy/d.x, is written as
\
the tenn f' (t )
=0
I
I /
';.
in El:). (4- 11) ,
occurs~
Since an
one
X
= COS;J
y = sw.J
--
I I I"' // / """
: ~/
the s l ope dy/dx is infinite.
infinite slope
I
I I
(4-11) ~-b!n
I
'lbe slope of
0
0
./
_..--
--
10
10 t--...._
h
c:atp:>nent
of the tangent vector is equal to zero, )
it ci:>es not introchx::e cx:mp.ttational difficultics.
Since a point on a para:netric curve o ...._...__ is speci fiEd t:Jy a single value of the
parmreter, the parametric fornl is axisindependent.
0
"'
Figure 4-1
c
I0
Circle r.::prese.ntatia'\5
'lhe curve length is fixed by the range of the pararreter.
often convenient to noonalizc the parameter so that the
It is
par~t.r:ic
curve falls between 0 ~ t ~ 1.0. Due to its axis-independent nature, a parametric curve is easily transforned into a curve of the sarre shape but wi. th various orientatioos. '!his is aooatpli.shed by using the tratrix multiplication techniques discussed in the previous chapters to perform rotatioo, translatioo, sca1ing, and perspective
projection. A ooopa.rison of the parametric and nonparametric form of a circu.l.a.r arc is sllOn'n
in Fig. 4-1.
'1'hls figure was used by W:lodsford (Ref. 4- 4) to sln4 that
a choice of pararretric functions can be used to inprove the distritution of
gra,;hlcal repr-esentation. The gtadrant of the circle shown in Fig. 4-la is represented by the non~tnc fouu y = h - x2. Equal increments of x were used to obtain the FOints on the arc. '1his form is inconvenient to catpJte because a square root algorit:bn is necessary. It also produces fOQr results since the arc lengths are unequal. Figure 4-lb is obtained fran the hr1sic parametric form given by (4-U) P (t) • [oose sine] 'lhis produces good output since the arc l engths are equal, but the calculatial is inefficient because trigJa'I'Ctric functioos require several operations to catpJted points on a curve and thus give a better
Figure 4-lc is an alternate paranetric representation gi'V'Cn by
2 (l - t )
2t
+ t
(1 + t
rd l Where t • tanS/2 .
2)
2
This choice requireS very little
cx:Jil'lltation t.ine , but it is
a oatp:anise si.J·x:e the arc lengths are not exactly equal . ~,
consider the problan of determining the value of y
x is known.
\>~hen
the value of
As an ex.anpl.e assure that x • 0. 5 and that it is required to deter-
r:u.Ite y a1 a unit circle.
For the explicit representatioo this is straightfor-
ward:
y
= A:
- x2 "' {0. 75 = 0. 866
For the para.rrotric representaboo it is first nece~ to solve for the p!U"a-
rteter t in terms of x and then use this value to obtain y .
Specifically for
the parazretdc representation of Eq. (4-12) we have
x • cose y =sine
Thus,
e = cos-lx =
ros- 1(0. 5) = 60° y = sin(60°) = 0.866 Alternately, far Eq. (4-13) X
2 1- t - .;;.........-"" - 1 + t2
y -
2t
1 +
t
2
Solvin9 the first of these equat.ions for t yields t •
1 - X l/2
1
= /'1 =
0. 57735
and thus
y =
2
~~ =
4 = 0 . 866
For rrore
4-4
tbPARAiflRIC
REPREsENTATIOO
a: Coorc SEcn oos
COruc sections fonn useful planar curves for rrany applicatialS. S\J'!'r"drize sonc useful relationships frau analytic gcarotry.
~'Je ro-;
Figure 4-2 ill-
ustrates hc:Jw the intersection of a plane and right circular oone defines the various conic sections, i.e., the CliCle, paralxlla, ellipse and hyp!'rtx>la. The general definitial of a cadc section can be given as:
.. PLANE OJRJES 95
, /
d ( 11
Figure 4-2 A
de
Cooic Sections
conic section is a curve described by a p;:>int which rrovcs
such that its distance fran a fixed p;:>int (cal led the focus) divided by its distance fran a fixed line (cal 1 ed the directrix) ~
a constant
£
(called the eccentricity) .
All the c:ooic sections can be described by the general i.rrpl.icit
sec:::oro-
degree curve
ax2 + bxy +
C'i + dx + ey + f
: 0
'rtle cross product term bxy in this equatjoo oootrols the orientation of the
resulting
CCIIUC
with respect to sorre reference axis system.
If the reference
axis system is selected such that one of the a.~ is an axis of symretcy of the curve, then b • 0 . Since we can always rotate the curve such that. the axis of S}ITI"'retcy is one of the axes of the reference systan, we need ooly consirler the equation a •x •2 + c •y • 2 + d ' x ' + e ' y ' + f ' = 0 2
a ' - acos a + bcosasinet
(4-15)
+ csin2a.
c' = asin2a - bswcosa + d ' = dcosct + esino e' • -ds:i.na + eoosa
2
ect:JS
a
f' = f and
cot(2a ) .. a - c b
(4-16)
Alternately i f the curve exists such that cne of the refereooe ~ is an axis o f symretry, then iL can be rotated through any angle a to a new on.entatial.
If the coefficients a', c ', d' , e ', and f' are known far the original
axis of symtetry orientation , then the coefficients a, b, c, d, e, and f o(
Eq. (~-14) in the
rcw orientation are 2 2 a = a'cos a + c ' s1n c b
= 2 (c '
- a' )cos~in,)
(4-17)
• 2 c = a I s1n o ... c 1 cos2o d :: d 'coso. + e ' siro e :o e'coso. - d 'sino
•
-.
f - ... Equatl.On (4-14) em then bo used to generate the requued curve.
'Ihe character of
~
conic dcscritwi by Eq. (4-14) is an ellipse, a para2 oola, or o hyperbola, de~~ing on t.l)e value of the di.$:r.i.minatc b - 4ac (cf Ref. 4-5) •
..
c
•
Narrely,
a parabola if b
2
th~
- 4ac
c:urve is
=0
an el11p5e or a circle if b 2 - 4ac < 0
a hyperbola if b 2 - 4ac > 0 Considen.."lg Eq. (4-15), a CJ.rcle occurs if a' = c ' .
'1l'le equation of the
cuclc with ce."lter at n, k and radius r is given by
(x ' - h)
2
(y' - k)
T
2
~ r2
(4-18)
h = ~· 2a'
~· k = 2a'
r•
r ~O
4a' 2
If r = 0 , the circle degenerates to a point .
A parabola occurs if a ' :; 0 and c ' = 0. equatlal of the parabola is (X ' - h)
2
\\'.ri t ten in the camon fox m the
(4-19)
• 4p (y ' - k)
h:: - d' 2
e' p=4
k
= 2f '
- d'2 2e'
e 'JO r
Equatioo (4-19) represents a paratola "-OOse axis of syrrtl'ntry is y ' .
If
PIA'IE ClJRVES 97
the roles of x ' and y' are interchanged, the x'-axis will oo the axis of
If p > 0, the parabola will open up or to the right; if p < 0, .1t
S}'ll"l'l'et.ry.
will open do..n or to the left.
occurs when neither a' nor c' in Eq. (4-15) is zero and both a ' the sane S.lgn, • • . ...___ I 1 > 0. 'Ihe equatioo of the ellipse with .l.e. 1 wucu a c
An ellipse - -' and c I 'nu.ve
oenter at (h,k) is then (x' - h)
2
2
..
(l
> 6
(4-20)
(l
h
-d'
= 28'
k"" (l2
-e '
p
= ~
a
e2. ~ c c;ao
g = -f' \~
g =
o,
a point results.
'!be eccentricity of the ellipse £ is
£-
I Q2 -
B2 < 1
a
A hyperbola is generated by Eq. (4-15) when neither a' oor c' is zero and
both have opposite signs , i.e., when a'c ' < 0 .
'llle equat.ioo of the hyperbola
with center at (h, k) is then
(x ' - h) 2
cl
(y ' - k) 2
- - 82
~1
(4-2la)
or (y' - k}
(l2
2
-
(x' - h) 2 - l
B2
-
(4-2lb)
g
d' 2
= - f' + 4a •
e'
+ 4c Cf?-O
If g - 0, the hyperbola degenerates into a pajr of intersecting straight lines. NoW that in the
case of a hypertx>la the
d.irect..i.on of opening is determined by
the signs of a ' and c' in ccnstrast to an ellipse , \oo'here the orientatioo is
cootrollro by the relative signs of a ' and c '.
'lhe eccentricity o f the hype.r-
tx>la is given by
c ...
lcl + r?- > 1 (l
•
4-5
fb.wARAt-ETR IC CIRCUlAR
AA.cs
•
Hany of the early techniques for curve definition were based on the use of
nonparrumtric cx:nic sections.
As long as relatively short curve segrrents
~
defined, coordinates CX>Ul.d be chosen which produoed rea.scnably satooth c:urves ,
even through the point distri b.Jtion was not tnifonn. The shape of ships and airplanes were adequately defined in this manner. One advantage of this ~ was the relative ease in ca.lculat.inq the intersectioo of boO curves, each defined by nonpt\I'atretric, algebrAic cquatioos.
COOsider the ooterminaticn of a circle through three paints given by [1 1) , [2 2] , and [3 2] respectively (see also Sec. 4-11).
•
'lb generate a
circle through these points , let a ' = c ' = 1 and write 6::). (4-15) as
where
\..e
2
i
+ dx + ey + f = 0 have drop{led the prines for coownienoe. x
+
Evaluating this equatioo
clt the three points yields
2+d+o+f""'O 8+2d+2e+f=O l3+3d+2e+f=O Solving these three equations for the three unknowns gives d • -5 , e • -1, and f • 4.
Thus, the equation for the ciro1l ar arc through the throe given
points is X
2
+ y 2 - Sx - y + 4
= 0;
1
:S_
x
~
3
The omter of the ciroJl ar arc is located at x • h
e/2a - 1/2.
Using Eq. (4-18) the radius is r
tiation of Eq. (4-18) yields 2xdy+2ydy-Sdy•O
=
~2a
= I!0/2
= 5/2
and y • k
"' 1.5811. Differen-
=
r ~ -
dx -
5- 2x
<2Y -
1)
I t follows that dy/dx • 0 when x .. 5/2 and dy/d.x - a. when y
= l/2.
However ,
oote that this infinite slope docs not occur within the arc ~ent oo~ [11] and [3 2].
When
a1e
creates curves for use in
be given to how the user will define the curve. t.O
create a circular arc ,,n.th a specified radius
For exlUTple, u . may be ~ ~ end
~sctry
poUlts, rather
than passing through three specified LX>ints as in the preVicus ex..:mple.
Exarrple 4-2: Let the
o.o
CIK"ULAR 1\fC WI'1li FIXED AAD.ItS
end points be [1
quire a radius of r = 5.
11 an1 [3 2) as before.
In addition
we re-
Fran Eq. (4-15) we obtain the three equations whidl
result ftom these three cocilitions: 2 + d+e+f=O l3+3d+2e+f=O
25 ;;
d
+ e 2 - 4af 2
2
4a 'Ihe solut1on to the
a}:x)ve
eqwUons
terms in the third equation.
l.S
nnre difficult due to the non.l.inear
An .tterative techni~ l11.lSt
oo
used.
By trial
and error, it can be sh:::Jwn that an approximate solution to the above set of
equations is
d - -8. 4
e = 5.7 f
=
o. 7
for a = c = l.O Thus, the center of the cucular arc which r:rects the .irrposed oonditJ.ons oocurs at x = 4.2 and y - -2.85. 'lhe arc no 1onqer passes through the int:.ermed.iate LX>int (2 2].
on the other ha.rd, it may be necessary to oontrol the slope of the c..L.rcular arc at the initial and/or final powts. 'lhls can be done by choosing an
alternate set of caditicns.
E:xanple 4-3:
CIR:lJI1\R Me l..zt'IH A FDCFD
1u~
NG£
:If an init1al slope of +1 is required at (1 1) and a final slope of zero
is necessary, then the final point oo the circular arc cannot be spew..fied. For this case, we require
.11'll~lendelltly
at: X • 1, Y • 1
at x; 1, at x
'lhese
~
~~
dx y. 1
= 3, ~;
"" 1 •
0
ocnditicns lead to 2+d+e+f=O 4 +d+e:::O 6 +d . 0
\olhich has the solutioo d = -6, e = 2, f = 2.
The equation for the circular
arc is thus x
2
+l
- 6x + 2y + 2 = 01
Not1oe that x "' 3, y ~ (-2 ~ 132>/2
= 1.828;
1
~x
.S.. 3
i.e . , the final point is [3 1.828) .
If the final point an a circular arc truSt be spectiied, then only cne end
slope can be specified in additioo to two end points.
•
For e.xanple , to produce
a fillet between two straight lines, one might specify the initial point, the
final {Xlint, and the final slope.
Exanple 4-4:
CI10.1l.M .Ale WI'ni GIVEN END SUlPE
Given the end points [1 1] and (3 2) with dy/dx • 0 at x ,.. 3 yields
2+d+e+f•O 13 + 3d + 2e + f "" 0
6+d:::r0 '1hen d • -6 , e • 1, and f • 3, with h • 3 , k ~ l/2, and r -- 5/2.
luticn prcx:h:es a slope of 4/3 at the initial {Xlint.
'lhis so-
'Ibis may be t.maeeeptable
and an iterative so1ution nay be requi.rea to find a final {Xlint ~ch allONS
an acceptable initial slope.
An alternate approadl
is to require the user to
specify the radius wch will pnxluce acceptable slopes at the two end points
and to proceed as in Exanple 4-2.
PIAN£ 0JRVES 101
'lbese
~les
illustrate the problems encountered when trying to provide ~
a catJlUter software (or haJ:d\..'are) interface between
usPr and the graphlccll
'Dle creatioo of the desired curve may be straightforward, or 1.t rray
output.
require iterative techniques, depending oo the known or de!Sired charact.eristicn of the curve.
Even 1Nhen the required equation for the
tions which produce \tlell-distributed (X>ints aloog the
Kno..m, calculamy be expensive in
C\.1IV{? ,l.S
C\l.t'W
Further, ~ nooparrureLric curves are created, care nust be taken to find the real roots rather than iJMqiJl.'1ry ooes, and i.ndcpandent terms of
variable step sizes between consecutive data points may require upcc1
<::ha.niJes based
the local curvature and ooordinate system.
c:Dlsider again the problem of pa.ssinq a circular are through three pomts. If the points are far renoved fJ:Cil\ the origin of the coordinate systEm, l
values of the coefficients can occur. To avoid this, translate the origin or the original coordinate system to the first point and define a new ooordi.nate syst:en x'y' as shoNn in Fig. 4-3.
The coordinates of the three points, relative
to this transfOl:'lred system, are then p
0
= (0 0]
((Xi- Xo) (yl- Yo))= (xl Yil [(x2- xO) (y2- Yo)) • lxi Yil Since x
f'
= 0.
0• Yo = 0 due to the location of the transfonred coordinate system,
'lhe other t-..-o equations, obtained by evaluating the general equatioo
at P and P , are 1 2
x•12 + y1•2 + d'~l + e ' y 1' • 0
Xi2 + Yi2
+ d'Xi + e'yi
=0
SOlving for d ' and e ' yields
d'
2
2
;
e' ;
2
Yi )
2
2
Y2 (Xj_ + Yi ) - Yi (Xi + <~i xiy~)
-
Xi(~
2
2
+ Yi ) - x]_
Yix2)
2
Yi )
It foll.ONS that the center of the circular arc, relative to the
nate system, is
x' _ c--
d'
:za' -
Yi<~ l
2
2
+ yl' ) - Yi(x' 2
2
2
+ Yi)
x'y ' -coordi-
'lbe center ooord.inates relative to the
origir.a.l coordinate system are silrply
xc = x'c + x..u
>I
Yc .. yc' +Yo \<.'hen f.Oints of intersection, offsets
for cutter tools , enclosed areas or wltr"eS , or other such geatet.ric infor-
mation is needed, the required calculations nay be si.nplifl.ed by expressmg the equatioos in explicit form. • •
The explicit
form of the circular arc is given as either y • f(x) or x • g(y) .
•
In t:erms
of the ooefficicnts of the general secondorder p:llyrouial equation, the first foon
•
is
y' = -
e' !
k•2 -2 4(x• 2 + d ' x ' )
and the second form is fl.gure 4-3
Translated coordinate
x ' .. _ d ' ·
system for a circl e
,4•2 -
4 (y '
2
+ e ' y')
2
In these fams calculations such as an area given by /y ' dx' or intersect:.i.ats
of the cucle with the ll.ne y '
a
ltD('
+ b can be rrade using rrat:henattcal tec::h-
ru.qucs. As an exanplo of the generation of a na1paranetric conic sectioo, an al-
•
gorit.~
c.
far generat.ing a m1iplete circle nonpararretrica1 ly is given in Appendix
'Ihe ba.sic oonoept behind the algorit:lJn is that near the x....axis equal
incrcm:mts in t.y are used.
'111us, between y
!JX are used, and between x ... t r
12 and x
=0
and y
= .t
r 12 equal increments
== 0 equal 1.nc:rercnts in t:.y are used .
This algOrithm allows only multiples of eight f.Oints .
Further , to avoid prcbl ems
with the square root of negative nurrbers , the absolute val ue of the square root ar~t
is used .
'lhc algorithn is carplicated and inefficicnt.
An exanple
of a full circle generated. with this algoritl'ln is shown in Fig . 4-4. erates reasonable circular
se<;Jrel1ts
very cnrle segrrcnts elsewhere.
It gen-
in the middle third of each quadrant , but
t-bre points can be nsed to i.nprove the result.
If a sttt:X)th gra}:hical output is the prirre consideration, the pararrotric repre-
sentation subsequently discussed in Sec. 4-7 is rec:anronded.
4-6
PARNURIC REPRESEtiTATIGI OF WHC SECTI
PIA'IE OJRVES 103
A pararretric rcpresentaticn of conic sections is of interest bx:ause it prodUces axis-independent curves.
curves for soue applications.
'111ese can have udvantages over nonparanetric
~vcr,
neither type of curve repr:esentat.ion is
a panacea, since both have advantaqes and disadvantages and both find uc.cful applications in CXX'!t'Uter graphics.
'lhe conic sections can be s.inply represented in param:!tric form. origin-centered circle can be paranetrically repxesentcd as X = ro:>SB
(4-22)
y : : : rs.ine
(4-23)
where r is the radius of the circle and e is the parcuroter.
In sec. t1-3 ~
remarked that although equal incrementS of 6 \Qlld produce gcxxl output
a\
a
display device , the calculation was inefficient because the trigonaretric functions rrust be repeatEdly calculatod.
Followin<] Ref. 4-6 we shall show below
that a nore efficient algorithn is possible.
4-7
PARAt-1ETRIC RePRESENTATIGJ OF A CJRCI E '!be circ:L! can be efficiently represented pararretrically by eliminating
the neoessity for calculating the triganretric functions at each step.
'Ihis
can be accouplished by using the double-angle fornul.as: cos(9 + d 9) = cos9cosd9 - sin9sind9 sin(e + d9) = cosesinde + cosd9sin9 Noting that the circle will be cxxtpletely swept out for a range of 0 fran 0 to 2n and assuning that a fixed numer of p:>ints will be used , then dO will be a
constant.
TOO cartesian coordinates of any point on an origin-centered circle
are then given by
Xn+l
yn+l
= rcos(O = rsin(9
+ d6)
(4-24)
+ d9)
(t1-25)
using the double-angle f orrrulas allows Eqs . (11-24 ) and (4- 25) to be rewritten as
xn+l = x 0 oosde - y0 sinde
(4-26)
yn+l = >
c4-2n
I f the circle is l ocated at h, k r elative to the origin of the coordinate system,
then we have k)slrlde
(4-28)
+ (xn - h)sindB + (yn - k)cosd9
(4-29)
>
(~
- h)cosde - (y0
-
A BIISIC subroutine incorporating this algorithn is given in Appendix
c.
since dO is constant, the values sinde and cosd8 rx:ec1 to be calculated only
once.
nus
ptoduces an efficient algorithm.
'lhe results of using this algorith:n
are stn.n in FUJ. 4-5.
4-8
Notice the equal spacing of the points.
PAEWEIRIC f?EPRESENTATIOO OF N-4 ElLIPSE Fran the results £or the circle
note thAt a fi.Xl:'d
,.~
n~
of points pro-
Vlded a tp:)d representation of the circle w1len
are ns«'' to calculate the display points, an unacceptable rosul t will be obta:ineC 'lhis ls particularly true for el 1ipses with high eccentricities (E: = 1) , as stn.1r
m Fig. 4-6.
Here we see that the ends are not lldequately represented.
'lhl.s is
because ncar the ends of the ellipse the curvature is too large to be representee by a f(!.N points.
.. •,
An alternate rrothod might be to use equal perirretcr lengths . cjent. n\.J!li:)cr of lengths this ~d give a better representation.
ell1p5e would be overspecified a.l.ong the sides, with a resultant waste of cxxtputer t:ine.
"nere
For a suffi-
Ha.--eve.r, the
the curvature is stall,
Further, as pointed out in Fe!. 4-5,
deten'ninatioo of the equal per.im::ter l engths involves calculation of an elliptl.c
•
Fiqure 4-4 Crude nonparametric representation of a circle
Figure 4-5 Equal-angle representation of a high-t>ccentric.1.ty ellipse
PIA'lE aJRilES 105
\
I
/
Figure 4-6
/
Equal- angle representat.ioo of a hiCJh-ecccntricity ellipse
periJreter length near the ends , "''here the curvature is l.arqe, and larger
pe.ri-
rreter incresrents along the side, where the curvature is smll.l . To ove.rcx.xre the problEmS of (Xlint distri.butioo, consider the pararretric
representation of an origin-centered ellipse of san.ilrajor axis a and semiminor axis b .
'Ihen
"''here e is again the paraneter. ellipse.
X = acos6
(4-30)
y = bsin6
(4-31)
Varying e beboleen 0 and 2n 5\oo'OOps out the entire
A specified fixed number of p:>ints can be used to represent the ellipse
by t:aJd.ng fixed incrcrrents in the parameter
e, say,
increments of 2n/ (n - 1) ,
where n is the nl.llt:Jer of p:>ints . £xamininq the derivatives (4-32)
dx - -asin6d6
dy
=bcosed6
(4-33)
shc7.ols that the desired perimeter i.J1crenents are aut:aratica l ly ootained.
e is near 0 or n, 1.. e ., near the ends, \ole have ldxl ;. 0 e is near Tf/2 or 311/2 , i.e . aloog the sides , jdxj .: ad6 near the ends
~
have mre p:>ints and alOJYi1 the sides
and and
fe~~~er
~
jdyl • bde~ and ~ jdyj o. 'nlus,
=
(Xlll\ts.
In fact
the ratio of the peri.Jreter l.ncrerre'lt size at the ends to that a lon
is approx:irrotely b/a.
Further, note that in the case of a circle where b = a
the optinun representaticn also results ,
i.e., equal peri.rreter or equal angle in)
crements •
••
Since equal l.OCrcrrents in the para-
meter
e,
i.e., a fixed nurber of points,
is to be used , an efficient algorithm can be developed by again using the double-
angle formulas (Pef. 4-6) .
I'
Here the ell i(&
;. is a5S\.IIred to be inclined at an angle i to the horizcntal , as sJ1o..m in Fig. 4-7.
Hence
x' .. xoosi - ysini
x'
y'
•
= xsini
+ yoosi
(4-34)
(4-35)
or
0 ••
._I
x ' "" aoos6cosi - bsin9sini
(4-36)
y ' = aoos8sini + bsin6oosi
(4- 37)
Introdu::ing the coordinAtes of the
'•
•
center of the ell ipse yields (4- 38)
y'n+l = yc + x'sini + yn' cosi n
(4-39)
n+l
Figure 4-7
Ellipse
+
y'sini n
X'
a
X
c
X 1 00Si -
n
For the recursial fOD'I'lllas, note that the ellipse algoritlTn in Appendix C uses
a SOI'I'ew'hat different technique than used previously in the circle alqorithm. In th:Ls case a teri(X>rary variable T1 i s used in the inner loop.
Snith I
(~f.
4-6) has sb:1h'Jl that this alqorithm can be made still nore effi-
cient such that ally four additi.a\s and four nultiplications are required within the inner loop.
Snith has also sb:7h'Jl that this technique yields the inscriOOd
polygc:n with max.irrun area.
'nlus, the additions and nul.tiplicaticns give an
efficient representation of the el] ipse , such as the CX81Tple shown in Fig. 4-8 .
4-9
PAR#ETRJc
REPResooAnON OF A PARABCX.A
Consider an origin-centered parabola opening to the right, .i. e. , with the axis of S}'n1'retry the positive x-axis, as shown in Fig. 4-9.
In rectangular
cxxn:dinates such a ~l.a is repl:'escnted in l"aaparaJretric fonn by
l = 4ax A parat~Ct.ric
(4-40)
representation of F.q. (4-40) is
= tan2e
(4-41)
y = • 2/itAniJ
(4-42)
x
' P~
CURVES 107
•
Figure 4-8 "''here 0 ~
e ~ r/2 .
Pa.rwetric representation of an ellipse
Alth::)l.¥Jh this provides an Nlequate representatias of a parab-
ola, Smith (Ref. 4-6) points out that it does not yield a figure with ma.xlrruml inscribed area and thus is not the rrost efficient rcpresentatioo.
An alternate
parametric representation which does yield ma.'Cirrun inscribed area is X=
a62
2ae 5"'-eeps OUt the entire parabola. Y
~'here 0 ,S_
e s_
(4-43) (4-44)
'Ihe parabola, ~Vcr 1 unlike the ellipse, i.s not a closed curve. Thus, the arrount of the parabola to be displayed must be limited by ch:losing a nruci.murn value for a • emax. '!his can be done in a variety of ways; e.g., the range of the x-cootdinato c:ou.ld be l..uru.ted. If this is done then CD
e
max
=/~ a
(4-45)
If the range of tha y-c:oordinate is limit.ol, then Yrrex emax; 2a Once
emax
(4-46)
is established, an algorithm to calculate N rcpresentatl.ve points for
the parabola in the fl..rst. quadrant can be de~loped.
Once the parabola is obtair
ed m t:re fl.I'St quadrant, the rest of the parabola Wich appears in the fourth
)
•
•
f
ocu~
Figure 4-9 ~ant
Para):x)la
can be obtained by reflection al::out the x-ax1s (cf Sec. 2- 10) .
An
appropriate alcprithm to obtain the required points in the first qu!ldrant is given 1n Appendix c . In the algorithm a fixerl ntri:ler of points is specified
•
e is
and a constant i.ncreme.nt in
Parabolas with displaced centers or at
used.
other orientations ca.'"l be obtained using rotation and translation. For en+l
= an+
d e , Eqs. (4-43) and {4-47) beoare
•
~ 1 ~ aen2 + 2a9nd~
I
+ a(de)
2
yn+l ~ 2a9n + 2adS o,..hl.ch
~ray
be written as
~1
..
=
~
+ y0 da + a(d9)
2
(4-47) (4-48)
Yn+l .. Yn + 2ad9
F.lgure 4-10 s~'S relationships. For
s~ifl.c
a."\ ~le
of a parmx>la generated using these recursion
purposes other pa.r21r00t.ric representat.Wns can be rore useful .
1his generally depe.."lds on the info:rmation specified by t.~ user.
a parabolic arc is to
oo drawn
bet~
For ex.a:rple , i f
t:"..o points, and oontrol of the end slopes
is necessary, then the folla..t1.ng form .lS suggested: X (t)
= (0'"Y.
- -2R'):
+
P X) t
2
+ 2 (RX - PX ) t + PX
-
0 -< t < 1.0
Here the paraxreter is t, and the t:'NO end points of the paral:x:>la are P
arx:l Q • (~ ~~. end ta.'1getlts.
~
= (Px PYJ
11\c \-)
is the point of lntersectlOn of the t..o call this rethod the vertex definitjon of a paral::ola , since
The pomt R •
use of t.hrcc vertices P, Q, R defi:~es t.~ paral::ola as s.l-o.n ln Fig . 4- 11. A rrore general method defining a C"..!n't! by use of vertices of an CJFetl £01)'901"1 was developed by Bezier and is d.lsc.lSsed iT t:he nex-c chapter.
o=
Pll\NE ClJRVES 109
Figure 4-10 Parabola generated by the PARAOOIA alc;priilin
4-10
PAIWIEl RIC RePRESENTATI~
a= A HYPERBOLA
As was the case for the parabola, we asst.~te that
we want to generate a
hypertx>la which is origin-centered, with the axis of symretcy being the positive x-axis.
'Ihe rectangular nonpararretric CXX>rdinate representation of this hyperbola
is
2
2
b
which inplies that the vertex is at a and the asynptotic slopes are th/a.
A
parametric representatia'l is given by
where 0
x "' tasece
(4-49)
y • fbtane
(4- 50)
s e ~ n/2
hypet'OOla.
yields the desired
smith (Ref. 4-6) points out
that with this pa.rametric representation the inscribed p:>lygon is not of rrax.inun
area .
Ha..oever, it is of nearly rnax:inun area, and t:r.e dcuble-angle fonrulas can be used to yield an efficient algorithn. To see this recall that 1 sec<&+ de) = cos (e + dO) -
1
- cos660Sde - si.neSillde and
tan(e + de) ~ tane + tande 1 - tanetanae
,,
't R ( R~ Ryl
1\ I I
'
I I
\
\ '
I
1
'I I
\
\
\
'\ \ \
I
',
I
\
II
'
~
I
'1hls we may write
'b+-1 • !asec
(e + d e)
= ..- bOOseab/_cose - btaneswe
l I
;. - ~cl
a
(
Figure 4-ll Vertex definition of a parametric parabola
and ~·n+l c ~ btan (5 + d 6)
Yn+l 'l'he algonthm HYPERBl
D
+
=
btan" + btarrl6 t
l - tan6tand"
b(y + btand>-) r--~"'---::::-:---....,--
(4-52)
•
- b - yntand6
Appendix C uses tl'w:!se rclatiooships.
1.n
A"l alternate para11etric representation of a hyperbola which yields the polyqt:Xl
.,.."1 th
max.i.nun inscribed area is
X
•
= acosh6
(4-53)
•
y = bsinh& 'Ibe
(4-54)
h~~rtx>lic functions are defined as coshe ..
(o 6 - c - fl)/2.
l\s
e varies
(e
6
+ e -e)/2 and sinh& =
!:tun 0 to co the hyperbola is traced out.
'll1e double-
angle fonnulas for CDsh and sinh are
•
'
CDsh(& + de> = oosheooshde + sinhesinhde sinh{6 + d6) = sinh6coshd6 + cosh6sirurle 'nlese allow wrib.ng tqs. (4-53) and (4-54) as ~
1
= a(cosh6coshd6
+ sinh'-sinhd6)
YOTl = b(sinh- coshd& + c:osh~sinhd& )
or •
~l = x0 coshd• + ~ y0 sinhdt1
(4-55)
~ x0 sinhdfl + y0 coshd6
(4-56)
yn+l =
was the case for the paralx>la, the trax.iJtun value of e = enax nust be set in order to limit the extent of the hyperoola. I f we coosidcr the branch of the hyperoola in the first and fourth quadrants and wish to plot the portion of the hyperbola for a .S. x ~ a + c, then 1\s
e
=cosh-l f-+c) max a
SWlarly other l.ilnits can be det.e.rlnlnErl. C uses F.qs. (4-55) and (4-56) .
(4-57)
'Ihe Algorit:J"'TT ~2 given in Appendix
Since nost cx:r:puter systems do rot have hyper-
OOlJ.c nn:tions, the cxpone..'ltial relatl.ons haw been used, i.e. , -dj:j d6 coshd6- e + e
-
2
dtl sll\hdB = e
-dr• - e
2
and ths .:..nverse hypc:rbolic cosine has been .,..Titten as
PLANE aJRVES 111
cosh-lx = log(l + x) - log(x - 1) 2
An exanple of a hyperbola using this technique is shC7...n in Fig. 4-12.
4-11 Af>Rocen.RE Foo niE UsE OF Coote SEcrtCR; In Sec . 4-5 the nonpararretric representation of a circle through t:hrc!e
points was cx:nsidered.
An essentially brute force
awroach
was adopted.
uere
we cx:nsider the sane problem; i.e., detexmine the location of ~ center and the
radi.us of a circle "hlch will pass through three specifl.c point.s.
However, here
"e divorce the problem of drawing the appropriate circle fran deteiJI\i.ning
~t
it is. in the x,y,-cx:xm::hl'lilt~ sysl811, 1 2 3 'l1le general , na1pararretric equatl(Xl for the circle
COO.sider three arbitracy points P , P , P
as shown in Fig . 4-13. t:hrou;}h these points is
(4-58) + (y - k) 2 ... r 2 in the xy-plane, direct substitution into
(x - h)
2
Since P , P , P IJ'ay be an~e 1 2 3 Eq . (4-58) rMY yield cxtrernely large rn.nbers . resulting equations difficult. nate
'Ihis ''"Ould rna.'
\'E! therefore translate the origin of the coordi-
syste:n to the first point P • 1
As we shall see, this will also make the
,,
Figure 4-12
Paranetric representation of a hyperl::ola
•
'
..
-
•
\ ~
•
.
•
l I'
I
l 0
0
-
\.
'
Fl.gure 4-13 Circular arc throogh three powts (gecr-etry)
solution rore oorwenient,
f'iqure 4-14 Circular arc through three points (pararretxic)
Lett:in:J
x ' = x- m and y' • y - n \>tlcre m ar¥1 n are t.~ translatjon factors (cf Sec . 2- 27) , we have m: ,_, n,.. y . 1 Equation (4- 58) may then be written in the x ' y ' -cxxmiinate systan as
(x' -h ' )
2
+ (y ' -k ' ) 2 • r2
h ' • h - m and
(4-59)
k' = k - n
Dl.rect sub6tJ.tutl.on of the o:ordinates of P , P , P in the x'y ' -ax>rdinate 1 2 3 systen yields
h ' + k' 2 - r 2
a
0 for
p
1
'lbcse oguations represent til.ree equat.ioos in the three
an:1 thus may h.:! solved .
~,
~
h 1 , k 1 , an:l r
the systan of equations 1.s non.l.inear ard
quires an iterauve solutioo, l.lhl.ch is undesirable .
Notice,
~,
~
that becaus
the origin is at the first point, the fi.rst e:}Uat.ion can always be S\.btracted fran the sec::ond and thl.rd equaUoos such that the nonl..inear t:etms are el.u:u..n.'lted.
"Ire resulting equ3t.ions are
PlANE CURVES 113
= x! 2
2x 'h ' + 2y!k' 1 1
~ - 2,3
+ y! 2
~
~
(4-60)
These are ttNo linear cquatJ.oos in the tw:> l.ll'lknc1wns h' and k ' \ohlich
directly. 2 2 h' + k' .
be solved 'llle radius r can Ulen be det&mi.ned fran the first equation r 2 CdR
=
its rOOius ar,- I"'CMi kno..n. To draw the arc of the circle, we first translate diroctly to the center of the circl e, i.e., to h ', k ' or h, k , as si10n'n in Fig . 4-13. Hith '1hc locaLion of tl1c OUltcr of the circle and
e1
= tan
-1 yl* ,_
•
e2 =
and
-1 y3*
tan
x3
,
(4-61)
kno.m, a standard origin-oent:ered algorithn, either para~rutric or na1paranutn.c,
can be used to generate the points which describe the circular arc fran P to P 1
3
through P • 2
CIK:UI.AR 1\0C PR:CE:I:XJRE
Exarrplcs 4- 5:
consider the three points given by P (3, 2) , P ( 4, 4) , and P (0, 6) • 1
2
lating to P yields Pi (0,0), P2(1,2), and Pj(-3,4) . 1
3
Trans-
F.quations (4-60) are then
2h' + 4k ' = 5
-6h
I
+ 8k
I
!!0.
25
Solutioo yields k' = 2, h' = -3/2, and r"" 5/2 .
Translation to the center
yields P1* (3/2 , - 2) , P2*(5/2,0), and P *(-3/2,2) . 3 01
Thus, Eqs. (4-61) give
tan-1 (-2/(3/2)) ~ tan-1 (-4/3) - -53.13°
82 = tan- 1 (2/(-3/2)) = tan- 1 (-4/3)
= 126. 87°
'Ihis provides the angular input for the arc drawing S\lbroutine.
A parzuretric form is crosen to inprove the point distribution along the
arc.
Figure 4-14 shows the resulting pax-arootric rcprescntiltion.
used to generate this
~
'llle algorithn
is called 3PCI..RAR:: and is given in Appendix C.
The
pararootric circular arc subroutine was discussed in Sec. 4-7.
4-12 CJ RCULAR AAc I tfrERPa.AT I 00 The snoothness or fcumcss of a curve is generally a quahtative judgrrent
based oo t..t\at is pleasing to the huron eye.
Nevertheless, it
1.S
possible to
select a quantitative treaSUre of sztmthr1ess and qenerate curves according to this critenon.
One such criteri.al used successfully by 1-\ehl.ll!! (Jef . 4-8) is
to minimize the inteqral aloog the arc length o£ the square of the cw:vature.
Curvature can be expressed as y"
K""
(4-62)
.,here the
pr~
indicate dif£erentiatioo with respect to x .
Thus , the p.rcblem
is to rnini.'1lizc I, 'Where I is given by 1
= Jilds
(4-63)
The prcblem of rninimi.zinq the value of I by use of variational calculus
solved by
~eh.l~
(Ref. 4-8).
was
kl iJtp;rtant result of this solution , expressed
as a theorem, is stated as follows: •
When a picce..rise curve is drawn
~'hich
mi.nirn:i,zcs the integral
al.crq the arc length of the square of the cu.r:vat.ure, then the
curve segrrents lie in a direction aloog ...nich the cw:vature varies linearly bet\o.een the end points of each se
several interrrediate data points . mic;ht loo.lc like that K
s~n
drawn~
0.0 specified end points and
The curvature profile
in Fig. 4-lSa.
~.
I '
• \OHJ;;ItOil
K
r• I
"·
these points
as pointed out by t-hhl.um,
•
a Ltner
be~
k.,_,
F1.guro 4-15 CUrvature profiles
PlANE aJRVES 115
the acblill direction of the lmear curvature variatioo is chfficult to dct.ermi.nc.
It requires the solutioo to elliptic integrals whose wtegratioo cx:nstants are dependent in a catplicated '"'W'f oo the specified boundary conditions.
t-~un
suggested that an approximatioo to the piecewise linear curvature be rro.dc by
replacin<] the linear variattoo with a series of disocxltinoous steps, each with constant curvature, as indicated in
F~g .
piecewise cx:nstant over the curve length.
4-lSb.
'll1e curvature profile is then
Since a ccnstant curvature descrires
a circular arc, the piecewise curve u; represented by a series of ci.rcular arcs with various centers and radii of curvature.
1\ large
nt.111ber of steps, i.e.,
circular arcs, will produce a closer fit to the ideal. curvature. The required nunber of circular arcs depends upon the separation distanoe between the dat..a };X)ints, the dlange in curvature over the curve segrrent, and the required ~ of the graphical output. Detai 1s of this technique for generating snooth curves by circular arc interpolat-ion are given in the cited reference. The output. of the itcraoon process gives the value of racHus and center };X)int for each circular arc, as well as the specified x-values where the curvature c:hangcs. '1hen sd:>routines for generating parametric c.ircular arcs can be used to produce the actual pieoewise curve. REFEROCES 4-1 07\TASAAB, "Fornela General Descripcicn, " Reg. No. 917-E, 51\1\B 1\KTI.EOOV.G, 58188 , Li.nkopi.rq, ~ . 1965. 4-2
"cat:puter Graphics Arrangerrent
Progrcm~, CCGAP Systems
Manual," NAVSIU.PS
090o-OJ7- 9030, e:AEll1>C 233083./MFTS, Naval Ship ~ing
center,
Washingtor
D.C., Nov. 1972. 4-3
Gottfried, B. s ., PrC?gramning with Basic, Schallll' s Outl.ine series, McGrawHill Book COrpany, Ne" York, 1975.
4-4
~sford , P. A.., "Mat:herootical Metrods in c.atputer Graphics- 1\ Survey,"
Gesellschaft f\1r Inforrratike , vol. 5, Synposiun on catputer 4-5
Gr~cs,
Berlin , OCt . 1971. 'l'hc:rnas, G. B., Sr., calculus and Analytical Gearet:J:Y,, J'ldll.son~ley Publ.i..shing Carpany, carbridqe, t-1ass., 1954.
4-6
Snith, L. B., "Drawir¥} ellipses, hyperbolas or parabolas w1.th a fixed n\J't'bcr of };X)Ults and nwc.iJtun .in.scribed area, " COJipUt. J . , vol. 14, p . 81, 1!
4-7 COhen , D. , "Linear Difference CUrves," in lldvanoed Cu!uputer Graprics , Plen\1'1\ Press, New York, 1971. 4-8
l-fchl\1'1\, E., "CUrve and SUrface Fitting based on Variational Criteriae for
Snoothness," Central lnstl.b.Jte for Industrial ResearCh (CIIR), Oslo, Nor!o.'ll}' Dec. 1969.
I
THIS BOOK BELONGS TO
IKE HAYDEN
CHAPTER 5 •
SPACE CURVES •
•
•
I~
' 5-1 Itfi"RROLCT ION ll\::lst objects encountered in the real world are l~ver,
t:.h...~ia\al
in nature.
nost drawul
on a plane surface such as a drawing board. represent.at.loo of
three~ialal
or sectional d.nwings.
'11lis frequently results in the
objects by
I'J:!allS
of t:wo-diJrensiona.l
Vl.L'".TS
'l1le use of a digital conputcr to ge.-,erate curves ndds
a new d.ireru.ioo to drawing techniques . ional spaoo C'l..lrVeS can be defined, gC!I'lC'!raLL..od , stored, rrenipulat:.ee , and plXldt.red as output. Sp;!CC cur\11"5 can be displayed oo a tw.'t>-d~ooal plane by use of ~tnc and perspective projectioriS, as d1.scnssed in Chapter 3, or three-
t::Y! prodooed directly by using a nu:rerically oont.rollf'O cutting tool. present chapter extends the previous discussions for
CUIVO
The
dcscriptioo to three-
d imens.J.ona 1 space curves.
5-2 R£PI ESENTATICU
OF
SPACE (aVES
As was the cage for plane curves, sp.~ ~ rray
mnparaztw:·trically or pararretrically.
oo
reprosc.>nted either
'11l.ree-ditramsional spaoo c:urves expressed
•
'•
SPJ\CE
~
117
in nonparaJtt:t.ric form a.re g.1.ven explicitly by a set of cquotions of the form (5~)
x=x
y • f(x)
= 9 (X)
2
Alternately a space curve rray be expressed in a nonp.trMPt:.nc, ir.phcit fonn. In this case the space curve is represen~ed rnther'atically by the Lnter-sectioo
of the t"-0 surfaces g:Lven by f (x,y ,z)
0
g(x,y,z)
=0
{5-2)
As an exan-ple, consider the t-..-o second-degree surfaces given by
y - z2 = 0 g(x,y,z) - zx- y2 ~ 0 f(x,y,z)
{So-3)
e
Fbr z ~ 0, x ond y can be expressed
terms of z to obtain the explicit Conn
l.l'l
of the curve defined by the intersecting surfaces 2
L,_ z
x =
y
z3
(5-4)
= z2
Notice that tho intc.rscct.i.oo o£ two secc:nd-degree surfaces l oads to a th.ud-degree ~ curve.
'l1u..s techn.iquc of solving for z fran the ir.plicit
surface functioos is valid provided that
2!. ll. i4X oy
det
1
0
~~ X y
at a poi.'1t (x,y,z) which satl.Sfios tho rrent applies when sol vi.ng for x In
c.~
= f (y) ,
%
general, a pararrctnc space curve
x
~
surface equations.
= g (y) or y - f (x) , z .lS
A similar argug (x) .
expressed as
x(t)
(5-5)
y ;: y(l) z = z(t)
where the pararrotcr t varies over a given range
_ t ~ t 2 • Roconsidaring Eq. (5-1) we see that x itself can be considered a paraneter, X,. t, and the sarrc curve is then expressed in param:!tnc fotm by
x=t
~
(5-6)
y = f ( t)
z • g (t) Further, i f we reconsider the oonparat"Ctn.c l.rrplicit representation given in
FJ::i.
(5-4) we see that
curve arc
we can let z
t:.,;
t, and the paranetric equations for that
)( - t3
=t2
y z ~useful ~ic
(5-7)
= t
space curves have 1cno1,..n analytic solutims .
exa.-rple, a para!:et.ric space curve that is shaped like the seam ~1 ~ s
a1
For
a tennis or
expX'eSSCd by the following equations: x = >. (acos {co +
~) - bcos3 {0 +
i> )
y =
i->
if) I
+
IJ (asin (6
+ bsin3 {6 +
z • csin(2E)
( 5-8)
•
•
" • 1 + dsl.n(2fl ) - 1
.c .
.. l., I
' I
v
=1 -
dsin(2e )
=1
21ft , where 0 ~ t ~ 1.0. space curve lies on a sphere o f radius a +b. and the parameter 1
= 0,
and d
then the curve lies
a1
a
s~
+ d(!) c - d(!)
c
If d • 0 and c 2 • 4ab, then the I f a = 1.0 , b = 0 . 5, c .. /2,
of radius 1 . 5.
J\rother exa.-rple of a parametric space curve is the ci.rcular helix.
pararrotric equatims are given by
x = acost y ~ asint
z for a and b
~
0 and _..,. < t <
cular cyli.n.dor of radius lal .
•
'Ihe
ao .
(5-9)
= bt
'Ihis curve lies oo the surface of a right c irThe effect of the equatioo z
= bt
is to nove the points of the curve uruforml.y in the z-di.rect.lal. After each 211 interval in the pararreter t , the variables x and y return to their initial values, but z increa.s es or decreases by 2n b l , depending upon the sign of b . z i.-; callro the pitch of the helix.
'lhis change in
l'lben an ahalytlcal description for a curve is not known , an interpolation scherre
may be used to fit a curve through a given set of
data p:>ints .
This
invol ves specifying boundary conditioos for the space curve in order to determine the coefficients for a given polynomial curve form and establ ishing a s•oot.hness criterion. r-'.any considerations enter into the decision of ho.l
curve.
to represent a space
'l1le form of c:xrrputer input desired, the type of nanipulat.ials required,
the ava.1lablc interface ~ user and cx:rtpUte.r, and the display device used
for
c;ral=hl-cal output can
all introdoce oonsideratims and limitations.
the input is a series of po.mts which lie
a1
t:llf!" desired curve , spl ine se
arc often used to forr a szrooth curve through the p:>ints .
in the next sectial.
~
'nlese are discussed
SPACE aJRVES 119
5-3
CusiC SPLINES The mat:hcrratical spline derives fran its {ilys~cal counterpart - ~ l ofts-
spline is a l.a1g narro.~ strip of "'COd or plastic used by a loft..sman to fair in curves beO."een specified data powts. ~ s plines are
rran's spline.
A physical
shaped by lead weights called "d\.Cks."
By varying the ni.ITber and pos ium o £
the lead "-eights the spline can be rMde to pass t.hrou.;;h the sp.."
points si.Ch that the result.ing curve appears srooth or "fair."
If the physical spline is considered to be a thin elastic bcilrn, then Eulers equati01 (cf Ref. 5-l) y.lelds
EI
M(X) = R(x}
where t-l(x) is the banding na1ent, E is Young ' s m::xlulus, i.nert.La, and R(x) is the radius of cw:vaturc.
r is the J:l'al'Cl'lt of
For small deflect.Lcns the radius
o f curvature R(x) may be replaced by 1/ y", where the pr.irre denotes differentia-
tion with respect to x.
Thus, we have y" (x) =
Assuning that
Ule
loti';>
d\.Cks act as sinpl e SliftQrts, them M(x) is a linear function
beb¥een the supports .
Letting M(x) = A +
ax
and integrating the above equation
twice shows that the physical spline is descriOOd by cubic polyncmials between supports. In general the mathenatical spline
is a piecewise polyro1ual of degree K
with continuity of derivatives of order K - 1 at the camon )Ol.nts between ~ts .
Thus, the cubic spline has second-order continuity at the joints.
Piecewise splines of lo..r degree polynanials are usually rrore useful for forming The use of lo..r~e
a curve through a series of points.
the ca•putatJ ona 1 requirements and reduces mm-erj cal instabl.l i ties that arise with higher order curves.
'D1esc instabilities can cause an undesirable wiggle
when several pomts nust. be jowed in a
CXJflTIDO
curve.
fb..lever, si.ncc l<:M-
polync:rnials cannot span an arb~tracy series of points, adj aoent polynomial segJl'el)ts
are nooded .
Based on these consideraticns and the unalogy with the Fhz'si-
cal spline, a camon technique is to use a series of cubic splines with each
sec:rent spanru.ng on1y
boo points .
Further, the cubic spline is advantageous
since it is the l~t degree space curve which allo..rs a ~int of infl ection and
has lhc dbility to tl
111e equation for a single pararretric cubic spline seqrrent,
paraneter t ,
~
1.n
terms of a
qiven by P(t) =
4
. 1
L a.e; . 1
l.=
l.
(5-10)
\."here P(t)
= {x(t)
y(t) z(t) J.
any p:lint oo the spll.ne.
P(t) can be considered the p:>Sitioo vector of
z ( t) \oohlch of the posit.icn vector . The coeffi-
It has tllree carp::n:mts
rruy be o::nsidercd ~ cart..esian O::Xll:di.nates
cients B
l.
x ( t)
, y ( t) , ard
are det:.errni.ned by specifying four boundary oonclitl.ons for the spline
se
In expar.ded fonn Eq.
s 3t 2
P(t) = Bl + B2t +
+ s 4t
3
(5-11)
a given pair of points throu<;h \ohlch a curve ~'lt passes be the vectors P , P.. (see Fig. 5-la) . COrresponding tangent vectors at these given points 1 are indicated by the derivatives with respect to the parazreter t . WithU1 the cubic segncnt the pararreter t varies betw-een t:'"'=> end-point values ,_ and
let:
Pi, Pi,
~.
To s.irlplify t.h.e calculations ,
we can assign
= 0.
~
The required boundary corrlitions for each cubic segrre.nt o::nsist of the b.'O
For the single seqrrent
end points and the tangent vector at each end point.
beb.'OOn P and P , these conditions are given by 1 2 P(O) = Pl
(5-U)
P(t ) = P2 2
•
~~ dt t=O
• P'
1
~lt=t =pi 2 Four relationships follow fran ~ . (5- ll) and (5-U): P(O)
dP
-
dt t=O
= Bl ~
~
4
L:
(5-D)
P 1 (i-l)ti-zB.
l.t=O
i=2
=B
= P' 2 1
(5-14)
(5-15)
dP ~ = L dt ,t=t i=l
· 2B. l)tl.-
(l. -
l.
2
SOlVl.ng tor
s3
and
s4
(5-16)
t=t
2
yields
B3 and
I
=
3(P - P ) 2 1 2
~
2P'
1
P'
2
- t2 - ~
(5-17 )
SPI\CE OJRVES
U l
(5-18)
along with 8l
= Pl
P' . 'nl(>Se val'lf>S o f 8l, 8 , 8 , and 8 fix the 2 3 4 1 curve for tho cubic SC."
2
Furthe r, notice that the value of
the erXI-point (X)Siticn and tangent vectors. the pararreter t
=~
at the end o f tre sec.;nent oocurs
.Ln
the results.
SlllCe
each of the end-point vectors and end tangent vectors has three m•p:Jncnts , the pararretric equation for a cubic space ~
and the pararreter value
C\ttVO
depends
at the end of the
up::lO
~'Ct.Or
12
component s
S~e
Substituting F.qs. (5-1 3) , (5-14) , (5-17) , and (5-18) into l:q. (5-ll) yields the equation for a cubic spline segment:
P (t )
=
)(P2 - Pl) Pl + Pi t + ( 2
2Pi -
P2
t.._
t2
-
2
t)t +
l
2 (Pl-P2 )
2
-l
Equation (5- 19) is for one cubic spline SC
~ts
+
) t2
PK(t) and PK +
1
P' 1 I
(5-19)
~2
J
It can be genera.h :r.ed
(t) , 1 ~ K _ n - 2 ,
wrerc
n is the nll!l'ber of data pomts throlgh \oohlch the curve J'IU.1St pass (sec Fig. 5-lb).
'the gencralizerl equations are of the form p {t )
C
K
+ pIt. •
(
K
K
)(PK+l - PK) [
t
2
-
2
2PK PK+l] 2 [ 2 (Pl<- "K+l) PK I'K+lJ 3 - t + + '""! • 2 t
(5-20)
j
~
t2
t
t
2
t..
2
-~
and PK•llt) - r K•l ~ I'K+l" •
.HI'Y•2- Pl<•ll
r
2
.. 1
2
-
''~•1 t
l
•
PK•2J 2 t t. -
[ 2 (1"1<•1- Pt<+21
)
l
-
Here we assl..l!te that the paraneter variat1.on is 0 ~ t ~ t
and 0
2
+
Pi:•l 2 tl
•
r ;.•2] 3 (5-21) 2
t
ll
for the h.rst segrrent
for the second secpent, etc . 3 For ex;:urq:>lc, if only three position vectors (data points) are specified , ~ t ~ t
the Jov..tn conditions for the entire curve betl.'eCO P
arc the position 3 and the tangent vectors at the ends o f the curve, i.e. , 1
and P
vectors P , P , and P 2 3 1 Pi and P)· To ensure second-order continuity for a cubic spline, we inposc the conditioo for
we have 4
P" (t ) ..
E
(5-22)
( i - 1) (i - 2)
i =l At the
end o f the first cubic spline segrrent, P" = 68 t._ 4"G
~re t = ~·
+ 2B3
and at the beginning o f the secxn:1 spLine se<.J•ent, where t-0 ,
122 Mr\11iD'Wl'lCAL EIWmi'S FOR cx:MPl-TER GRAPHICS
P" • 2B
3
~t.lng
6~ [
these ~ results and using ~ · (5-17) and (5-18) yields 3 (1>3 - P2) l (Pl - P2 Pi] [l(P2 - Pl) 2Pi Pj ]
pl
t3 2
•
2 +~ • 2 t2
f.lll.tiplying by
t:fi +
~t
2(t3 +
~ -~
2
3
~ -
-
2Pi
pi] [ t; • 2
t2 3
-
~ (P2
- Pl)]
(5-23)
t3 - tl
and collecti.'lg terms gives
~)Pi
+
¥3 = ~3t3 [ t~(P3
- P2) +
(5-24)
...,h lch can be solved for Pi, the unk:rloNn tangent ·vector at the internal joint. Again
nouce that the end values of the paran-eter t, i .e . ,
the resul ti.N3 equat; oo . For n data points the results given above can be
cub.tc spline
•
~ts
~
and t , occur in 3
g~eralized
to yield n - 1
wit.l) position, slope, and curvature continuity at all the
internAl JOints . The narenclature for multipl e spline se
The secx:n:i-derivat.ive continuity condition Eq. (5- 24) ~
alizcs to t.,-. 2r; • ' t ,. l ' '1 -1 "K•l ' ly.lr;. l •
rl'.:"l<•:1'~...:, 1P)'•2- PK•l '~•2(1'1'•1. P"'ll .s, K1 11 -.,
(5-25)
(5-26)
,
..
•
•
•
I~
•
•
•
•
l''
•
•
n
rt~l(Pn . pn·ll .. t~ F'n-1 - l'n-2] n-1 n ~
t) t
'!he expansion of Fq . (5-26) gives r. - 2 equatioos in n unknown tangent
voctors .
\-lhen the t\oJO and tangent vector s Pi and P~ are specified, then the
systan of equations is dete.rminant .
Thus, to generate a curve tt:e position
vectors P , 1 ~ i ~ n, al on; with Pi and P~, are specified. Then Elq. (5- 26) 1 is 11SE!d to calculate lhe i nt..enrediate tangent vectors Pi, Pj, . . . , P~-1' Tlus calculation is rrost conveniently aco::c1plisr.ed by natrix inversion (cf
sec.
5- 5) .
'l1le tangent vectors are then used to calculate the B. coeffl..1
cients g.tven by the 9f!f1eral ized focn of Eqs . (5-13), (5-14), (5-17), and (5-18) ,
for each curve segrrent, e .g . , (5-27)
B = P'
2
K
(5-28)
SPJICE CURI1ES 12 3
, .~
-
PK-+l
- 'i<.. l-
~+1
+
----::-P~
2r;
r,
-
(5- 29)
P'
K 2
(5-30)
lK+l
)----- ).
Finally, each etbic
is gen-
SC<}"'\!nt
erated by use of Eq. (5-10 ) , with
3
0 ~ t ~ tmax.
Before t:.he c::urvr can be
generated, the JTa'
,'\
,._
(':
tmax for each seqn-ent., i.e. 1 t. , L31 2 • • • 1 t , must be chosen. 'I'his do1ce 0
,,n ,..n
'n
will affect the curve snoothncss.
Continuity of second danvaUves at the internal joints docs rot in i tscl f
.,.,. '
produce a srooth spline in the
P. P.' ,, I I
S
ro.iniJ:run curvature along the curve .
/ y
of
To
obtain a mininun, and hence nwd11tJm SlOCXJthness, the ex>efficients 8
3
and 8
4
oust be minimi?.ed for each seqtYmt by
Figure 5-1 CUbic spline .
cl'x:losing tre correct wloos of the par-
aneter range within each SC
c:OOosing
This
Swpler nethcds
tmax can be used to generate cuxves SII'OOth coough
tor rrost practi-
cal purp:>ses. One approach is to set the rrax.im.Jm parazret.er valms equal to the chord
lengths
be~
sucx::cssive data FCints.
srrooth curves for graphical display.
This has proven to 91ve acceptably
A se<:x>nd approach is to rormalize the
variatioo by choosinq trrax '"' 1. 0 for each cubic segnent. the previous cquatioos, each choice of t;,
rnax
As can be soon fran
will prodooc
values and , hence, different curves through the given data fX>ints.
As the
M2Jnitw:iP of the tanqcnt vectors is changed, the slope of the cubic
5eC}tents
bet;:...JPcn data FC11lts is changed.
On the other hand, the di l'Cc~il'm of the
vectors controls thc shape of the cli>ic seqments at their cmd
5-4
t.aJ19ent
p...i ,. ~s .
~IZED PARA."£TERS In spite of the approxunate sr.oothness of the resulting curve, normali7.ed
paraJreter
defining a
ranges for al 1 cub1c se
\'hm this approach is
Eq. (S-26) ca.n ba .roarran
u'?ed, 0
~ t
s. 1
for all spans and
'
(5-31)
P3•
l 4 1 0 0 • •
•
•
•
•
•
•
•
•
•
•
• • •
•
l
•
•
•
;(P - P > - P' r. n- 2 n-2J
•
'i.'le four ccxllf1cients for each paranetric cubic equation, givcn by F.qs . (5-27) and (5-30) , can be expressed in z:utrix form
r
B4 ., •
a3
2 -2
-3
1
1 r-PK
3 -2 -1
0 0 1
01 PR
BlJ
1
OJ
0
0
1 ~ K ~ ·n - l
(5-32)
PK+l
B2
l
l
QS
_PK+lJ
\',"ith the nonr.llizcd fo.arulation the tridia<;Utal matrix in Eq . (5-31) need only
,
'
be i.•wtj...--ted
~
to dete.rmi.ne the ta.ngent \:ectors at the inte.mal joints.
a set of n positioo wctors , tlu.s represents a cxnsiderabl e savmgs of m 1p.1ter ~.
tire.
l.f the positi.cn vectors are oot un.i!or:rnly distributed, experience
uxlicates that t:ht! resulting curves l.lreter length ~"~
fi.'ld use
..
For
J..S
m:lde equal
to 1.ts
a_Y'@
not as st:mth as w-hen each segrrent. par-
local chord.
The nomallzed forna t 1an does
creating cubic bo\.ll'XB:ry curves for three-<:l.lmmsiCXlal surface patches
(see Chapter 6) • Although
par~l.c
C\i:>ics have rr.any advantages for representing and
nanipulating curves, there are also disa&rdl1tages \.hlch limit their usefulness . Parametnc cub1.cs never reduce exactly to a c:u:cular arc , and so true circles can only be approximated.
Also,
curves whlch are asynptotic.
t.~y
do not closely approximlte rrat:herretical
Nevertheless , they are widel y used for many
appliCliUons, expccially in the shipbuilding and aircraft industries.
5-5
13ruJlARY CcfiDJTJrJ~S h~
Eq. (5-26)
l.S
used to determine the tangent vectors at the internal
JOl.nts o! a curve definod by n points, a nonsquarc rratri.x with n - 2 rows ar.d
n coll.lmS results.
'It.c tangent vec:---....or colum matrix contains n r:o..rs and the
coefficient cohr.n rratrix in~
tx:Jnta.ulS
n - 2 roo-'S.
Thus Eq. (5-26) tndY be written
forrn
rm IP' 1 -
1a1
(5-33)
SPN:E 0JRVEs 125
- is an n M
2 x n matrix
P ' 1.S an n x 1 nattix
B is an n - 2 x 1 rr.rt.r.ix
'Ihe only nonzero terms
at
each ro.~ o f the nonsquar:e [l-l) -1rat.rix arc
M(.J, .J-1), M(J, J) , and M(J, J+l) for 2 ~ J ~ n - 1.
Hence the expanded form
of F.q. (5-33) is M(2,1) M(2,2)
P'(I<,l)
r:ic3,2> 11(3, Jl
•
P ' (1<,2}
•
•
•
•
•
•
•
•
R(l\,2)
(5- 34)
U(t\, 3)
•
•
•
M(N- 1, N- 1) ~(N- 1, N) P ' (l< , N) B(l<,N- 1) A squ
The required square mat.nx M can be created by spcci fy-
ing boundary c:xnlitions lit each end of the total piecewise cubic curve as
sho.m be}.o..o.
In this r..ase Eq. (5-33) may be \oi'I"itten as (M] (P. I
~~:: t-1 J..S
tOO required square rratrix.
(Pi , . . . ,
~
1 are given by
-
{5-15)
(B}
'Iben the internal tangent vectors
P~ >
(5-36)
[P' ) = [l1)-l[B]
[M]-l is the inverse of [M) .
cnce the
P '-values are known, B.-values for l..
each segnent arc calculated using Eqs. (5-27) and (5-30).
Vany chc)l.m5 exist for specJ..fyinq the end boundary CCI"lditions for a plcce-
wise cubic curve .
Several differo.'"lt choices may be desirable i f ally a few
data points are kno.rm or 1..£ P"~Ysical cn'lStra.ints require accurate ooot.rol of the curve shape at the ends .
'l1le rrost direct solutioo is ootained by specify-
ing the two end tangent vectors P' and P ' of the total picccoiliso spline. '11tis 1 n boundary condition is called tile clarrp?d end corrlibon .in Ref . 5-2, and the encastered spline in Pef . S-3.
1he CC111'lete matrix equatioo for
tro cubic
spline curve is then given i:Yy P ' (K,l)
1 M(2, 1)
~:(2 , 2)
M(3, 2) M(3 , 3)
•
-
(5-37) BU<,l)
p• (l(,2)
BIK,1l
P'(J<,))
B(l<, 31
•
•
•
•
•
•
•
•
•
•
•
•
M(N - 1,
r: -
11
11(N- 1,
1
Nl
P' (}(,N - ll
B(K,r; - 1)
P ' (K,N)
B(l<,N)
,
Eina."l'S FOR
126
t-11\'lli:o~TIChL
'Ihis
c:~ql~'ltion iJrplics that
~
P ' (J<,l)
rcsult..lng square (M)-mat.n.x
.LS
GRAPHICS
= B(K,l)
as is required.
nus
also t.ndi.aqonal.
NOtice that the
system of equauons can
ro.xiily be solV('d for the unl
Since
this (t-1)-nutrix is t.rid.iaqa\al, the solution can be obtainod very conciscly by rne.'Vl..c;
nus procedure is \lSed in the a l gorithm \o1u.ch
of recurs1on fornul..as.
appoars in llppCndbc c. Other CJ¥i cxn:htialS will increase the nlrl'ber of I'O'Izero terms lJ\ the first and ln'lt ro.15 of the M-matrix. Fbr ex
1bis oondition
.LS
called c1t:her relaxed or natural.
to zero for the first span (J< .. 1), wi th t "" 0 , and using Eq. (5-17) lwd!. t.o •
SettlJ"'C} Eq. (5-22) cqt.oal
P' +
1
... I
!
2
P'
2
- p )
(P
= 1_2
2
(S-30)
1
~
If th('> s
I
u
0
,
then with the help of f:):Js. {5-29) and (5-30) one obtains
2P '
n-1
'
+ 4P ' = ~(P - P
n
t
n
0
n-1
)
(5-39)
It follows that the 11002m'0 terms in the first row of the square (11)-mat.rix are
= 4.0.
P(N,N - 1) .,. 2.0 and M(N,N) s1.de of
The nonzero tcnns in the last row are
1.0 ard t-1(1,2) - 0.5.
M(l,l)
~.
In the [B)-natrix , B(J<,1) equals the right
(5-38) and B(K,N) equals the right side of Eq. (5-39).
Notice that
this relaxed (natural) end condition also produces a tridiagonal (MI-mltrix.
otmr C!l1d concli.tions of practical use are the cyclic end oondition and the antlcyclJ.c end candl.tion . A cyclic spl ine can be used to prodUce a closed curvo or a portion of a C\J.t"Ve which repeats at intervals . It is based on the 'l\oA':>
follow.t..nq end oonchtion specificatialS P ' (t )
(5-40)
P " (0) • P" (t ) n n 1
(5-4 1 )
Pi (O)
n n
1..c., the s lope .:tnd curvature at the beginning and end of tJle curve are eqtkll. Recallmq Eqs. (5-16) ard (5-28) to (5-JO) the condition expressed by
Eq.
(~40)
yields
• 2 P'- P ' 1 r -1
3 (P - P
n-1
n
)
2P' P' ] n-1 _ -E. t
_
t2
t0
n
ln n
+3
2(P - P ) P' P' n-1 n + n-1 + ..12 t cs- 42) t t2 t2 n n n
SinUlarly the condition expressed by Eq. (5-41) y1clds
J r)(l' -
lfl '1 - p I ) _ 11''1 _ r•2 • ~ t :1 l7 t2 1
r
7
n
t
n
I'n-1 I __ 2P'n-1 _ tn
I''] n
tn
• {
fl ll'n-1 - -"P ) • -Pi~ P' tl t n
n
1
I'']
~ ~
t"
-n
(5-43)
SPACE CURVES
127
These ~ equatioos can be cxxtbined to yield a single equation "'hlch when com-
bined with the l+iratri.x yields a (n - 1) x (n - 1) square nntrix. the order of the square rratrix has been x:ed•rerl by one.
Note that
'lhi.s is because the
sl.q)e and curvature oondi tioos inposed at t:\..'0 of the n JX>ints are no longer in-
'lfuls, there are only n - 1 independent slopes to be dctem.i.nod.
dependent.
We cxx1bine Eqs . (5-42) and (s-43) by nultiplying Eq. (s-43) by t
t..racti.ng it fran El:). (5-42).
n
and sub-
This results in ·~]
2
(5-44)
li n 1\
Recalling that Pi • P~ and rearranging the terms yields 2(1
t
~
t2 n )Pl
t
-1
P2f-2 + p~-1 •
Using this result allOw'S writing t
2(1
+ ~)
t-i(2 , 1)
~-
t
3(P2 - Pl)-1- 3(Pn-l - p )!_
n tn
t2
(5-45)
(5- 35) as
t
-t2n
1
P ' (1<, 1)
t-1(2,2} M(2, 3)
•
P ' (K, 2)
8(1<,2)
t-1(3, 2) f.1(3, 3}
•
P' (K,3)
B(K,))
0
•
=
B(l<, l)
•
•
•
•
•
•
•
•
•
•
t-t(N - l,N - 1)
P ' (K, N - 1)
(5-46)
B(K, N - 1)
t-bnnal Gaussian e.l.imi.nation or otro.r matrix inversion rreth:xis can be used to invert the nontridiagonal matrix in El';{. (5-46). f'.btice that ~ P ' and B rratrioes are
r'OII
(n - 1) x 1 rratrioes as required
for matrix nultiplication in Eq. (5-46). The anticylic spline is similar to the cyclic spline, except that P • (0) = - P ' (t ) n n 1
and
P " (0) = -P., (t ) n n 1
Following that sarrc prooedure used to derive El:J. (5-45) yields
t 2(1 + t;>Pi + Pit;-- Pn-l = 3(P2 t
t
P1 )~-+
1
l(Pn-l - Pn)t;;
(5-47)
Equation (5-47) s!nols that the only effect of irtt=osing the anticyclic end oonditioos is to change the one in the M(l,t~ - 1) position of the f.t-matrix for the cyclic spline boundary oonditioos (cf Eq. 5-46) to a minus one, as 'n"Cll
as the sign on the second term of B(K, N-1) .
'lhis type of spll.ne is useful for
produc.ing parallel end spans with end tangent vectors which are equal in magni-
US K\nm-MICAL ELEMEN1'S FOR CX:NFUI'ER GAAPHICS
tiXlc but opposite in directicn (cf Fig. 5-2b) . The cubic spl1ne end conditions considered above are S\.IIITI\arlzed m
Table 5-l. 'ilili1<> .._ l
Dld Ccnd.LtlCI!\9 for Cl:blC Sphll'.cz l'rllbtrl.X
U.J Q:J'IIilti oo
1.
~~t.
cpcc1ty P' !0) , P' It J
~ (1,11
~~ :urod
.Mil,ll - l
Ir.a t..ural n:
•
I
o
1
J. ()'r;L ~u.1n: ell /at. e-o C!l / dL t•t n
-
-
.ro 2p
d
2 ~ 2 / dt ~ ~ ~ / dt
K•1
'·
I
····rl-l
•;o.l
MU-cyclic
B(K,NI
•
Cl."'f¥1 (onc4Sbl.Z'Odl 0
2.
B(K,ll
Nonzero ElAn.lnts lt\ First and IA.It Polo'S.
• 1
M(N,NI
=l
B(l<,l) I 0(J<,l) BIK1 1>) ... U(K,NI
l (l/t 1 n nA(K,N) - 61Pn- P 1
2 M!N,N - 1 ) H(1,2 ) ~ 0.5 K
B(K,l)
M( l,l )
BIK, l ) - 3CP2 - P l (tnltzl 1
'4(1
,2 ,
~(1 ,1{-1)
211 • t,!t2 > - t n/ t 2
0
_
1
0
)
(1/~)
-liPn-1 - r n 1 11/tn l
= l
je-tn
(3/2)!P - P
~
!'I (I< • " ) :
IZidl!f i.nad
t..-..c'l-1
M(l,1 )
~e
di>1 dt ~ "' -dl'tdt ~
M(l,2 )
n
r-1
~( l,!i-1 )
t:...~,
.. 2 c1 • t / t > 2 - tr!tz
s cK,l l • 3
n-
; -1
1 - p ) (1/ t. )
n
n
and
2 d 2v/ dt
~-
t-o k•l
2
ii ::2
d
dt
BIK,~) :
undctinod
t~
n
K•r.-l
Cubic splines will have CXJntinuous first and secc.rld derivatives w.1.th any of these end ccrditions.
Hc7Never, if the nuroer of data. points is J.arge , 1:00
~
r;;utaticn t.in'O required to invert the (t-lj-matrix can be excessive when cyclic spline or anticyclic splines are generated .
Additl.onal spline end conditions
are discussed in .Refs. S-3 and S-4 . Ccrrparisons of the effects of the varicus erd conditions is shown Fig. 5-2.
1'hcse results
AppE:ndix C.
~
J.n
obtained with the SPLDlE subroutine given in
In Fig . 5-2a, the "claaped" curve has a specified slope of dy/dx •
(dy/dt)/(dx/dt) equal to -1 at the beginning and a slope of +l at the end. The other C\.I.I"Ve has relaxed boundaJ:y conditions so that d 2P/dt2 = 0 at each errl poi:"'t.
'Ibere is a s1.gnificant variation in curve shape, especially near the
bcqi..nrung. dates powts .
&:>t.h curves consist. of three ctbic spans ~ c::arp.1ter
for thl.S ptl.rt icular output .
throlXJh
the four indicated
progtarn qenerates 10 wterm.."
'
SPliCE
aJRVES
129
lr-----------------------------~-----------
• I
l't• I
t
•
I
·~--~--~--~--~--------~--~--~--~--~
• Figure 5-2a
l
2
J
•
I
COOpari.son o f cl.alrped and relaxed end oonditioos .
l l r---------------~--~--------------~----~--------~
u
•
. ...
•
'
_,
..,
e L---~----~----L----L--~~--~~--~----._--~----~ \1 -'l 2
-u
Figure 5-2b
' catplrison of cyclic and antiC}'Clic end corrlitioos .
F~gure
s-20 sb:lws bole closed curves, a curve with cyclic end coocli.tions
through the U data (X)ints, the first and last (X)utt being identical at x = 0 , y • 0, and a curve, also closed, with an anticyclic end condition at x • y • 0 .
Notice that the di.rectioo of the initial end tangent vector is vertically up.tard, while the cli.rectial of the final end tangent vector is vertically da.mward. f'iqure 5-2c OCI!p9.res a relaxed end boundary cx:n:litioo to a cyclic end or::nli-
tioo. curve.
This illustrates ~ a cyclic en:3 conditioo can be used with an open
Uoweve.r, the di.rection of the initial end tangent vector on the curve
with cyclic end conditions is the same as the direction o f the final end t::angent
vector. •
E:xarrple 5-1:
u•
'
alBIC SPLINES
Assune that the three (X)Siticn vectors P (0 OJ, P (1 2) , and P (3 2) are 2 1 3 J
We base trrax on the chord lenqths of each span.
To begin we first calculate the chord distances t
•
~
•
;(~
t3
= ~x3
- ,_,2 + (y2 - yl)2 -
~)2
+ (y3 - y2)2
Usi.nq fl:!. (5-26) and the relaxed
ero
= 1(1)2 = ~3
2
and t : 3
+ {2)2
= IS
- 1)2 + (0)2 • 2
conditicn F.qs . (5-38) and (5- 39) I the
2 is
rMtnx equaticn required to find the int.ernal. de.rivativc P
1 0.5
0
Pi
2 8.472 2. 236 P'
2
0 2
•
3
vS (P2 - P1) 2 3/5(5(P - P ) + 4(P - P l) 3 2 2 1
2
4
Solvi.nq for the derivatives by invert.inq the 3 x 3 square mtrix and rultiplyi.nq yields
l •
P2 Pj
1 . 1574 -0 . 0787 0. 0440
0.67l (P2 - Pl) - 0 . 3148 0.1574 - 0.08798 0.671(5(P3 - P2 ) + 4(P 2 - P1 )J 0.1574 -0.0787 0.2940 3CP - P2 ) 3
Thus,
Pi= 0. 5654(P2 - P1 l - 0. 132(P3 - P2l = 0. 5654[1 2) - 0.132(2 0] • (0.3013 1.1308)
SPICE CURVES
131
Pi • 0.2112CP2 - P ) + 0.264l(P - P ) 3 1 2 0.2112[1 2] + 0.2641(2 0) - (0.7394 0.4224] Pj; -0.1056CP - P ) + 0.618(P - P ) 1 2 3 2 • -0 .1056(1 2] + 0.618(2 OJ = [1 .1305 -0. 2111) Recalling EkJs. (5-11) and (5-27) to (5-30) allo,.,s calculatioo of the ci.Dic
spline segmmts.
fbr the first segrumt
P(t) = B + B t + a t 2 + a t 3 1 2 3 4 and B
1
• P
1
82 ;
•
pi -
(0 0)
,
(0.3013 1.1308)
3
8 3 "" S(P2 - Pl) -
2Pi
IS" -
Pi
7s
= o. 6(1 21 - o.a944IO.J013 1.13081 - o.4472[0.7394 o. 42241
= ro
01
• = 0. 17889(-1 -21 + 0.2(0.3013 1 .1308) + 0. 2(0 . 7394 0.4224)
• [0.0293 -0.0472) Oxx>sing
t/~
.. 1/3, 2/3 yields
t = o. 745 , 1.4907' and
P(0. 745)"" (0 O) + 0.745[0.3013 1.1305) + (0.745) 2 [0 0] + (0.745) 3 [0.0293 -o.0472] = {0. 2366 0.8227) arrl similarly
P(1.4907) = (0.5462 1 . 5289] For the second span, Ekjs. (5-27) and (5-30) yield
11. • p2 • (1 2) 82
= P'2 =
[0.7394 0.4224}
3(P2 - P2 )
2P ' P' 2 83. - ~a (0.19535 -0. 31685) 4 2 P' P' 2(P2 - P3 ) 84+ ~ + .2.; [-0.03253 0. 05283] 8 4 4 ~
choosing
t/t3 • 1/3, 2/3 yields
t •
2/3 , 4/3, and Eq. (5-11) gives
P(2/3) = [1.573 2. 1579] P(4/3) = [2.256 2. 125)
are plotted in Fig. 5-3.
It should be noted that it l.S not necessary to mvcrt the rratrix to obtain the internal derivatives . E>quatioos 'n1c re.sulLs
(5-24) 1 (5-38) I and (5-39) can be easily solved by sl:bstitutioo to yield the
I
~~ resw~. An algont:hrl for gene.rat.lJ"-9 cubic spl.L"te fl.ts for n kno..n data p:>i.nt.s
c;1ven L" AWC."ldix
l.S
c. ~· fl.caticns to the" a!:x:M!
It is possl.ble to nai
r ec"v-.ique "'t-.!.ch
may offer ~t for certa.1.n appll.cation.s. For e:thness for m
was rcp:>rted l:tj.• Detlt'a"\ (Pef. 5- 6) .
P-is technique requires an interactive
nt:reral Sl"...arch and the use of Gaussian quadrature mtcgration . Details are descnbcd m the c.1t.ed reference. 'lhe t:ni.q-E feat ure of this n:thod l.S d-.at the end c::ondltions are autaMtically selected based ~..{a' a rJ..nimiz.auon c:n.tencn. It hAs been ased
1.n
the autmobile industry.
A t.echn.:..que ''sed to Sl:ccth out undesirable oscillations oc:cur ~th cubic spl::.n~ mves
l.S
~o.hlch scr.et.b~s
the use of sp:ines ~ ter.sion.
tions occur since the cubic S?line
.l.S
OscJlla-
influer-..oed locally by each data p:>1r.t
I W' .... :'CJ
•
- J lie:
- I
-2 ~--~~--~----~----~----~----~----~----~----~--~
•
2 Figure 5-2c Cc:r.pariscn of open c.n::ves .
'
re~ and ~'Cllc
•
crd conditions for
11
SPJlCE CURVES
133
alonq the curve, and the lhin1 dd 1vativc is only pieoewisc constant.
Dis-
CXlCitinuities in third derivatives can
thus induce \JI"looanted inflection points at certain locations
a.l.on
the curve.
Consl.der a {tlysical spline supported
If the
by docks at certain points.
thin, flexible beam has small oscilla-
tions 1 one way to retove them ...ould be • to apply tension to the ends of the beam ., to SttOOth Out the spline.
-
-------
I']
•
A Jrathemati-
cal spline under tension is an appro.xi-
mation to this procedure. 'I'heory and applications for this technique are dis-
,•
1
cussed in Refs. 5-7 and 5-8 .
5-6
PllNDit-.G
PAFWn.IC
L
'n'le technique for paralx>lic blend-
Figure 5-3
ing presented here "'as first suggested
by A. W. Ovel::hauser (Ref . 5-9). utive points sinultaneously.
J.
J
I
J
Results of cubic spline fit for Exanple 5-l.
'Ibe interpolatioo scheue CXlCISiders four mnsec-
A srrooth curve between the 0.0 interior points is
generated by blend.i.ng two overlawing parabolic segnents .
The first parabolic
defined by the first three points, and the last three points of the
segnett is
set of four define the second
~lie
segnent.
consi der four oonsecuti-w points in space specified by the {X)Sitial vectors P , P , P , and P • '1\o,Q overlapping parabolas P(r) and Q(s) between these 5 3 4 6 points are sho.m in Fiq . 5-4. Each parabola qces through three points and each
is defined relative to its own local coordinate system. th.rolxjh P
3
1
P
ur-ooordina~
4
1
and P
5
The parabola P (r)
is CjO\I'Im'led by the follc:Ming equatioo, relative to the
system u = P(r)
= or(d
- r)
(5-48)
INhere as shown in F.ig. 5-4 r is rreasured alonq the crord length P P and u is 3 5 neasured perpendlcul.ar to r in the plane defined by P31 P 1 and P • The chord 4 5 length between P and P is d. A parabola can be oorpletely specified by two 3 5 end p;:>ints P , P51 and a th.i.rd p;:>.int P on the curve. 'Ibe value of the CXJnStant 4 3 a is chosen such that the parabola P (r) passes t.h.rough P . 4 In a similar rranner I the parabola O(s) is defined so as
the p;:>ints p 41 p 51 and p 6 •
The equatia'\ is
to pass thrO\.Xjh
v = Q(s) = 6s (e - s)
(5-49)
Here s 1s rroasured along the crord length P P , v is perpendicular to s in the
4 6
plaoo defined by P , P , and P , and 8 is d'osen such that the parabola passes 5 6 4 throU'Jh P5 • 'Ihe crord length between P 4 and P 6 is e . 'Ihe par.urctcr t is now chosen as the distance measured alonq the chord and P • A curve CCt>, which is a blend of the blo overlapping 5 4 po.ralxllas, is c::onstructed bet:l.'een P ~ P by use of an interpolalioo schete. length bct:l.-.;en P
4
5
1he blending curve C(t) is defined as
C(t) = [1- (f-)JP(r) + lf-JO(s) 0
where t
0
(5-50)
0
is the distance between P4 and P •
'nle coeff!cients of P(r) and Q(s)
5
act as blending functions , varying linearly respect.ively.
be~
1.0 and 0 , and 0 and 1.0
'11-.e {XlSition vectors P , P , P , and P
above are specified in teJ:rns of 5 6 tlle cartesian xyz-<:OOrdinate systern, whereas the blendin:J parabolas P(r) and 3
4
Q(s) are specified in terms of a local cx:ordinate system.
To derive the para-
rretric, parabolic equat.i.oos in tems of the xyz-<:OOrdinate system, the gearetry
'
sho..n in Fig . 5-5 is helpful.
I
bct'w'Oen P
3
In Fig. 5-Sa , P J is perpendicula.r to the cOOrd
4 Thus, the ur-plane can be defined by the ve;:tor cht prodoct
and P • 5
(P
4 -
(5-51)
J) • (P
5 - P 3) • 0
If J is lo:::ated in the ur-plane at r =
m,
then in the xyzwo:x>rdinate system
J and E>;J.
= P3
+ x(P
5
- P ) 3
(5-52)
(5-51) may be written
(P - (P + x CP - P ))} • (P -P ) ""0(5-53) 4 3 3 5 3 5
Solving for x in the xyz-<:OOrdina te sys-
)
tem yields
"I
(P4 - P3) • (P5 - P3)
I
\)'
,
_, ... ,
. .t-./
x=
'; --;.-;- \ ~
'6
P3>
(5-54
2
(P4 - P3) • (P5 - P3)
=
PI
d2
•
With this informatioo tho vector equation for a EX>int P on the parabola P (r) , rela-
tive to t:00 xyz-<:OOrdinatc system, is given by
,I Figure 5-4
Parabolic blendin:J.
SPACE; CURVES 135
P(r) = P
r + d(P - P ) + ar(d - r) (P - J ) 5 3 3 4
(5- 55)
or after using Eq. (5-52) for J, r P(r ) = P3 + d(P5 - P3) + ar(d- r)((P - P ) - x (PS- P ))
4 3 It remains to deteonine a arrl the pararretric equation f or r
~rdinate system
P(xd)
is
o
(5- 56)
3
( t) •
Since in the
P , it fo.l.la..os that the vector equation for P - J 4 4
or Q -
-
1 d .x(l - x)
(5-57)
2
'l1le required relatiooship for r = r(t) can be obtained fran the C}ealctry
shown in Fi g . 5- 5.
It follows that
r = J
tcose
(5-58)
Ps - PJ cosO= (P - P )·( t d ) 5 4 0
(5-59)
For the paratola Q(s) , similar equations can be derived using Fig . 5-Sb.
For
the parabola Q(s) , the relaticnship s = s(t) is p6 - p4 s = toos6 = t [ (P - P4 ) • ( t e )) 5
(5-60)
0
and
(5-61) Once the points are specified, the procedure is to calculate .x using Eq . (5- 54) and then
a using Eq . (5-57).
by Eqs. (5-58) arrl (5-59) . using Eq. (5-56).
For a given value of t, r is given
Finally , points on the curve, P(r) , are calcu.l.ated
This procedure is then repeated for the Q(s) parabola.
To continue generating a curve through aaH tional points, a blend.ing curve
C. (t. ) is foz:rned between each adj acent pair of points. ~
l.
'lbis creates a contin-
uous curve which is also oontinu:rus in first derivative at the internal data
points.
'lhese first derivatives can be easily detetmined i f Eq. (5- 50) is
rewritten as C(t) = P(t) + (~)(Q(t)- P(t))
(5-62)
0
+ (- l ) (Q - P) to At point P4 on the ble.'lding curve, t = 0 and P • Q. ck:: dP
p4
-
p4
(5-63)
'lbus
•
"nlat is, the slope of the blerv:ling curve
•
)
equals the slope of the parabola Plr) at Likewise, at P on the blending 5 cur.
P • 4
=
=
0
(~t)
p_
r
(~)
•
at: p
Ot
5
I
p
5
'Ibe equatioo for the blerrling curve
a r r(t l I
C ( t) is
'
cubic ..men expressed m terms of
the haslC car-...esian coord.lnate system. For t:his reason it can prodUce a p:>int ~
•
of .Ulflectioo within an interval.
\l
!4
~s
::.;1 -
c..._
J
ever, the blending cubic which defines
~ Ohl
the curve between p:>ints P and P does
- - f>6
-
4
terpolatioo
r)
~c
5
not pass through P 3 and P6 . 'Ihis behavior makes parabolic blending a different m-
I
b f>
H:Jw-
~
than one INhich passes
a C"Jbic through four p:>i."lts on a curve.
,
1
FJ ;u.re 5- 5
Ge:m:!tric relatu:nships r ft.) and s ( t ) .
wiqqles are not l.nt.roduoed
~g
According to Ove!i'.auser (P.ef. 5-9) , this characteristl.c guarantees that spurious
the curve defini.tioo.
Parabolic blc.:.'ldi..nq can be used o.>'lly for _nternal segrrents of a curve .
te a swgle parabola, defined through the fliSt an:i last throe data p:>l.nts respectively Cc f sec. 4- 9). 'Ihe di.st.anoe between these p:>.iJlts can te smaller than that for other sec;rents in order to c:are.f ully
The t:1o.r.> end segr;ents rust each
specify the shape of the curve at its end points .
can te
uSE(}
in reqion.s of high
Also, closely spaced p:>ints
cur<~ature.
'1he technique of parabolic blendi.nq offers a different approach to curve
defi.nitioo a.'Xl generation.
it
was necessary to
vectors before the
In the cubic spline tech.l·U.que discussed previously,
define the catplete set of points and two e.'ld tangent
~
...as ~ted.
polnts are needed to start a curve .
In parabolic interp:>l.ation, ooly three
These defl.ne the initial parabolic se;rrem:
t.etween points P and P passing through P • A fourth point can then be acXied 3 2 1 and the cubic blerdi.nc; seg:tent beb.-een p:>ints P and P detexmined as shoom
2
3
above.
Su::oessive points may be added, one by one, and these determine the cx:nunuous C"Jbic blendJ.nq cu..~ segmmts t.~t make up tr.e interior part of tlv>
cun~e.
If the shape is rot mnect, the last
{:011\t
or p:>mts can be
delct.OO and a re11 s.h""'f.C defined by usL""~g alte.rnate points.
When an artist, stylist, or designer sketches, he or she uses short, over.lappir.g strokes to produce the cootou:r desired .
This is not unlike tile tech-
SP~
Ct..'RVES 137
Qloe the sketch is defi.Iw::ld in
nique that can be used with paralx>lic blencling.
a OCill'Uter, then the vectors definin<] the points can be qwc..-. ly displayed in a variety of ways . In sare applicaticns it rray be desirable to sketch a shape by using parabolic blend.ing and then use the resultinq junclia'l (X)ints as data
for other techniques.
Exanple 5- 2:
PAAAOOl.IC
Bu:NDn~
Assure that the positioo vectol:'S P [0,0), P4 (1,1), P (3,2) , rud P6 (4,3) 3 5 are J
We wish to dcte.rmine a curve between t:.he t:o..'O po1.nts P
4
and P
5
The procedure is
parabolic b l e..,din}.
Fbr each span catp:)Se(i of three points
1.
a. calculate x
e
b.
calculate (lor
c.
calculate r or s for given t
calculate the blended curve
2.
fbr the first span E):j. (5-54 ) yields
3+ 2 - - - 0 . 3846 13
c~+j>·<3i+2jl
(9
+
4)
1
(13)(0.3846) (1- 0. 3846) ~l ing
Ek}.
(5-58) t
0
=
first calculate
\ole
fc2>
= 0 • 325
2
+ (1)
2
- IS
cose= =
(2i + j) . (3i + 2j) .. 6 + 2 =
IS l.i3
/5 10
8
=
o. 9923
IS li3
and
r = xd + taos~ ~ 0.3846113 +
8t
2
1.3867 + 0. 9923t
/SID fbr
t/t
0
=
1/3, t = 0. 745 , which yields r - 2.U6.
Equatioo (5-56) then
yields
P(r) : PJ + ~(PS- p 3 ) + nr(d- r) [(P4 - P3) - x(P5 - P3)1 P(2 .126) = PJ + 0. 5896(P - P3) + 1.0223 ( (P4 - P3) - 0. 3846(PS- P3 ) 5 • root + o.ss96(3 21 + l . o223lll 11 - o.J846£J 211 = [O 0) + (1 . 7688 1 . 1792) + 1.0223[-0.1538 0. 2308)
usmg
= 1/3
P(2.126) • (1.612 1 . 154]t/t 0
For t/t
0
- 2/3, t • 1.4907, and r = 2.8659, &J. (5-56) then yields P(2.8659) = (2.279 1.7487llt/t ~2/3 0
For the secocd span (P5 -
p 4) • (P6 - p 4)
2
X=
8
:::;
IT = 0 . 6154
(P6 - P4)
B• t
0
0.325
= ~ •
e = II3 (PS - P4 ). (P6 - P4 ) t e • 0 . 9923
oos6 •
0
s
I
I
= tcose = 0 . 9923t
o. 745,
For t/t .. 1/3, t • 0
and s •
o. 7392 , &J.
(5-61) yields
s =P4 + e(P 6 -
P4 ) + Bs(e- s)((P5 - P )- x(P - P4)J 4 6 • P + 0 . 205(P - P ) + 0 . 6887((PS - P4 ) - 0.6154(P6 - P4 )] 6 4 4
Q(s)
-= and for t/t
0
r1. 121
1. 2511 1t/t =l/J 0
= 1.479,
• 2/3 , t,... 1.4907 and s
Eq. (5-61) yields
Q(l.479) = (2.3882 1.584J i t/t •2/3 0
Equation (5-62) rray now be used to b l em the bNo parabolas to yield the desired curve between P4 and P 5 : C(t) .. (1-
f
+
) P(r)
0
0
for t/t0
•
<-£=->oCs)
1/3,
ceo. 745)
= ~P(2 .126) +
j.oco.7392)
= [1. 648 1. 361] and for t/t
0
= 2/3, C(l.4907) = j-P(2 . 8659)
+ ~0(1. 479)
= [2 . 352 1.6398) 'Ihe results are
stn.m plotted in Fig. 5-6.
An algorithm which will inpl.erent
dt:-scnbed above is given in Appendix
c.
tre parabolic blendi.ng technique
SP~
5-7
Be:ziER
ct1RVES 139
Cl.Rves
The previously discussed r.ethods
for
ional-curve generation
have been oonstrained to pass through all t.l'lo specified data points; i.e. , they are curve fitting ted\ni qucs.
3
1n
many cases excellent results arc achieved } with these rreti¥xis.
certain
~ks
'1be:re arc, however,
which render these
2
nethods ineffective for interactive "ab
initio" curve design.
'!his is due to
the fact that control of the
curves~
by nurerical specification of both direc-
tion and rragni.tude of tange.nl derivatives
does not provide the intuitive "feel" required for curve design; i.e. , there
..J
l
is not always an obvious relatioo be-
Results of parabolic blend.i.ng for Exanple S-2.
tween the nlDI'tlers and the curve shape. In addition, the cubic curve fitting
technique specifies a curve o f unique order, which d::les not vary fran spline
to spline.
In order to increase flexibility, nore points lTLISt be input, creat-
ing norc spl ines which are all sti 11 of cli>ic order. An alternate method o f curve description has been described by Bezier
(Refs. 5-10, 5-ll , and 5-U) which
all~'S
the relation between input and output.
the user a nuch cp:-eater feel for
'lhis enables him to use the pxogcaan
as an artist , stylist or desi9J1&, varying curve shape and order by the use of easily controlled input pnrarreters until the output rmtchcs the desired shap?. A Bczior curve is associated with the "vertices" of a polygon which uniquely define the curve shape .
cnly the first and last vertices of the polygon
actually lie an the curve; however , the other vertioes define the derivatives, order, and shape of the curve.
as sho.m in Fig . 5-7.
'lbus, the curve is defined by an open polygon,
Since the curve shape will tend to
follo.~
the polyga1
shape, changing the vertices of this pol}'9011 gives the user a III..ICh greater intuitive feel.i.ng for input/output relationships.
All that is necessary to
increase the order of any curve segnent is to specify another interior vertex. This qreaUy increases flexibility and owr:co1es many of the difficulties of the cubic fli>line fitting and parabolic blending techniques .
control is easy with Bezier curves.
FUrtherrrore, local
span will will retain
Any change in the -..ertic:es of a
only effect the curve within that span .
~ne
rest of the
o.lr'W
P.
---
•r.0
,. p
Figure 5- 7 N:xtenclature for Bezier
C'lll:'VeS .
unaffected . 'The mathematical
of the Bezier curve is a i:olynomial blending func-
basl.S
tion 1
'The Bezier p;>ly-
related to the Eernstein polyranial.. 'Ihu.s, the Bezier curve is said to have a Bernstein basis. 'lhe basis fuoct.ion is given by
nanial
l.S
J
. (t)
n,~
where
,n
1i)
I
= (~)ti(l ~
~
- t)n-i
(5- 64)
n! i! (n-1}!
(5- 65)
with n being the degroo of the p;>lyocrnial and i the particular vertex in the ordexed set (fra11 0 to n) •
In general an nth order p;>lync:rnial is Sf'ECille::l by
The curve points are then give."l by n P(t) = P.J . (t) 0 < t < 1 i=O ~ n,1 where P . contains the vector COtfXJne.,ts of the various vertices .
n + 1 vertices.
L
~
(5-66 )
At the start-
1.n9 point of a curve segrrent,
3n,O(O) "" l\t the end point
nl (1) (1 - O)n-O n!
.
1
(5-67)
of a curve se<;rent, J
n
(l) = n! (1) (0) n,n n! (1)
n-n (5-68)
Equatioos (5-67) and (5-68) along with Eq. (5-66) sl'x::7N that P (0) = P and 0 P(1) = Pn; .1..e., the vertices P and Pn lie on the actual curve seg;rent at the 0 starting point and the e."ld point respa:tively. Another characteristic of the interpolatioo fln::tioo is tl-.at max:iirun
values occur at t
=.
i/n .
'1'his maxirrun is g'iven by Ref. 5- 12 as (5- 69)
FOr ex.ar.plc, for a cubic ,
-4 9
SPACE CURVES
2 4 J3,2(3) - 9
141
1',•
As an e.xa.'Tple, consider cubic
Bezicr curves, as sha.m in Fig. 5-8. For this c;.,se n = 3. Assune equal increrents :m the pa.ra:retcr
segrrent, sa:,.·,
t-=
o, l/3,
t
r
P,
for the
2/3, 1.
Then
--
using Eqs. (5-64) and (5-65) yields
3!
JJ,l(t) .. 11(2) I t (1 -
t
2
)
= Jt(l
- t 2)
= Jt2 (1
- t)
and
31
2
J3, 2(t) • 2! (1!) t (1 - t)
"u
'
'111e results are tabulated in Table 5-2. USing Table S-2 dl1d Eq. (5-66)
1
/fP,. / I
P,
/
8
4 2 l P(J) ,. 27PO + 91?1 + 91?2 + 27P3
/
I ')A I
P;;- ~ ""~.../I
'nlus, to create a cubic curve seq-
......... J
nent, it is only necessary to specify the
PI
the four p:>l;,:JOO vertices and then calFigure 5-8
culate JX>ints along the curve for
Bezier p:>lygons for
cubics.
0 ~ t ~ l.O using F.qs. (5-64) and (5-80)
Table 5-2 Results for Bezier Curve
n=J • 1
J n,i
0
l/3
2/3
1
0
1
8/27
1/21
0
JJ,O • 1
1
0
4/9
2/9
0
J3,1- Jt(l - tl
2
0
2/9
4/9
0
J3,2 = Jt2(1 -
3
0
l/27
8/27
1
J3,3 .. 1
as shc1...n arove.
tives .
2
t)
lt is not necessary to explicitly ccnsider parametnc deriva-
A user can quickl)• learn to predict the shape of a curve which will be
generated by a certain p:>1}1901'\al shape.
Various Bezier cubic curve seqzrents
arc stn.m in Fig. 5-8 . Although tt 1s not necessary to calSider curve derivatives in generating
142
l~C\L
EIDIENI'S FOR C<:M='ln'ER GRAPHICS
i.rrlividua.l Bezier curves, rraintaininq continuity of slope and curvature "tlen
JOiru.ng bolO eezio.r curves requires consirleratioo of these derivatives. rth derivative at the st:.artJ.ng point
is given by It
I
pr(O) ..
The
•
L: (-l)r-l.(:)P . 1. 1.
n.
(n - r) •1 i=O
(5-70)
and at the end point by
n! - (o - r) I
r
L: i=O
(5-71)
'Ihus, the first derivatives at the end powts are P'(O) = n(Pl- P ) 0
(5-72)
. •
(5-73)
'lhls illustrates that the first derivative of the Bezier curve at the initial and fi.nal points is tangent to the initial and final pol}'90n segtrents, i.e.,
the end s lopes are fixed by the directiat of the line jo:i.n.ing the two end vertl.ces.
,
Similarly the secC>l¥3 derivatives are P"(O) = n(n- 1) (P - 2P + P ) 0 2 1 (5-74)
I
P"(l)
I
= n(n-
) l ){P - 2P n n-1 + Pn- 2
These results illustrate that the second derivative of the Bezier curve at the
initial and final points depends en the nearest two polygon segmnts or on the nearest three polf9C11 vertices .
In general the rth derivatives at an end point
or start.ing point are detenni.ned by the end or start.ing point and its r neighboring poly
If
one Bezier curve of order n is defined by vertices P1 and an
adjacent Bezier curve of order m by vertices Q . , then first derivative oontinJ.
uity at the joint is obtained when P' (1) ,. g:J' (O)
o r where 9 is a scaler .
Using F.qs. (5-72) and (5-73)
01 - 0o =
~) CPn
- Pn-1>
Sinoe the curve must be ccntintDUS at the joint, ooc:; sec}'l'ent
0o "" P0 •
(5-75)
Thus, the end of
is equal to the starting slope of the next segrent when the three
points 0 , P , P are col.i.near . FOr the particular case of two adjacent 1 n n-1 cubic Bezicr curves (cf Fig. 5-9) , n = m = 3. If both diroctioo and JTa911itude of the JOint tangents are to be equal, Eg. (5-75) yields 0 1 - 0 o = P 3 - P2 = ~ - P3
SPJICE
Figure S-9
First-order cxmt.inuity in
~ier
~
143
cubic segrents.
Thus 0 1 + p2 - 2P3 '!hat is, P is the midpoint of P o . Note that continuity of slope c:nly requires 2 1 3 that P lie on P o1 sare..nere bet\.-een P and 0 • 3 2 2 1 For the sane P and Q curves above, seoond derivative continuity is given by
m(m - 1) co0 - 201 + o2 > = n(n - 1) (P _ 2 - 2P _ + P ) (5-76) 0 0 1 0 As a practical matter it may be necessary to increase the order of the
curve, i.e., .it.crease the nl.llber of p:>lygon vertices to rraintain higher order derivative continuity.
This is frequenUy eas1.er than splitting the overall
curve into smaller setiJieJ'lts ard m:rintaining lower orders, say C\Dic, within the ~t .
nus
flexibility of increasing the order of curve
~ts to
get bet-
ter control of the curve shape is one of the major advantages of the Bezier
rrethod .
• BEZIER aJRVE
Exanple 5-3 :
Assume that P0 (1 1], P (2 3), P2 (4 3) , and P3 (3 11 are the position vec1 tors of a Bezier p:>lygon .
R£lcall Eqs . (5-64) to (5-66) n
P(t) ..
L P.J . (t) i=O ~ n , ~
n i n-i .n n! where J 0 , 1. (t) = (.)t (1t) and \ 1) • . , (n _ i)! • 1 1 1. have 4 vertices . Hence n 6 (i) = i! (3 - i)!
and J ,
ct> 0
0 = (l)t ct
-
3 2 J3 , l(t) = 3t(l - t) 2 J3,2(t) - Jt (1- t)
t>
3
= (1- t> 3
Here n • 3, since
we
Thus
P(t) • POJ3,0 + PlJ3,1 + P2J3,2 + P3J3,3 table of J n, 1 for various valoo.s of is given bela..~ .
A
t
Table S-3 )
1',
P,
J
•
,
.L
pl
I
t
JJ,O
0 0. 15 0. 35 0. 5 0. 65 o. 85
•
I
Coefficients for a Bezier CUrve
1 0~----~----~------~----~ I
figure 5-10
P(O)
~
J3,1
0 0.614 0. 325 0.275 0. 444 0. 125 0. 375 0. 043 0. 239 0.0034 0. 0574 0 0 1
•
J3,2
J3, 3
0 0 0. 0574 0. 0034 0.239 0. 043 0. 375 0.125 0.444 0.275 0. 325 0. 614 0 1
~
Results for Dezier curve segrrent for Exanple 5-3. Finally
= P0 = (1
1}
P(O. lS) : 0. 614P + 0.325P + 0.0574P + 0. 0034P • (1.5 1.765] 2 1 3 0 P(0.35) =0.275P0 + 0.444P + 0.239P2 + 0. 043P3 = (2 . 248 2. 367] 1 P(0.5) • 0. 125P + 0. 37SP + 0. 37SP2 + 0. 12SP • (2 .75 2. 5) 0 1 3 P(0.65) : 0.043PO + 0. 239P + 0.444P2 + 0. 275P = (3 . 122 2. 36) 1 3 P(0. 85) = 0.0034P + 0. 514Pl + 0.325P2 + 0. 614P = [3 . 248 1. 75] 3 0 P(l) ; P3 (3 1]
=
'Incse p:>ints are plotted aloog with the defining p:>lygon in Fig. 5-10.
An algoritr.n which nay be used to calculate Bezier curve segments given
t:re p::>sition vectors of the po1ygcn vertices is given in .Appendix c .
5-8
8-SPLit~ CLRVES From a m2.tl'.ematical point of view, a
ver+ioes of a defining
p:~lyyat
C\lnE
wtu.ch is qenerated by us:ing t:.M
is dependent on sare interpolation o r approxima-
tion schcfrc to establish tre relutionship bcO..'eCJ'l the curve arrl the p:>1ygon . This
schal~ 1.S
providod by the croice of a basis or
~ight..ing
functioo.
As
SPK.'E CURVES noted in the
prev1.0US
145
section, the Bernstein ba.eis prod11ces &...-z.ier curves gen-
erated by Eq. (5-66) .
'lle;e
have several useful properties.
'1\.o chAracteristics of the Bernstein b
of the resulting curves.
First the nll!Tbar of spec1fied polygon vert1ces fixes
the order of the resulting polynomial which defines the
cw:ve.
For exnrrplr>, n
cubic curve nust be defined by a polygon with four verttces and three spans. A polygoo with six vertices will always produce a fift:h-dcqree curve.
'Iha only
'Wa¥ to .reduce the order of the curve is to reduce the n\.'Ulber of vertices, and
cnwerse.ly the only way t:o increase the order of the cur:vc is to ux:reasc the m.1tbe.r of vertices. 'Ihe second limiting characteristic is due to t..he global nature o( the Bernstein basis .
'Ihls rrcans that the value of the
\
function J
. (t)
n, 1 given by Eq . (5-64) is nonzero for all pararreter values over an entire span of the curve. Sinoe any point on a Beder curve is a resul.L o! ~<~eighLing th!= values of all defin.ing vertia-s, a change in one vertex is felt 1:hrougl'xlut the entire span. Practically, this elim.inates the ability to produo:! a local chango within a span. Fer e.xaJTPle, siJ1c:e the end slcp:s of a Bezier curve are established by the
directialS of the first
aoo
last polygon sides, it is (X)Ssible to change the
middle vertex of a five -point pol}'900 witOOut changing the di1•ection of the end s l opes .
However, tha shape of the total curve
nature of the Bemst=Pin basis.
.LS aff~
duo to the qlobt,l
This lack of lcx:a1 span oontrol can be detri-
rrental in sate applications. 'n1are is an:>trer basis, called the B-spline basis, \orhlch C
'lbis basis is generally nooglobal.
The
nonglobal behavior of a-spline curves is doo to the fact that each vertex P. is l.
associated with a unique basis functiCX\ .
'111us, each vertex affects the shape
of a curve only over a range of parMeter values where 1.ts associated basis function is nonzero .
'lbc a-spline basis also allows the order of the resulti.nq
to be changed without changin9 the mmber of defin:ing polygon vertices. The theory for a-splines was first suggested by Sc:hoenbe.rg (Ref . 5-13). A C'\.lt'VC
recursive definitioo useful for nt.Drerical cmput.ation was published by Cox ~-
5-14), and by de Boor (Re£. 5-15).
Reisenfeld (Ref. 5-16) applied the
B-spline basis to curve defiru.t.lCI'I. If
\
again let P(t) be the positioo vectors alon
of the parall'cter t , a curve gererated by the use of the B-spline basis is given by n
P(t) =
L P.N. k(t) i-o l. l.,
(5-77)
146
l·~ICAL
EII:'0ll'S FOR
en~
GRAPIUCS
where the P . are t:re n + 1 defining polyqon vertices. 1.
FOr the ith oornalizcd B-spline basis curve or order k, the w.ighting func-
tioos N. k (t) are defined by the recursion fornulas 1,
N
1,1
1 i f x. !:. t _ x.+l
{t) =
1
1
J0
0
othexwise •
N . k(t)
• (t- xi)Ni,k-l(t) +(xi~- t)N~+l,k-l(t)
1'
xi+k-1 - xi
(5-78)
xi+k - xi+l
'lb:l values of x . are elerrents of a krDt vector which is cliscussed belo..r.
'lhe
l.
along the curve P(t) . rrax . An additional variable rrust be uc:ed for a-spline curves to aocount for
pa~ter
t
var1.es fran 0 to t
the i.nherent added flexiliility.
'lhis 1s achieved by use of a knot vector.
A
~ xi+l for all 1 Examples of knot vectors are (0 1 2 3 4) and [O 0 0 1 1 2 3 3 3] . The
krDt vector is s.i.nply a series of real integers xi, slrll that x
xi.
values of xi are considered to be pararretric knots .
'lhey can be used to indi-
cate the range of the paraneter t used to generate a a-spline curve with
'
O -< t -< t max . Recall that the pa.ratTeter range for a Bezier curve was arbitranly chosen to be 0 .i t ~ 1 for a Bezier curve of any order.
'~ f'OI
deviate fran this con-
vention and use a knot vector to specify the paraneter variation for the
cut""'.l'e.
Fbr cx.a."tple, the vector (0 l 2 3 4) indicates that the par.uroter t varies fran 0 to 4.
'lhe m.aTber of inte.nred.iate knot vectors depends on the nmbe.r of spans
in the defi.n.i..ng polygon.
A d\~licate interrrcdiate knot
value indicates that
a nultiple vertex (span of zero length) occurs at a point, and an int:ei.1rodiate
knot value in triplicate indicates three coocurrent vertices spans) .
(bot>
zero-lenqth
The actual point on a B-spline curve which corresponds to the value
of a paranetric knot (t • xi) is called a gearotric knot.
It is convenient to
use evenly spaced knots with unit separation between roncoincident knots. gi VI!S integer values for the
a:J"IPC)nents
of the knot vector .
'lhls
In addition to the
knot vector values, the order of the curve must be specified.
I f the order k
equals the nunber of polygon vertices, and there are no multiple vcrt1ces, then
a Bezier curve will be generated.
As the
lies closer to the defining polygon.
order dccre-1ses , the curve produced
'~
k
=2
the generated curve is a series
of straight lines which are identical to the defining polyqon . 'l'hc order of a curve is reflected in the knot vector that ls used to gen-
erate the curve.
It is necessary to specify knots of nultiplicity k at both
til" begl.M.i.ng and the end of the knot set. pol~'qOI"l
For exarrple, oonsider a hve-point
(n + 1 - 5) with no duplicate vertices.
h1lcn there are no duplicate
vertices, the p.u-aneter t varies fran 0 t..o n - k + 2 over the entuc curve.
SPJ>a: CURVES
For a th.i.xd-ordcr curve defined by fiw vertices, 4 - 3 + 2
~
value of t
nwc
14 7
•
The conplete knot vector, using rrultiplicity of 3 at each end,
3.
is then given by (O 0 0 1 2 3 3 3).
A second-order
curve for tro sarrc defining
p:llygon has a knot vector (0 0 1 2 3 4 4), a."td a fourU.-ordec curve has (0 0 0 0 1 2 2 2 2} .
If seven polygon vertices are used a thircl-ord,.J" curve
lS
desired, then the koot vector .is (0 0 0 1 2 3 4 5 5 51•
a final cxanplc, consider a polygon of four djstincl vcr.t..locs. 'lhe knot vector for a seoond-ordcr curve is [0 0 l 2 3 3}. Nc:M, if the 0.0 center verAs
tices are rMde to coincide so that a rrultiple kn:lt occurs between the two end
krx>ts , the knot vector for the sec:ood-order curve is I 0 0 1 1 2 2] • If suoothness is based on continuity of higher order derivatives, the order of the curve determines how "srtooth" the curve is.
For exanp1e, a fourth
order curve is continoous in first and second derivative, as well as p:>sitlon,
along the entire curve. The 8-spl.i..ne curve is matherrat.ically defined as a polyromiAl spline func-
tion of order k (degree k - 1} since it satisfies the folla.r.ing t:\..o oondiucns: The function P (t) is a p:>lyncmial of degree k - 1 on eACh interval (xi~ t ~
xi+l }.
P(t) and its
derivatiws of order 1, 2, . • . , k - 2 are all continuous
over the entire curve. 'lhus, a fourth order a-spline curve is a piece.ri.sC! cubic spline. Due to the flexibility of B-spl.i.ne curves, different types of control
"handles" can be used to change the shape of a curve . by changing the integer order k for 2
~
COntrol can be achieved
k !:.. n + 1 , by usa of repeating vertices,
or by changing tOO nurber and/or p:>sition of nc:nrepeat.i.ng vertices in the defining polygoo.
These effects are illustrated in the follo.dng six figures.
Figure 5-11 sho.Ys three &-spline curves of different order, each de..fined by the sarre four polyc;pl vertices given by
0 0 •
The second-o.rder curve
3 9 6 3
9 6 creates three straight lines bet\.Jeen the four vertices, the fourth-order curve corresponds to the Be'.der curve for the p:>lygat set, al'X1 the t:hi.Id-order curve produces a looser curve bet-..-een the two end points.
Notice that all three
curves have the sane end slopes, determined by the slope of the first al'X1 last
spans of the defining polygon .
As the order of a curve increases, the resulting
sha?> lodlygou shape. tightens the curve.
'!bus, ioc.reasing the order
! t r---~--------------------~--------~------------~
•
•
'
'
I"
•
•
'•
•
•
•
Figure S- 11
'
B-spline curves .
•
u
• •.----~--~----~----------~---------------~--------~
• 5/S
' '
fL---._--~--~----~--~--~--~----~--~--~
•
2
F'l.gUre 5-12
4
'
•
Multiple vertex B-splinc curves.
"
SPliCE CURVES
149
Figure 5-U shows the effect of rrultiple vertices in the defini.ng polyyoo. For each of the four curves shown, the order of the curve is equal to the n\.r.be.r 'lb! lower curve m fig. 5-12 is identlc-al
of vertices in the defining polY90fl.
to the lo..er curve in Fig. 5-ll, a fourth-order curve defined by four polyqa1 vertices.
'Ihe seo::::nd curve
vertex at [3,9) .
at (3,9).
l.n
Fig. 5-12 is a fi.fth-arder curve with a double
'Ihe third curve is a s.ixt:h-ottler curve with a triple vertex
The fiMl seventh-order curve has a defining pol yqon given by
001 3 9
3 9 3 9
3 9 6 3
6 e . g., four nultiple vertices at [3 91.
'lhis figure clearly shows how a curve
can be pulled clo$er to a speci fjc vertex positjoo by use of nultiplc vertices while rraint:aining the sa~re end slopes for each curve.
()) the other hand,
decreasing the order pulls the curve closer to all polygoo vertices. In Fig. 5-13 the defining vertex lS
l l r---~----~--~r----r----~--~----~----r---~----,
I
' •
e L---~----~--~----~--_.----~--~----~--~--~
I
2
Figure 5-13
•
'
•
Multiple knot B-spline curves.
II
150
wmaP.TICAL El c:.V.ENTS FOR cn!PL"l'ER GRAPHICS
ro o 2 5
I4
8 6 3
6 3 •
8 6
l1o '!1lat is,
for each curve.
\>'e
1
keep a double vertex at the fourth ele!l'ellt in the
polyr;p1 . Hen! the curve is al t.ered by chanqi..ng the order of each curve, ~ ing the defining p:>lygon constant. The first curve is of order seven , equal to the m.:'li:er of vertices in the polygon . The second curve is of order five . 'lhls curve shape is closer to the polygon shape, expccially O('ilr the double
vertex.
The third curve is of order 3 .
Notice that a "knuckle" occurs at the double vertex since the slope and curvature are discontinoous. A duplicate vertex is required to create a knu::kle in a third~mer curve . A triple vertex creates a kmx:k.le in a fourth order curve. This ability is a mmLll"1 requirement in stup desl.gn.
• I I
Figure 5-14 deronstrates the enUre shape of a curve.
~
local changes can be .rrade without a£fecting
Each curve is a
seven-point polygol"' with oo multiple vertices.
fi.fth~rder
curve, defined by a 'llle only difference bebolee.n
2
I
r~cu.
l:lnZt
""...........:lr. .....t )"~ r.ulll:.lle ~
~
•
1
Fiqure 5-14
'
•
Local c::allrol of B-spline c~ .
ll
SP/tCE CURVES
151
each curve 1.s that the fifth vertex is rroved to a new posiUoo, as shewn in the figure.
It can be seen that the first
part of each
curve is 1.11'\Chanqed.
'1llis behavior is a result of the nonglOOal. (local) nature of the B-spl ~ basis.
Figure 5-15 shcJo...'S curves of order 2 through 5, generated by a six-point polygcn with no nultiple vertices.
'the fourth order curve is a cubic spline.
'1lle &-spline curve technique all.oNs a orne spline to be generated witll three
or nore pol ygon vertices . fif~rder
Since there are five spans in this polyc:pt, the
'Ihe third-order curve may be of special
curve is the Bezier curve.
interest since it is tangent to the midpoints of the internal polygon spans .
nus characteristic is also sh::Jwn in Fig. 5-16, where a third-order curve is generated with an eight-point polygon.
Ex.anple 5-4:
&-SPLINE CURVE
Cl:lnsidcr lhe sarre four polygon positioo wctors
tLc;ed
in the previous
'lbese are P (1 1) , P [2 3), P (4 3), and P (3 11 • 0 1 2 3 set for a second-ordP..r curve &!fined by four polycJOn vertices
exanp1e for a Bezier curve . The knot ~tor
is [O 0 1 2 3 3) , where we denote
>
~ =
0,
X:z"'
1, •• • ,
\ohm using Eq. (5-78) the conventioo 0/0 = 0 is applied.
>
3. N. k values l.,
needed are (t - O)NO
N0 , 2(t}
(t)
1 ~- xo
c:
+
N2 , 2(t)
N3 , 2 (t)
c
+
X:z-~
c:
=
t)Nl,l(t}
X:z-~
(x
(t - O)Nl l (t)
N , (t) 1 2
(X:z3
- t}N , (t) 2 1
x3 -
X:z
(x - t)N (t) (t - l)N2 ,l (t) 4 3 1 + x3 - x1 x4 - x3
x4- ~
5 - t>N4 , 1 (t)
(x
(t - 2)NJ , l (t)
+
>
x4
Specifically, fior t = o, N , 2 (0) = 0(1)/0 + (1 - O) (1)/1 • 1 . 0 and 0 N , (t) a 0(1)/1 + (2 - 0) (0)/l = 0. Fort= 0 . 5 we find N , 2 (t) = 0.5(1)/1 + 1 2 1 (2 - 0.5)(0)/1 = 0 . 5 , and fort= 1 we calculate N , (l) ~ 1 (0)/1 + 2(0)/0 = 0 3 2 other values of N. k are found in a similar manner. J. ,
plete results.
Table 5-.4 gives the COir-
By usmg tl1e values of N• . given in Table 5-4, points oo the l.,)
seoond-order curve can be detennined.
P(t)
Ftun Eq. (5-77) , in general,
PON0,2 + PlN1,2 + PiN2,2 + PjN3,3
~TICAL E:I.EME:Nl'S
152
FOR CCt1Pvn::R GWIH!CS
1 ~ ~------------------------------------------------~
' •
• •
I S
t L---~--------~------------------_.
•
•
• F~gure
'
____~___.____J I
u
5-15 Varying order of B- sp1ine curves.
8
'
I L---~----_.----~----~--~----~----~----~--~-----J I ? ll 4 ' •
Fl.gure 5-16
'lhird order a-spline curve.
SPACE O.I'JM:S
153
Table 5-4 'oJeight.i.ng f\mctions for a second-Order B-Spl.tne
0
0.5
1.0
1.5
2
2.5
J
N0,2
1
0. 5
0
0
0
0
0
Nl,2
0
0.5
1
0.5
0
0
0
N2,2
0
0
0
0. 5
1
0.5
0
N3,2
0
0
0
0
0
0.5
1
t
N
l. , )
For t
= 0,
Similarly
+ P (0.5) + P2 (0) + P3 (0) • O. S(P + P ) 1 0 1 P (1) + P (0) + P (0) = P 1 2 3 1 P (0.5) + P (0. 5) + P (0) ~ 0.5(P + P ) 2 3 2 1 1 In a like rranne.r the last three points are P(2) ,. P , P(2 . 5) • 0.5(P + P ),
P(0.5) • P (0 . 5) 0 P(l) = P (0) + 0 P(1 . 5) = P (0) + 0
and P(3)
= p3 ,
2
2
3
'l1lese points obviously fall on the straight lines joining the
polygon vertices. Now, i f the order of the
curve is increased frau two to four tor the sane
four defining polygon vertices, a Bezier curve will be obtained since lhe order l'lCW
equals the nt.nbe.r of vertices.
'lb! knot vector set (with tmax •
3 - 4 + 2 ... 1) is [0 0 0 0 1 1 1 1). by Eq. (5-78)
Values of the weight.ing function.s given
are (t -
O)NO 3 (t) 0 - 0
t> •
Nl, 4 <
N (t) 2 14
N3 ,4 (t)
a
=
+
(1 - t)N , (t) 1 3
l-
0
(1 - t)N2 , 3 (t) (t- O)Nl , J(t) ---.rr----;:o~- + 1- 0 ( t - 0)N , (t) 2 3
1 - 0 (t - O)N) , ) (t)
1- 6
Proceecling with the requ.u:cd N.
l. ,
3
(l - t)N) , J (t)
+ +
1-0 (1 - t)N (t) 4 3
1 - 1
evaluations,
(t- O)N , 2 (t) 0 0 - 0
(1- t)N , (t) 1 2
+
0- 0
N , (t) 1 3 ~
=
13 (t )
II
N3,3(t)
:a
(t - 0)N , ( t ) 1 2
6- 6
(1 - t)N , (t) 2 2 + 1- 0
(1- t)N , (t) (t - O)N2 , 2 Ct) 3 2 1- 0 + 1 - 0 (1 - t)N , (t) (t - 0)N , (t ) 4 2 3 2
+
1- 0
1- 1
(t- l)N , (t) c1 - t>N5 Ct> 4 2 12 N4 , 3(t) = +- 1 - 1 1-1 'lhese N.
~.
3
equations in turn requi.n! N.
~.
(t - O)NO,l (t)
N0 , 2(t ) "'
(t - 0)N ,l (t)
1 0- 0
(t - 0)N
N2, 2{t)
=
uJ I
2 0 - 0
''
(0 - t)Nl,l (t)
(1 - t)N)
+
l- 0
o.
0 -
{0 - t)N , ( t) 2 1 + 0 - 0
l (t)
( t - O)NJ, l (t )
N3 , 2(t) =
values given by
+
0 - 0
Nl,2(t) "'
2
1
(t)
1 - 0
(1 - t)N , (t) 4 1 + 1- 1
(1 - t)N , ( t ) (t - l)N ,l (t) 4 5 1 + N4, 2(t) = 1- 1 1- 1
N5 , 2(t) •
t)N , Ct ) 6 1 1- 1
(1-
(t - l)NS , l (t)
+
1- 1
Let us evaluate the point on the curve at t • 0 . 5.
Since the pararreter
variation as specified by the koot vector fa ll s between 0 ~ t ~ 1 for this the value P (0 . 5) slx>ul.d oorrespcnd to the value P (O. S) found in tho Bezier
example in sec. 5- 7. and N3 ,l • 1 . 0.
For t • 0 .5, NO , l • N1 , 1 • N2 , l • N4 ,l • N5 ,l Thus , N0 , 2 = N1 , 2 = N4 , 2 = N5 , 2 = 0 , and N2,2(0.5); 0 + (O . Sl(l) ~ 0 . 5 N3 ,2 (0 . 5)
tt~rk.ing backward
= (O.Sl(l)
it follows that N , 3 0
+ 0
= N4 , 3 =
Nl,3 = 0 + (0 . 5i(0 . 5) (0 . 5) (0. 5) N2 , 3 = 1 N3,3
= (0 . 5 )1(0 . 5)
N0,4
D
=0.5 0 , and
= 0 . 25
+ (0. 5) (0 . 5) - 0. 50 l
+ 0 = 0 . 25
Likewise 0 + (0 . 5)10 . 25)
= 0.125
N1,4 = (0 .5) 1(0 .25) + (0.5)1(0 . 5) - 0.375
&
case,
N6 ,l ~ 0,
SPN::I:. C'tJR\.1&S
155
(0 . 5)1(0.5) + 0.5(~ . 25) : 0.375
N
3,4
~
(0 . 5)i0.25) + 0
Finally, the point an the curve at
= 0.125
t = 0.5 is
given by Fq. (5-77) as
or
P(O.S) • (l 1]0.125 + [2 3)0.375 + [4 3)0.375 + (3 1)0.125
= [2 . 75
2.5)
'Ihe x~t is then P
X
agrees with the
ee~:ier
= 2. 75
and the y~t P
y
= 2. 5, \oohlch
curve results.
Algoritl'lm9 which will generate the required B-spl.ire rusis knot vectors and the corres{X>nding B-spllne curves are given m Appendix
c.
REFERENCES 5- 1
w.
Rigdon, A., Chlsen, E., Stiles,
2d editioo, John Wiley
& Sons,
and Weese, J ., t-t!chanics of Natcrials,
Inc., New York, 1967.
5-2 South, N.E. , and Kelly, J.P. , "An.llytic SUrface l'ethods," Ford Z.lotor Catpany N/C Dewloprcnt. ll'rit, Prodl.d: Engineering Office, DeoerTber 1965. 5- 3 Nut.boume, A. W., "A Cubic Spline Package Part. 2 - 'nle Mathematics," Cc:xrput . Aided Des., vol. 5, no. l, January 1973.
5-4
Adams, J . Alan , "A Ccnparison of l-:ethods for CUbic Spline CUrve Fitting," Carput . Aided Des., wl. 6 , pp. 1-9, 1974.
5-S
t-~,
J . R., "Co!puter-M.ded Footwear Design:
A rrct:.hOO of c::a1stnlcting
srrooth curves , " SJ\TW\ (Sb:le & Allied Trades Research Association - SATRA
House, lb:::k.ingham
~d ,
Kettering Northants, England) Research Re{X>rt
R.R. 251 , December 1972. 5-6 Denman, H. H., "Stooth CUbic Spline Interpolation FUnctions," Industrial Mathematics, J . Ind. Math.
soc.,
vol. 21, Part 2, Rl· 55-75, 1971.
5-1 Cline, A. K. , "Curve Fitting Using Splines U1der Tension,"
ll.trro.s . Tech.,
no. 3, pp. 60-65 , 1973. Sclr..oeikert, D. c., "AA int.eq:olaticn Ctlr\A:! usinq a spline in tension," J . Math. Phys. , vol . ~5, pp. 312-317, 1966. 5-9 OVerhauser, A. w. , "Analytic Definition of Curves and SUrfaces by Parabolic 5-8
Blending,"
Technical Re{X>rt No. SL68-40, Ford f-btor Cl::ltpany Scaentihc
IJ'OOratoty 1 1'1ay 8 , 1968.
5-10
Bezier, P. £., l:.;P10i des t-'dcirines
.JlC •
S-11
1
~iml.
Control .r-utherratics and Flpplicatials , John \'Iiley & Scns,
London t 1972 •
~zier,
P.£.,
•e:xanpte of an Existing
Ul · surf Syst.£!'1\," Proc. Roy. S-12
Nurerique, twasson et Cie,
Transl.dt.ed by FOrrest , D. R. and Pankhurst, A. F. as Bezier,
Paris, 1970.
P.F.,
a~
soc.
Systan in the t-btor Industry:
The
(I.ondon), vol. A321 , pp. 207- 218, 1971.
Forru;t, A. R. , "Inte.rpolaticn and Approx:irraticn by Bezier Polyncmials, " CNJ C".roup Doc. No. 45, l!-U.., catrbridge university, Q:tobcr 1970.
5-13
SChoc.nbcrg,
r.
J ., "Contributions to the Pl:cblm\ of Approximaticn of
Ek)w.dist.ant Data by Analync
F\lnctions, "
Q. Appl. l-lath. , vol. 4, 1946,
pp. 45--99; 112-141. 5-14
•
Cox, M. G., '"lbe Nurerical Evaluation of B-Splines , " Nathicnal Physical Iatx:lratory
D!l..~
5-15 de Boor, Carl,
4, August 1971. "()1
Calculat.i.ng ,.'ith B-Splines," J . Approx. 'lheory, vol. 6 ,
pp. So-62 , 1972. 5- 16
Rl.esenfeld, R. F. •aerstein- Bezier l'ietrods for
of Free-Form C\l.n-es and SurfNYS . \
l-arch 1973.
t.~
COrputer-A.ided Design
Ph.D. 'llll!sis, Syracuse tkriversity,
CHAPTER 6 SURFACE DESCRIPTION AND GENERATION
Surfaces and surface representatim play a cntical role in rost design and manufacturing prcx:csses.
'lb! design and manufacture of aut..ard:liles , air-
craft, s':ups, nachlne parts, glass..rcue, and clothing are obvious exant>les. ~ b:cldi.tiO'\al way of r<..-prescntir.g a surface is to use multiple orthogonal projections .
lb.-ever, l"'&ly couplex surfaces cannot be effic1.ently represented
Ul
this
rm.nner.
rather the rrathmt.ltical description of shape fran which either drawings or ic.U controlled rmchine tool tapes can be obtained.
nt~Ter
The feasibility of produc-
ing a surface chrectly from a c:x:r:p..rt:..e stored representatioo of that surfo!lee
has been denpnstrated by Bezier (Ref. 6-1), Sabin (Ref. 6-2) , Peters (Ref. 6-3), and others .
Various techniques of anal.ytically and nurerically representlng a threedin'ensicnll surface are discussed below. a SlJ!'Ple spherical surface .
To
begin the discussicn we oons1.der
158
6-2
Nl\THEM\TICAL EI.F2'.ENTS FOR
SMRJCAL SuRFACES SCire
properties of curves which are useful in surface defiru.tion can be
derronstrated by considering a surface of Jcno.m analytical description.
To this
end cons1der a spheriall surface .
Specific curves on t:he surface are defined by
planes wilich 1.ntersect the ~e .
As an exanple consider the intersectioo of ~l
• a . 'Ihl.s is 1 The resulting curve lS a parallel of latitu:3e. 'Ihe equa-
a unit sprore and the plane defined by the surface z ..
sho.-m in f'ig. 6-la.
t.ioo for the parallel of latibrle is oot..ained by solving the two surface equ.a-
ticns silrultaneously.
The ncnpa.rairetric equation for a unit sphere is
~ + y 2 + z2 • 1.0
(6-1)
Thus,
zw )
'
defines Ule intersectioo. In Fig. 6-lb, the plane at
r : e0 which
intersects the sphere is defined
by
or
cx - by=O 1 1
(6- 2)
l
• •I
• •I
..a I
-.
'\ \
-I
,~ I )
/
~I I
/
I
I
,
~"
•
, "'
I
b
l'•nallcl ol latuudc
Figure 6-1
I
\l truloiln 111 lc'n~11ul1c
Int.ersectioos of a plane and sphere .
I
'!he resulting intcrsectioo of the plane and tOO sphere yields a para l lel o f
longitude .
am
Solving F.qs . (6-1)
(6-2) sinultaneously yiel ds the equatioo for
• e. , the resulting curve, l..
b
2 il<_!> cl
+ 1] +
i ::
1.0
1\ spherical surface patch can be foxued by four planes int.ersectl.l'¥} the ~.
as shc:1wn in Pig . 6-2.
'Ihis creates bolo parallels of latitude and two
parallels of longitude for the patch bolrdaries.
'llle vector equatioo f or the
result:.l.ng surface patch 5(9 , ~) is
s
-+
e2, am e) are unit vectors
where ~ ·
s
•
= (cosesin$)e 2 l.Il
•
+
(oos~ >e
3
(6-3)
the rectangular coordinate system, and
is a p:>sition vector 01 the surface of the sphere as shown in Fig . 6-2.
'11\is
surface patch can be thought or as the locus of a p:>int in three-ditmnsional space whi ch rroves with two degrees of freec1an cxmtrolled by the two pararreter
e and
vari.llhles 1\
~.
pararretric representation of the un.it spherical surface is X •
(6-4)
cos6~
y = sil16si.nQ z
Notice that two pardlmters tioo of a surface.
1
~;4 ~
v ~ 11/2 .
and ¢ are required for the pararretric representa-
The patch shc:1wn in
Fig. 6-2 is defi..ood betw'een 0
am
= cos¢'
~
9 ~ 'Jl/ 2
Fran Fig. 6-2 tNe
see that the surface paten is defined by the four boundary curves AB, oc, CD, and Dr\.
FOr the spherical patch these 1
are cil:cular arcs.
Further analysis of
Fig. 6-2 shows that the shape of a b::lundary curve can be descrHvxl by specifylJ
ing the tangent vectors at the end of
each curve, i.e., the eight tan<]ent
.
vectors at the end of each of the corner p:>ints A, B, C, and D.
For a
()
/
~
analytical surface these end tangent
vectors can be obtained by d.if ferenti-
D
y
ating the equatim describing the surface.
t
II
2
For the spheri cal surface
shown in Fig. 6-2 these are
as
.
..
as : -(SJ.n'ISi,n
-+
(COS6sin¢)e2
Figure 6-2 (6-5)
Spherical surface patch.
$as ..
.. +
. • - {siM)e• (sl.n°oos¢)e 2 3
(cosecos¢)~
'Ihe rragn:i.t:u.'ie of each of these tangent vectors is unity . ~ =
(6-6)
At comer A,
(1 =
O,
Tr/2, and •
as .. ai = -c3 At oomer C,
~
• n/4, E = r:/2 , and
as
1 ....
'aif .. -If~
as
3i = as;ae
Here the rragnitude of
is
1
n
1/12, and
•~l//2)2 +
" 1 + e2- 12 c3 the IMgnitude of ilS/il4> is
{1//2)2
e
1.0
'Ib! l.angent vtctors at all four comers are shown in Fiq. 6- 2 and are given in Table 6-1.
Table 6-1 Tangent Vectors for the Spherical surface of Fiq. 6-2
Point
as;ae
as;ae
e2
1
-e3
1
1 ....
1
1
1 .. 12 2
1
...
A
CIS/CI¢
....
B
-e li 2
-
1... 1 .. -e --e 12 1 12 3
c
1+ -;
-e- -e
12
-12l
-el
1
12
....
0
1 • ~ 3
+
-e3
Iasr~l
1
The cross derivatives, or twist vectors , at the oomers are also useful
in describing the boundaries of a surface patch .
For the unit S{:tlerical patch
these are (6-7)
'Ib! magnitu::ic of the twist vector is
Ia2s;aea¢.- o a2s
aea~
Note at oomer A,
-
.C2eoos2¢ + and at
c:os2Poos2~ .. COSQ
(6-8)
cm::ner c, a 2 sh"CI~ = -<1N2>~1 • '!he twist
vectors and their rr.agni. t' !des for the comers of the spherical and surface patch are sho.m in Table 6-2.
St.JRF'ACE DESCRIPI'Irn AND GFNEPATIOO
161
Table 6-2
Twist Vectors fur the Spherical surface of Fig . 6-2 il/a~
Point
I /lB(ClS/
0
0
-·
B
1/12 e 2
1/l'i
c
-1/12 e 1
l/~
0
0
~>I
0
SUt'face normals can also be u9¥1 to indicate the shape and orientation of a surface patch .
'lt.e surface nonrals are given by the vector cross prc<}t.ct:
of two tangent vector s .
as
For the SFherical patch ,
as
N=aexa¢ "'
It follo,..ts that at comers A and +
.
~
e1
e2
-sin6sin9
cos9sin4
ocx~
sin6cos~
+ 0, N • ~ ,
2
t
...
(6-9)
e3
0 -sin¢ ~
+
at corner B, N = -l/2(e1 + ~) and •
a t comer C, N a - 1I 2 (e + e ) . 'nle no:mals at c::o~rs Band Care slno.n l11 2 3 Fi g . 6- 2 . 'nle out"...'al'd drawn normals for the surface patch are the negatives of these
values. A unit s urface
nornel is defined by ;:; •
N/INI .
Expanding F.q . (6-9) gives
(6-10) and (6-11)
'lllus, (6- U) where
lnl = 1.0
(6-13)
A f.i.nal vector operation which is useful for surface descdpt.i
dot procb::t.
For the surface patdl under cxnsideratioo,
;;s · ~ as • ( C-uinl.llin01e• • (001i:t~le• J • ( ((X)Gf.C015;)e .. .. (5.Ul'c.oa.tle · .. - (sin~le• 1 • 0 (6-14) ~ 2 3 1 2 1 Si.nce the dot product of these tangents vectors egual.s zero , the b.o tangent vectors are perpendicular to one another.
'll\us, lines of ¢
r
constant
(par-
to lines of 9 = constant (rreridians of
longJ.tu;!e) on the surface. HaVl.nCJ used a spherical surface to illustrate a nmber of useful oouoept.s
for surfaces, e.g. , tangent vectors, twist vectors and surface normals,
"-'E!
now
consider rrcthodc: for describing rrore arbitrary surfaces .
6-3
PLANE
~FACES
Sate of
tr.e techniques for surface reprPSentatiCI'l are rrost oon\A:!f\iently
described by considering a plane surface . In thls sectiCI'l we restrict ourselves to a four-sided plane surface. l1'le four sides def.i.nin<] the surface are four pa.rarretric curves or lines which o:>nnect: the four corners and form the shape o f • the boundan.es. As sha..m by El:] .
(6-4) in the previous section, it is possible to express
points wtuch fall on a surface in terms of
w 1
,
pa.ra!l'eters be u and w.
~
parcllOOters.
Let these ~
The expression for the plane surface is then x = f(u ,w)
(6- 15)
y • g(u,w) 'Ille funct.i.als in Fqs . ( 6- 15) are chosen such that the four bouOOacy cuzves are
described by the follc7wd.."lg four oonditi.als : u=c
c
w
~
c
u=c 2
c 1 .5.. w
~
c2
w·~
c
w .. c
c1
1
2
1
1
~
~
u
~
c2
~
u
~
c2
'lhls defines a square in the tr,..._plane, as
~
(6-16)
2
in Fig . 6-3.
The functiCI'ls
"''hich represent the actual surface are t:OOse which map this square in the
u..~
plane into the desired surface in the xy-plane . The derivatives of the point functions x and y as defined by Fqs . (6-15)
are
oXd + ~ XA.. dx • auu ~
dy=fuou•~ These equations can be written in matrix form as
[dxl = ax ax du1
r
au~
I'
(6-17)
ldyJ = ~ ~ awj Equatioo (6-17) relates a differential change in the trN'-plane to the corres{Xltldinc; different •al change in the xy-plane.
PlANE SlJRF7ICE
E-::xmtple 6-1 :
Consider the surface in the xy-plane defin
x=Ju+w
y - 2u + 3w +
\.5oo'
Wl.th bomdaries specified by
u
=0
0
~
w
~
1.0
= 1 0 .!. w ~ 1.0 w = 0 0 -< u < 1.0 w = 1 0 < u < 1.0 u
-
'.Ihls surface is shoom in Fig . 6-4a.
-
-
'lbe equatioos for tm boundaries can be
obtained by substuting the appropriate u and w values. u=O
ya3x
u = 1
y = 4x-10
w • 0 y,..
'lhey are
x
w=l y=x+2 'l11e derivati\1\:.:t which appear in the F.q . (6-17) are a~au ay;au • 2
+ w, and 1y/f!w
= 3
+ u.
of the boundaries can be found.
= 3,
~~aw
Co.
1,
Front these partial denvatives the slopes
It fol.lc::Ms that ~ • ";)au • 2 + w
dx
~
au
3
along w • oonstant, and
al0119 u = constant.
These s lopes can be verified in Fig. 6-4a .
I f desired, additional pararretric l.inBs may be a
the surface patch.
For exanple, when u • 1/2 the equatiat for this parametric
line is y • (14x - 17)/4. (l Ox + 13)/U.
hben w • 1/2, the pararretric line is y •
These 0.0 surface paranetric lines are shcJ..m as dashed lines
in Fig . 6-4a .
Similar calculations can be made for an expressiat for a planar surfc'!IC'e written as u = f(x,y )
w=
g(x,y)
Figure 6-4b sh:lws the surfc'!IC'e defl.ned by 2
u -L X
(6-18)
,2-
w:y
'!he l::loormries are 9iven by u- 1
u- 4
I
- 1
w=1 1 -< u w- 4
". ( ' C]-----+---------·-----+---
-<
4
l.i.u.i.. 4
'Ibe equations for each boundazy curve are
indicated in Fi9 . 6-4b .
Many other types
of curves, su:h as o:::nic sections , C\bic splines, Bezier polynanials , o r B-spl.ine •
curves can be used to de fine the boundar-
ies of surfaces.
c,
.c,
~--~-----------------~
L
~VED ~FACE REPRESENTATION
•
Before~
c.,•
with the details
of the rcprcscntatioo of curved three-
F'i.
d..i.rmnsiala.l surfaces, it is necessary to make several deM s1ons. First, we ass1..11re
a surface representatloo h:tserl oo a vector-valued para:retric oonoept. are several reascns for this.
'lllere
'lhis type of representat..l.cn is axis-.in:iepe.ndent:
it avoids infinite slope values with respect to sare arbitrary axis system, it all~'S
the unarrbiUS representation of llllltivalued surfaces or space func-
t.ials , it facilitates the repz:esentatioo of space curves in ha ogeneous coordin-
ates, and it is oonpatible with the use of the
iooal. harogeneous
ooord.il\ate t.ransforrnatioos discussed in Olapter 3. We will assme that a surface may be represented in a
i.e., a surface
l.S
~se
fashial;
cac!V)Sed of a nlreber of patches which are joined together
in sore fashioo at the boundaries .
No
attenpt will be rMde to represent a
oatplete surface analytically. In order to efficiently represent a surface, sane notaticnal dctcils are
necessary.
For consistency we represent a curve as a vector valued function
of a SiJ19le variable; i.e., the cuxve P(t) is P(t) - fx(t) y(t)) for a planar curve and
P(t) = (x(t) y(t) zCt)) for a space curve.
Here t is a paraneter.
As we have seen
in the previous
sections , n:presentatioo of a surface requires two parmreter values . 'lllus, we assurre that a surface nay be represented by a bivariate ~-valued two
•
y
l
b
y
• • J•••
4
y•1'U •"" •\I._ •'
f
n
•
•
b
Figure 6-4
functioo P(u,w) A
I
b
= (x(u,w)
•
J
Plane surfaces. y(u,w) z(u,w)J
curve on this surface m!IY be represented by fixing either u or w, e .g. 1
P('\ ,w) represents a curve along u = ' \ • constant, and P(u,wi) represents a
curve along w
a
w1
m
cono;tant.
A curve an the surface P(u,w) rray also be
represented by specifying sare relationship betl<.leen the
~
pararrctcrs, e.g .,
'I11is relationship might define the intersection of a plane witl, a surface patch, for exarrple. f (ulw) • 0.
A poinL on the surface rre.y be represented by specifying values of roth pararretcrs, i.e. , P ('\ 1 wi) , or by specifying ~ independent relat.i.cnshi.ps beO.'eell the pararetcrs1 say, f(u,w) = 0 and g(u,w) - 0. Note that the latter representation does not necessarily define a single fixed point. We will consider that a surface patch can be built up ftorn knoNn data:
i.e. , we oonsider a surface to be constructive.
With this in mind we adq>t
the fol..lowi.ng convention:
data. 0 is the vtX:tor defining the surface coost.ructed from the input data P .
P is the vector obtained fran kr'lcMl or input design
'Ihus , P and 0 coincide ooly at the specified data points.
6-5 BILINEAR SuRFACES One of the sinplest surfaces is the bilinear surface.
Coosider that the
four corner points of the surface are given in the u ,w-plane by P(O,O), P(O,l), P(l,O), and P(l , l), the corners of the unit square. tte wish to construct a bivariate functioo or bilinear surface Q(u,\.,t) where (u,wc(0,1)) allows us to
linearly-interpolate any p:>int on the surface . 'Ihis yiel ds Q(u,w) = P(O,O)(l-u)(l-w) + P(O,l) {1 - u)w+ P(l,O)u{l-w) + P(l,l)u,.; or Q(u ,w)
(1 -
=(
u) u J rP(O , O) P(O,l)]
LP(l , O) P{l,1)
[1 w
wl
J
(6-19)
,..hlch is sha.m in Fig. 6-5 .
Here it is easy to venfy t:hllt
Q(O,O) : P{O,O,, etc. sur~ace
Also, a (Xlinl in the center of the
(u = w = 0.5) is sirply
(P(O,O) + P(O,l)
QCO . 5, 0. 5)
PCl,Ol + P(l,l))
+
4
general, a bilinear surface and t-'le (Xlsi tion vectors of t!-.e ccmer points of the surface are three dJ..""eJ'\Siona.l. 'Ihis is perl'>.aps easier to sec i f "-"e write Ec;. (6- 19) in a rrore o:xtpaet IT'dtrix forr: . In particular , tn
Q(x(u,w), y (u,w), z(u,w)) =
where IJ is a (XlSiuoo vector on •
~
L~
interpolated surface , P iS tile ll\3tri.x of )
• •
PI(! II - - -- - --,l"illl
Ql
PtU 1Sll:!5t
•
p ( I 01
Pl0.01
.. ..
•
a.,
•
,_ .
•
-
l
• QtfJ.O)
u
••
. ..
Figure 6-Sa
•
•
""L '
•
-~~01.11
•
Qr flJ
..,
•
-•
• •
Planar bilinear surface.
r (1
positiOO \.'eetOrS o: the comer points, a"ld U •
- U) (! - t:.)W U(l - W) u..-) .
An exan-plc using t:-..o-dilrensional posit.J.on vectors is shown in Fig . 6- Sa.
J..n exanple of a th.-ee-dilrensional bilinenr surface 1s sha.m in Fig . 6-Sb .
Fig. 6- Sb tr.e
as~
In
three-dir'er.siooal position vectors were four of the el.CJht
corners of a unit cube. /In algorithm
dix
5-6
,..;uch will create a bilinear surface patch is given m Appen-
c. loFTED 00
R1u.o
SuRFACES
Lofted cr ruled surfaces are freq..lently associated with the aircraft or
shipbu..l.d....r.9 ..n::lw;tries. bor.Jndary
c-.JI:VeS
To obtain a lofted or ruled surface, asS\m'E! that bo.o
asscx:utted with the opf()Site s;dtts of
u ,w-pl..ane arc knc;r..n, say, P(u,O) and P(u,l). t-lm t.E<:h.ruques discussed
tained by
1
.i.r. Chapter S.
t.~
11\
the
'lllesc li\3Y be defined using
a."l}'
A ruled or :ofted
u:early 1nterpolat.l.ng between these
t"-'0
unit square surf~
of
is then ob-
curves, as stown in F'i.g. 6-6.
The intcrpol.atioo ~ is
Q(u,w) • P(u,O) (1 - '-') -. P(U, l)\o.'
(6-20)
1'<1 II
I O.S / )
\\•COnSI:lOI
10
I'll 01
I
u-
con~tant O.S
'
II lj
I
•
If
z I
I
i
)
I
I I
Pcu.IJ
Q f)l.tnc Plu.lll
l>fO.(ll
I0
[x(u,w), y(u,w), z(u,w)J •
((1-u)(l-w) (1-u)wu(l-w) w) P(O ,Ol
p(l I l) ((1-u) (1-w) Cl-u)w u(l-w) w) 0 0 0
0 1 0 l 0 1
01l Figure 6-Sb 'lhree-dirrensional bilinear surface. and P(l , \,.) are knc1Hn.
O.St
Figure 6-6
P(O,l) P(l,O) -
Ql~
lofted Surface.
again note that
0(0,0)
= P(O,O),
etc.
F\.trther note that the edges of the mter-
polated s\lt'f.ace and the given data curves
are coincident, i.e., O(u,O) = P(u,O) Q(u,l) = P(u,l) Alternately
we can assme that P(O,w)
'Ihe lofted surface is then given by
Q(u,w) • P(O,w) (1- u) + P(J,w)u N}ain it
(6-21)
is easy to shc:M that
0(0,0) = P(O,O), etc.
Ex:anple 6-2: 'I\.'0 curves
sec.
U>f1w SURFJ\CE
P(u, O) and P(u,l) are given by cubic spline se<.JrellW {cf
5-3) .
Ne wish to find the equation for the lofted surface bebo.U!ll these two l.xlundarlcs. 'l1'le cubic spline 5Cg~Ml'lt.s are assu-ned to have cl.arrped ends. 1\.ssi.M:! for
P(u,O) p1 -- (0 p2 .. [0 P'1- [0 P'2- (0
and
0 l 1 1
0] O! 1) 1)
P(u,ll pl. [1 0 p2 = [1 l P'1- [0 l P' • (0 1 2
0] 1] 1] -1]
1-l..ence, csi."\g &i . {5- 19) ,
P{u,O) = [0 0 0] + [0 1 l]t ~ [0 0 -3]t2 + {0 0 2]t3 P(u,l) • [l 0 0) + [O 1 l]t • (0 - 0.6213 0.7929)t2 + (0 0.2929 -0.707}t3 Equation (6-20) then yields
ocu,w) . lw 0 0] • (0 l l)t 'Zhi.s is the surface
fr-7
Ltt-lfAR
+
~
{0 - 0.6213w f- 3 + 3 . 792~~)]t 2 - [0 0.2929w (2 - 2.707W))t 1 in Fig . 6- 6.
Qxx.Js S~FACE
If the four bounda...-.y curves P(u, O) , P(u , l), P(O ,w), ar.d P(l,w) are I
then it
rni~t
be assuned that a si.'Ipl e
and (6-21) in the
~
S\:1\
of the lofted surfaces
di.rections w::>uld yield "t:re desired result.
r..qs .
(6- 20)
If t:lus
1.5
dale, then
Q(u,w) • P(u,O) (l - w) + P(u,l)w + P(O ,w) (1 - u) + P(l,w)~ (6-22) tk::Me\.-er, examinatioo of F.q. {6-22 ) at the ex>rners of the surface patdl yiel.Cs Q(O,O) = P (O ,O) + P (O ,O) , etc . arxl at the edge.s,
Q(O,w) • P(O,O) (1 - w) + P(O , l)w + P(O,w) neither of which oorrespoods to the original data as they should. Examination of F.q . (6- 22) sh::lws tl-.at the problem occu..""S because t.~e corner points are oounted bdce, since P(O , O) is contained in both the P(u ,O) and P(O,w) be~ curves. 'lhe oorrect result can be cbtained by suhtracting the excess contributicn t~
surface due to duplication of the oorner points. 'Ihis yields Q(u,w) • P(u,O) (1 - w) + P(u,1)w + P(O,w) (1 - u) + P(l,w)u to
(6- 23)
- P(O,O)(l - u) {l- w) - P(O ,l)(l - u)w - P(l , O)u(l- w)- P(l , l)uw An exarrple of such a surface is ~ in Fig. 6-7. It can be readily stn.n that at the oorner, Q(O , O) = P(O,O) etc.
ard Along t"le bo..ndaries,
Q(O,w) ==- P(O,w) Q(u, l) • P(u,l) etc.
\\hlch .inplies that the interpolation is oorrect at the oorners and bo\.t\daries.
'Ihis linear blend of the four boll'ldacy curves is +Xe s.inplest Coons surface. A rrore general COOns surface is discussed in Sec. 6-17 and Jef. 6-4.
notation,
at.
In vector
(6-22) may be written ctS
Q(u,w) • Ill- u) ulfP!O,w>J + (P(u,O) P!u,l>)[l-wj- 11- u u}rP(O,Ol P(O,ll 1 [1
LP
IPo,o> PU,uJ
w
-w] w
or nore
P(O ,w) 1 - w -P(l,O) -P(l,l) P(l,w) w 0 1 P(u,O) P(u,l) 'ltle functioos (1 - u), u, (l - w), and w are rolled blending fmct.ions because t:Ney blend the boundacy curves to produce the internal shape of the surface .
Exarrple 6-3:
LINEAR cx:x:Ri SURFACE
Let the four boundary "curves" be the d.iagc:nals of four of the sides of
a unit cuve as sha.m in Fig. 6-7. Coc.ns
'''e wish
surface between these boundaries. For the oenler point oo the surface, ~(0.5,0 . 5) •
[0 . 5 0. 5 1)
to fiM the center point on a linear
~· (6-24)
0
0
yields
01 0.5
-1 -1 lJ 0 . 5 0. 5 0
0.5 l
Other points en the surface are calculated in a similar manner.
An algorithn
which will
~ate
a
linear COOns surface as described by Eq. (6- 24) is given in Appendix c.
6-8
BIClBIC Ste=ACE PATOi '1lle previous sections have discussed
the representatioo and descriptioo of
s.inple surface patches fmn a CO'ICEptual
point of view. Ne l"'f:M turn our at tention to sane practical aspects . One of the nore useful patch descriptions uses para-
rretric cd>ics for the edge curves P (u,O) , P(u, l) and P(O ,w) , P(l ,w) and the blend-
It
z
ing functions .
IJJ
'l'he blending fU'Ictions
for this schene are derived below .
)
Fran our previous work (cf Sec .
''
s-3)
recall that the fotm of a parcmetric C\bic is (cf
BJ.
(5-ll))
P(t) ~ B + 8 t + B t 2 + B t 3 (6- 25) 1 2 3 4 Figure 6-7 Linear Cbon' s surface . "'here P ( t) is a vector-valwd fmct.ion having x-, y-, ~tpcue.nts. For cawenience we limit the range of the paranenter to 0 ~ t ~ 1; i.e., we use a nox:rna1 ized pararretric cubic. For the bo\J'ldary curves of a surface, t "" u for one pair of curves and t • w for the other pair.
We
no,..t
derive the blending functions which will allow interpolation of
the points oo the unit square beb.~ the bolo paranetric Cl.bics describing the edge curves , i.e ., bebo."een P(O) and P(l) . '!here are four \l1'lltnam coefficients i.n BJ. (6-25) , and thus we nquire four i.ndepenoont equations . Pecalli.ng that the derivative of the pararretric cd:>ic is P' (t)
= 82
+ 283t + 3B4 t 2
(6- 26)
and letting t • 0 or 1 i.n B:Js . (6- 25) a.nd (6- 26) yields P(O)
(6-27a)
= ~
: 8 + 8 + 8 + 8 4 1 2 3 P ' (0) = B 2 P' (1) = 8 + 2B + 38 P(l)
2
3
Writing Eqs. (6-27) i.n matrix fonn yields
4
(6- 27b) (6- 27c) (6-27d)
SURF71CE OESCRIP'rim h.\10 GENF.:AAT!Gl 171
1·
P(O)
P(l) p. (0) P' (1)
J
rs4
0 0 0 1
(6-28)
1 1 1 1 83 0 0 1 0 82 3 2 1 0 (J31
or p.,. MB Solving~ .
(6-29)
(6-29) for the coeffid ents, l.• .e . , !or B, yields B !.f"lp -1
M
=
2 -2 -3
1
l
(6-30}
(6-31)
3 -2 -1
0 0 1 0 1 0 0 0 'lhis agrees with F.q. (5-22) .
S\Dstituting F.q. (6-30) into F.q. (6- 25) tlnd re-
arranging yields
P(u) = F1 (u)P(O) + F2 (u)P(l) + F3 (u)P' (O) + F4 (u)P ' (l) where in general, ~i.th t as the paraneter, (F (t) F (t) F (t) F (t)) 2 3 4 1
= (t3
t
2
t 1)
f
2 -2
1
(6- 32)
1
(6- 33)
-3 3 -2 -1 0 0 1 0 1 0 0 0 'Ihe F i ( t) ' s CM be oonsidered as scalar blending fU'lct.ioos for the pararretric
ctbic.
'lhey blend the quantities P(O), P(1), P'(O), and P ' (l)
a cx:nti.nuous
C\l.t'W.
to !onn
Figure 6-8 sho..rs these blencll.ng functions.
toJe nt:JW utilize these results to gererate a b1C\bic patch. ~ure
~ther
~·as
'lhe sane pn:r
used to develop the fonn for the bi 1 inear patdi is enployed. ~~ first oonst.r\ct surfaces which will satisfy one family of boundaxy ccnditioos along the edges where u"" 0 and u = 1 and than construct another surface whidl will satisfy boundary cxmditions ala1g the edges where w = 0 and w ~ 1. ~ then carbine these results to obtain the bicubic patch. Before prooeeding, sare notatiooal details are islportant. For oc:nvenien~ in later work we int..roduce a superscript notatioo for the derivatives of positioo wctors . In particular that
Pa (u) .. a
aua
\F\li
p (0,11
P(l,O)
P(l,l)
0
r 1CwJ PO,l(l,O) ~O,l(l,l)
1
1 P '°Cl,O) P1 ' 0 (l,ll r 1 •1 c1,01 P1' 1 Cl,l)
(6-36)
f";> (wl
p-• co,OI P '°CO,ll rl.lCC',CH .,t,IIO,II
l
j
f
3
Cw)
r 4 (w)
Alternately Fq. (6-36) may Le written tJore oonpactly as Q(u,w) "' (u 3 u 2 u l)N P NT [w3 w2 w l)T
(6-37)
~re N is tho squaro I"Adt.rix given previously in Eq. (6-31), ll1ld P
matrix given in Fq. (6-36).
is the
~r
NT is the t.ransposc of N.
Examination of Fi:JS. (6 36) and {6-37) shaRS t.hat is defined by the Cl.bic blending functions
biai:lk surfac:r patch
r 1 , F2 , r 3, F4 , four Cl.bic
bo\a'ldary
curves , four oorner points , eighL tanCJent vectors at the oorncrs, and four twist wctors at the oorners . 'nle 4 x 4 P-rnatrlx may be considered as a boundary oondition rr..).trl.x.
It
cxntains only infonratioo aboul oome.r points, oorner tangent vectors , and oomer twist ~ctors. For a given surface patch all thP elennnts in the P-
mrtrlx are kno.m oonstants. can be c lass1tied as
Examination of the P-matrix
r
CoTJ"Pr
p;:
I
s}n..os
that the t:errs
w-tangent
ooonlinates : vectors
_ _ _ _ _ _ _.,. _ _ _ _ _
u-t.angcnt
1
vectors
1
I
Twist
vectors
Sinoe for the surfao:> palc:h to be physically ~le , adjao>nt boundary curves nust intersect at a oomer, calStraints are placed on the specific func-
tioo used to represent the bou'ldaty curves . For eY.anplc , at oo~r A in Fig. 6-9, P(O,w) ... P(u,O): at corner B, P(O ,,~) = P(u,l) 1 at oomer C, P(u,l) • P(l,w); and al corner
o,
P(l
,~r1)
= P(u,O).
A carp t ete deri vatim of the oonstraints inposed on the blending functialS
in order that they satisfy certain booodary CXX"'ditlons was derived in
~f.
6-4
and is giwn in Table 6-1.
The
sh~
of a bicubic patch is oontrolled by the ca'\tents of tho 4 x 4
P-rratrix, i.e., by tho l.ocatioo of the oorner PJSitioo vectors and the fT\agl\itWe and direction of the correr tanqent and bast vectors . Figure 6-lOa
stn.os the effect of c:hanging thr> magnit\XP but not the dirccticn, of the oorner tangent vectors along the (O ,w) ari
Figure 6- l
the effect of chanqinq the direct.J.on , bul not tl'lr> JTagnitude , of these sare comer tangent vectors.
Figurr.s 6-lOc and 6-lOd show the cff<'Ct o f addi.ng a
twlSt. ~ to an initial surfacr; patch wit.h zero b.tist \JIC!Ctors .
174
NM'liEM\TICl\L EIJOJ-1ENI'S FUR OO.v.Pt7IER GRAPHICS 'lhe bicubic surface patch has been suc-
cessfully used to design surfaces (cf P.efs. 6-5 and 6-6). 1 p 11.1)
Ho..-e'lleJ", several
discdvant.a9es exist when usmg bicubic surfaro patdles in a carputer-aided design system.
'1he rrajor disadvantage
exists because three different quantities,
position, tangent and odst vectors all generally of different orders of rragnitu3e
nust be specified
to describe the surface .
For ex:anple, for .a patch 10 units long,
the tangent vectors mi¢\t P
an order
of n-agnitu.Oe of 10 units, the twist vec-
zw
b
ha'v'e
tors 100 units, and the position vector 1000 units.
FUrt:har, the user typically
lacks intui tioo when creating surfaces by changing elemnts in the definiJ'Ig rratrix.
For ex.oop1e, the oorner tangent vector P 0 • 1 (0,0) p:>ints toward the interiol. o f
Figure 6-9
Table 6-1 Bounda!y Corxli.tion Inposee Constraints on
Surface Blending fUnctions
u
•
=0
or w = 0
u
= 1 or w = 1
Fl
1
0
F2
0
l
F3
0
0
F4
0
0
F' * 1
0
0
F'
0
0
F'3 F'
l
0
0
l
2
4
*Prirres indicate differentiatioo with respect to the indepe.r.dcnt varicble either u or w.
0
a
t>
VU utll o n o r IOO f:e OI \'Cli
P0 .1(0.11
\ a rt
ma,n uudc
~
dared tun
...-......
pO.I(O.Ol
[)
c
Ze ro
I\\'"' a t C
Figure 6-10
I
11\ II C ' " ' ' ' B l (
COntrol of bieti:>ic surface patdles .
0 1 t:.angent vector P ' (0 ,1) points ;rway f.ran the inte.rior of the boundary curve
as shom in Fig . 6-ll . Also note that increnenting the y-cx:xrp::l')ent of the ex>mer tangent vector P0 ' 1 (0 , 0) causes the (0 ,w) boondary CUlVe to bend q:> , 0 1 "hereas incresnent.i.J'lg the y-cx:rrponent of the corner tangent vector P ' (0,l) an equal anount has the opposite effect at the (0 ,1) corner. pected behavior occurs when inc:rerenting twist vectors .
'!be sare
ll"teX-
EquAl dlanges in all
twist vectors cause qJpOSi te surface beOOs at the (0 , 1) and (1 ,0) ex>rners cmpared with the effect at the (0 , 0) and (1, 1} oorners .
the
~ts
Thus, manipulating
o f the P boundary ex>ndition matrix can be cxm.fusing.
An algorithm which will generate the bicl.lbic surface patch described by
F.q. (6- 37) is giwn in Appendix C.
6-9 THE
F-PATOi
In order to eliminate sare of the prcblems lolhidl oca.tr when manipulating
bic\bic surface patdles, a si.JTplified version originally due to Ferguson (cf
Ref . 6-7) , called an f'-pat.ch , is sorret.:i.Jres used. surface patch with zero blist vectors .
An F-patc:h is silrply a bicubic
Ql.ly first-order CXXltinuity is possi-
bl e across the boundaries of an F-patdl, because the surface is oonstrained to leave zero cross derivatives at tile oorne.rs . This on occasion may lead to
176
l~J\!HE.l>oo-ICAL
EUMNI'S FOR C'G'.PUTER GRl\PHICS
surfaces which are not sufficiently srrooth for sene appllcat.la15 . Nevertheless , it ~-~6-:-:-
pO.Iw o
is easier to oonstruct and rodify an F- patch, and they are suitable for many types of surface definition.
pO lw 1
In partic-
ular, F-patdJeS have proven to be useful \
)
for axisymnetric surfaces, e . g ., vases ,
\
cups , bottles, aixa:aft bodies , etc .
a lnattal cunc
6-10
BEZIER SuRFACES 'lhe previous sections have dis-
cussed a n\.Dibe.r of aspects of surfaoe
z
description .
In all cases the tacit
as5\ltPti.on has been made that the necesA
sary mathematical infoxmaHoo such as
1
position, tangent, or 0.1.st vecton; or blend.in; functicns is available. l:b..>Figure 6-11 Effect of change in ever , providing this infoorat.ion may be end tangent vectors an ext:.renely difficult for the user, espethe bo\.l'daxy curve shape. cially a nonmat:heiMtical or 11011-Caq:lUbeJ:oriented user. A mnber of sc:he!nes have been devised to overocJTra this di ffib. f1 nal tunc
culty .
Prd:>ably the rrost S\ICQ?Ssful o f these is the extension of the Bezier
curve (cf Sec. 5-7)
to surfaces (Ref. 6- 8) .
'lhc surface equation for a Bezier surface patch can be written in the
form of F.q . (6- 36) .
•
Osing the biocmial rep~entatioo of a Bezier curve given
in Sec. 5-7, the equation for a surfac::E patch with boundary
CUI"VeS
described
by four point Be-zier polygons, i.e . , a 4 x 4-point Be2ier s urface patch is
given by
3 P(u,w) = [(1 - u ) 3(1 - u) 2u 3(1 - u)u2 u 3)18)
(l -
w) 3
3(1 - w)
~
3(1 - w)w2 w3 8= ~( 1,1 )
8 (1, 2) B(l , 3) 8(1, 4) 8(2 ,1) 8(2 , 2) 8(2 , 3) 8(2,4) 8 (3 , 1) 8(3,2) 8(3,3) 8(3,4) LB(4,1) 8(4,2) 8(4,3) 8 (4,4) 'lhe B-tcnsor is O::Cip::>scd of the position vectors of the defining polygon points , as sho,..rn in Fig . 6-U . In the Bezier fornulation , only the oorner
points A, 8, C, and 0 octua..lly he en
surface. Thc {X)int 8 , 2 dafines the 1 slope vector fran the first to the secthe
end p:>int on the first Bezier p:>l}'9XI in the u-di.rection .
The {X)int 8 ,
de.f.ines
2 1 the slope vector fran the first to the
)
seocnl p:>int on the first 8ezier tx>lygon in the w-clirectioo.
The p:u.nts • 8 1,3'
8 2,4' 83,4 ' 84,3' 8 4 , 2' and 8 3,1 are defined in a similar manner.
'lbe tx>ints
s 3 , 2 interior to 2 3 3 3 the defining £X>lygonal surface are used
~, , 8 , , B , , and
2
to define the twist vectors at the cor-
---
ners A, B, C, and 0 respectively.
"
( 1 - .....
As shown in F'ig. 6-12 the Bezier
surface is cnuplete1y defined by a net of design {X)ints face .
the p:>lygoo.al sur-
Cl'l
'Ihese £X>l.Ilts
--· Unundar )
serve to define a
-
,-,r Ocl ttt
Llc ~li!O
Sur (;aH
•ur(act
ntt
two-pararreter family of Bezier curves on the desiCJll surfac:o.
Figure 6-12
To generate a para-
4 x 4 Bezier surface ~ture .
rretric curve on the Bezier surface shown in Fig. 6-U, let the pararretric curve be defined by 0 ~ u ~ 1. 0 and w =
'i.,
where 0 <
'i. ~ l. 0.
Before this curve
can be generated, the elerrents in the s-tensor given by Eq. (5-53) DUSt. be known.
For w =
S,
the matrix product [81 (1 - w)
3
(6-40)
3(1 - w)~ 3(1 - wlif w3
w=C
1
produces four position vectors P , P , P , and P for the Bezier polyyon which 4 1 2 3 defines a Bezier C'UI"Ve with a pararretric value of w "" The point P lies 1 on the surface bo\D'ldary OW, and the p:>int P lies on the surface bourdary 1w. 4 The points P and P are slq:e des1gn points which lie o ff the curve. 2 3 'nlen, the rMtrix prodoct
'i..
P(u,c ) 1
~
3 [(1- u ) 3 3(1- u) 2u 3(1 - u)u2 u ) P
1 p2 p3
p4
(6-41)
giws the e>cpressioo for a point oo tbe surface along the curve w .,. eadl value
U1
c1 .
For
F.q. (6-41) gives the oorresponding position 1.>ector for the point
on the surface. The bol.lldary C\.IJ:'-~ for a Bezier surfaoo can be defined in an analogous
mar.ner to the Coons surface boundaries.
In fact, a Bezier surface can be placed
within the family of generalized Coons surfaces described in sec . 6- U .
exauple, aloog the boundary u = 0 1 0
c
=
~
w
~
For
1.0, the Bezier curve is
(6-42)
(51, 1' 52, 1 ' 53 , 1 1 84 111 (1 - w) 3 3(1 - w)Zw
3(1 - w)wl w3 or alternately, (6- 43)
z. ~
'Ihe Bezier boundacy curve for 0
~
u
~
1.0 1 w = 1. 0 is
1
c • [(1- u) 3 3(1 - u) 2u 3(1- u> 2 u3J 54 , 1
(6-46)
8
412 8 41 3 84 , 'Ibe equations for the other two boUI".dal:y curves can be written directly 1 with
the aid of Fig . 6-12 . It is saret.iJres desirable to create a surface patch that is rot described by a square tensor. •
a couplex surface.
'lhi.s can be used to al lo,.r sinple curves to be bounded by As an exanple, the surface sho.m in Fig . 6-13 has three-
point Bezier curve boundaries along u • 0 and u
curve l:x::undaries along w ..., 0 and w • l. 0.
=1. 0
and five-point Bezier
The equation for the bo\.U'&ty along
w • 0 is given by
C
e
[(1- w) 2 2(1 - w)w
'llle equatioo for the bouOOary along u
C
a
=0
w2J
(6-47)
is
[(1- u) 4 4u(l - u) 3 6u2 (1- u) 2 4u3 (1- u) u4 J
(6-48)
--
I
)
7
a,,
---v. -oFigure 6-13 3 x 5 Bezier surface.
'lhe surface equatioo is then given by
P(u,w) = ((1- u)
4
2 4u(l- u) 3 6u (1- u) 2 4u3 (1- u) u4 ][Bl f (l- w) 2
t~l
(6-47)
-
w)w
A five-point Bezier boundary curve has an advantage over a four-point
curve, since a change in the central design point does not effect the slope at either end of the bouroacy cu.rve. 'lllus, t.l)e surface shape can be changed without disturbing the slope design points, which may then be independently spe<:J.fied to satisfy continuity requi..rerents with adjacent surfaces. '1lle Bezier surface discussed above can be represented in the fonn of a cartesian product surface. In particular the surface .l.S gl.Ven by
m
m
L L: Bi+l , j+lJn,1. (u)Km,J. (w) i • O j-o
Q(u,w) •
{6- 48)
\o.t lere by anal ogy with F.q. (5-63), •
•
Jn,i- ~)ul.(l - u)n-l.
(6- 49)
f.!iwi (1 - w)m-j
(6-50)
K
. •
n,J
J
and m and n are one less than the mll'ber of pol}'9011 vertices in the w-
aoo
Bezier has been successful in using his technique in a production environ-
n-ent as described in Ref. 6-8.
In the Uni.surf system at Rclgie Pcnault a free-
hand curve is first drawn with a pencil in the xy-plane to give y • f(x) .
A
o::rrputer controlled drafting lT\!lChine used as a ctigither receives the input ooordinates of the polygon vertices which roughly define the curve.
'1lle result-
ing Bezier curve is then drawn by the CXZtpUter controlled drafting rrach.ine.
rn::rli ficaticns are needed, the
pol~t
If
vertices are adjusted and the curve is
redr1Jwn mUl the designer is satisfied.
A th:reE~immsional curve is oo~
by repeating the process to produce z : f (x) •
nus rrethod is used to define the four bomdaJ:y curves for a surface .frane-
The surface definitioo is then catpleted by specifying the coordir\ates
\o."Ork.
of t.ie rc:na.in.ing surface a CCITpUte.r prograrmed to described surface curves dice a ional
design
~rk
points.
'lhls data is then input into
prrrlx:e rotated, scaled, or perspective vieo..'S of the
or into a c:arputer ccntrolled cutting rnach.iJ1e to pronodel !ron a block of polystyrene foam.
A. P. Annlt (~f . 6-9) has also developed
which makes use of Bezier surface patches.
a design system called MJLTim
His system allows easy use of various
types of Bezier boundary curves, rangin
curw are concerned wi. th local control. This is because the Jl'O\Ie'!'ent of one point on the polygonal generating surface will affect the entire Bezier surface .
'lhls can be a distinct disadvantage to the designer, especially for detail
work.
In order to obtajn local control a patch nust effectively be sulx1ivided.
'lhe use of a B-spline basis, i.e . , by using B-spline curves as bl.endin<} fmc-
ticns, eliminates this problem.
This is discussed in the next sectioo.
An al9Qrit:hn which will generate a Cartesian product surface based oo
Eq. (6-48), using eezier curves as the blending f\.D"'Ctions, is given in AWendix
c.
6-11 B-sPLINE SuRFACES •
IITplernentatioo of 8 spline surfaces can take many foxms.
Perllaps the
si.nt>lest is the cartesian produ:t surface analogous to the Eozier surface of the previous section . Fbr a a-spline surface the corresponding Cartesian prodlX:t surface is rn
Q(u,w) =
rn
L: L i=O j=O
8.+1 ' +lN. k(u)M. L(w) ~
,]
~.
],
(6-51)
\o.'here by analogy with F.q. (5-78)
N.
l., 1
(u)
N. k(u) 1.,
l i f x~. !. u .!. x.~ + 1
l
0 otheiwise
"" (u - xi)Ni,k-1 (u) + (xi+Jc - u)Ni+l,k-1 (u)
xi+k-1 - xi
xi+k - xHl
and the xi are the elem:mts of the k knot vector discussed in Soc. S-8 .
(6-52)
Further,
i 1 i.f yi s.w s. yi+l
~ , 1 (w)
lo~ise =
(w -
y.) M. L-l (W) J
],
+ (yj+L - w)Mj+l,L-1 (w)
yj-+L-1 - yj
yj+L - yj+l
and the Yi are the elat\.!nts of the L koot vector discussed in and n are
ale
(6-53)
sec.
5 8, and m
less than the m..1rber of vertices in the dt:fininq (.Olyqons in t.~
~ and w-directions respectively.
def.ininq polygonal surface, as
The Bi+l, J+l are the pos1boo vectors of. the
srown
for exanple in fi.q. 6-12.
As with B-spline curves , knot vectors with vanous dL--qroes of rrultipll.C"ity
can be defined in either the ~ or w-directions. ~r , the above fonrat:ion requires that each defining polygon in a given di.J:ectia'l must h.Jve the sarre degree of mult iplicity in a given direction.
Alternate for:nul.at.u::ns could
allow various degrees of knot IT'l.llt.iplicity in a given di.rectl.on in order to i.ncrGase the flexilJility of local ea1trol. Further di.scussioo of h::w &-spline curves can be used to form surface patches is given by
s.
A. Coons, ~-1. J.
Gordon , R. F . Rieoonfeld, and others in Pef. 6-10.
6-12
teeRALI ZED ~S ~FACES Exatninatioo of the results di.scussed above SUJ9f!Sts a uore qeneral repre-
sentation for a surface.
Follc:Ming Forrest (Pef. 6-11)
scalar funct:.ioo B
'lbe
r,1. (u) .
~
introduce a
e.r ,1• •s serve to blend the boundary oanditions
and/or boundary curves to fonn a surface.
The r subscr ipt indicates that the general blen:l.i.ng functioo is TNeig.hted by a single-valued function representing a bourx1ary derivative of o.rder r .
For curves , r = 0; for tangent vectors , r = 1; and for curvature or twist
tors, r
==-
2.
vec-
'I1le i subscript indicates the position on the unit l.ll.'-square of
the function bel.llCJ weighted by the
sr,i . Here
we assi.Ure that u = u
1 is inplied:
e . q ., BO ,l (w) is the ble.ndi.ng function weighting the curve P(u,l).
s0 , 0 Cu)
ls the b l ending
f~oo
\\-eighting the curve P(O ,w) .
Forrest (~f . 6-10) has pointed out several properties of these blending func-
tions.
If a patch is specified by boundary curves alone, then
f ~ Oij lS the
.(j) =6 ..
0, 1
lJ
Krooecker delta , i.e., 61 j ::. 1,
l
= j and 6ij • 0, i -,J j ·
As an exanpl e , recall the cquaticn describing the bilinear surface, i.e.,
182
M'M"HEMb.TICAL EUM:mS FOR CXl-1PtJI'ER GRAPHICS
Q(u,w) • (1 - u) (1- w)P(O,O) + (1 - u)wP(O,l) + u(l- w)P(l,O) + uwP(l,l) (6-19) Now suppooe t.hat we replNP the linear blending functioo (1 - u) by
still require that
s0 , 0 CO)
.. 1 and
s0 , 0 (1)
o.
=
s0 , 0 Cu)
but
Further, we replace the linear
s0 , 1 (1) • 1, (1 - w) and w by s , (w) with 0 1
blending function u by SO,l (u) and require BO,l (0) • 0 and by
s0 , 0 (w)
s0 , 0 (0)
s0 , 0 (l)
-= 0 Equatioo (6-19) m:ty then be written as
and require that
= l and
B , (0) and s , (1) • 1. 0 1 0 1 O(u,w) • r 0, 0 CuJB0 , 0 !wJP(0,0) + e0 , 0 CuJ60 , 1 ~J P(0, l ) • a , 1 CuJ & , Cw)P (l,Ol 0 00 \\hlch may be written as l
Q(u,w) •
L s0
i-o
l
. (u)P(i,w) +
, l.
E a0
j~O
l
j (w}P(u,j) -
Noting that in Eq. (6-54 ) r •
o,
0 1 Cu)~O,l(w)P(l,l)
l
E E
i=O j=O
'
~ s ,
B
0 .~
(u}B j(w)P(i,j) (6-54) 0'
we see that thesla s.inple bilinear
COOns
surfaces admit only positional continuity. The bicubic surfaces discussed in Sec. 6-8 can also be written in a rrore general form.
First let's coosider the bicubic lofted surfaces represented by
fl:!s . (6-34) and (6-35) . In the
lfOre general notatic:n Eq. (6-34) m:ty be written as
and F.q. (6-35) as (6-56)
Otu,v) •
~<~here
the blendi.ng furx:tioos , i.e., the Bij ' s, cbey the relatioo.s p
6 (B
r ,1. (u) 1
p
= Br , l.• (Uj)
= 6 pr . 6l..j
\.F\1, )
Thus 80 , 0 co>
s1 , 1 CO)
• 1 , 80 , 0 <1> - s0,0 co> - s0,0 cl> = Bl,l (1) = Bl, l(O) = 0, Bl,l (1)
•
o
a
1
catparisoo of F.qs. (6-55) and (6-56) with E):}s . (6-34) , (6-35) and r ecalling D:j . (6- 33) shows that 3 (8 , Cu) & , 1 Cu) f\J,O(u) B , tu)J,. (u 00 0 11
i
u 1)
2-2 1 I • (F (u) F2 (u) P3 (u) F~ (u) J 1 -3
3 -2 -1
0
0
1
0
1 0 0 0
and 3 w2 w l)
(w
2 -2 1 1 -) 0
) -2 -1 0
1
0
1 0 0 0
(6-57a)
'l\lrru.ng no.t to the bicubic surface patch itself,
we see that
~.
(6-36)
nay be written in the general notatioo as l
OCU,v ) ..
l
L E
11
pr 'OU ,w)ll
• lui •
r,.
1-
By substitutioo
L ~
TO ~
11
1
1
PO,e(u,j)l!• •'"'' - L }' L L rr,all.)l r llul • •• l~ 1";1) roO ..0 ' '
J""' (6-58)
and/or differentiatioo it can be sho.m that at the oorncrs Q(i,j) a P(i, j) Ol' O(i,j) • Pl,O(i, j)
0 o,l(i, j) • PO,l(i,j) . ') -- pl , l(l., . J' ) 0 1 , 1(1.,) ard aloog the boundary curves O(i,w) = P(i,w) Q(u,j) • P(u,j)
0o, l(u, j) 0
N:)tice that since r
=1
~
PO, l(U, J' )
1,0(.1. W) -_ Pl'O(.l. W) 1 1
here, normal derivatives can be satisfied across patch
boundaries to first order i.e. , bicubic patches may be joined with first order CXI"'t.inuity acruss the boundaries. Forrest (Ref . 6-ll) has sho.om that it is p:lSS.ible to generalize U1e Coons
surface descriptioo su::h that higher order bollndary cx:n:litians can be satisfied.
Assuning that we wish to satisfy nonml derivatives to order m oo u "'
1
order
llt
and to
n . oo w = wj , then the surface boundary curves are given by the functicns
pr, O(i,~) ,
with 0
funct.ioos by 1
Q(u,wl •
L
1o.()
.l
~~
,:::()
~ r ~m1 ,
ar , .1.. Cu) r•
and B
0 ct,w)llr
. (w) . s,J 1
,1
(U) •
~ s ~nj ,
and P0 ' 5 Cu,j) , with 0
L
"1
Lo
~ ..
and the blending
The surface equation is then
0 1 P ' Cu,jlf 11 , M-
,'
1
l
01
.,
E 1: riO() L 1: Pr' CP()
110() }II()
8
6-59) (1.,)1&,. 1 cuHI8 M , ,1
where the blending functions have the properties p
Br, i C~l = opr • 6ik
for 0 ~ p ~ m1
p
as , j (~) = 6ps . ojk
(6-60)
for 0 ~ p ~ nj
Equation (6-59) represents the rrost general foon of a CCons surface.
It should
be noted that rrost inplenentaticns do not utilize this fonn, but rather ooe of
the special cases discussed previoosly. Forrest (Ref. 6-11) points out that i f in Eq.
1
PO , s(u, j) •
(6-59)
fi'i
L L
pr , s(i,j)Br i (u)
i•O r=O then the last two t:ems of Eq. (6-59) cancel
'
am the surface equatioo is
1
o =
rn.1
r 0 L: L: p , a . i=O r=O r,l.
which is a typo of ruled or lofted surface.
1
p r,O(.l.,W ) =
then the surface eguation
(6-61)
Siroilarly J.f
n.
L ~
pr' s(i,j)B ). (w) j=O s=O s,
l.S
1
Q(u,w) =
E
n.
:f: PO, s(u, j)f;s
j ::{) s::::O
J. (w)
(6-62)
I
which is a .ruled or lofted surface using the a1t.ernat.Et boundary conditions .
If both the condi.ti01S represented by Eqs. (6-61) and (6-62) are iJTposed, then the surface equation is 1 Q( u w) =
1
rni n.
L .L L ~ pr' s(i-j)Br i (u)Bs ieO J•O r-0 s;O
,
I
j(w)
(6-63)
I
which is the so-called tensor prodoct or cartesian product surface.
F.quation
(6-63) is .i.Jif.Orto!lnt because it represents the natural extension to surfaces of
the c:auron rrethods of curve approximation. by Eq.
The cartesian surface represented
(6-63) or the lofted surfaces represented by Eqs. (6-61) and (6-62) are
the nost frequently :irrplerrented forms of Coals surfaces.
=
Yhen m n. = 1 and the 8 . . 's are given by Eq. (6- 57) , Eq. (6-63) repre1 ) l. , J sents A bic\bic patch, i.e., 1
Q(u,w)
a
1
1
1
L: E L: L:
pr' s(u, j)Br i (u)Bs ). (w)
i=O j=O r-0 s-0 sho~~S
Ccmpilrison of Eq. (6-63) with Eq. (6-58)
form.
I
,
that it is of a rrore limited
In particular, the bicubic surface patch described by Eq. (6-63) depends
only en oomer infoanation - positicn , tangent, and twist vectors , alalg with
the cubic blending functions to determine the interior shape of the patch .
The
bicubic surface patch described by Eq . (6-58) a l so takes into acoc:unt the boundary curves which outline the patch when det:e.rmi.ning the interior shape of the surface patch.
The bicubic surface
patdl described by Eq. (6-58) is thus
rrore qeneral.
Notice thllt in Eq . (6-56) and its special cases, 5\.ITI'IUtion occurs for i
0 and 1, j - 0 and 1, i.e., fran one boundary curve to the other.
This
in'plies that the interior of the surface patch is govemee entirely by the boundary CUl"V'Cs, oomer points , oomer tangent and twist wc:tors , and the blending
functi.cn.
That is , the interior of the surface patch is controlled by the ex-
terior boundary conditions .
lf, however,
"'~ 5\.IITl
flan 0 ~ i ~ M, 0 S.. j S.. N,
thr>...n all curves P(u ,w) and P(u,wj) between the boundary curves, and in fact any 1
SURFACE DEOCR.IPtiOO AND GENERATIOO 185
nox::mal derivatives on these curves, cs.ro accounted !or.
'1hi.s a llows specifying
interior curves and bound!uy ccnditioos to assist in controlling the inter ~or In this case the surface is ~ represented by
shape of the surface patch. w
0 u,w) •
n
.,
1
"
)~ 1-o M
E
,.c.
0
(u1,VIt r •'"' •
•
IJ
0
l
~
l:
I" ,c (u,wjiO
)'
.J
'wl -
w
.,
~
l.. L ) ~ [
o_.,
"-
~
IMI
,,r,,l
)
r' &
•t li •
."'1
(6-64)
Here the blend.i.ncJ curves satisfy t..OO relatioos p
6r,i (~) •6 pr p
8S,J. (wk): 6pr · 6jk 0 -< P -< n). 0 ' k <- N Notice that the degree of the int:erp:>lated nomal derivatives is given by mi and Further notice that for N = N -= 1, F.q . (6-64) roduces to the COons surface described by Fq. (6-56) . Figure 6-14 il l ustrates the difference bet:\..Jcen the
nj.
surfaces described by Eqs. (6-56) and (6-64). blended sur£aoe.
Figure 6-14a sho....s an exterior
only the boundar:y cuzvcs P(u,O),
P (u,l), P(O,w), and P(l,w)
and the corner points, CO!i1er tangent, and twist vectors at A, B, C, and 0 are
considered in blending the center of the surface patch.
Figure 6-14b shot.-s an
interior blended surface where the interior curves P(u ,O. S) and P(O.S,w) are also consideruc:.
'!he generalized lofted surface and the cartesian product surface can be obtained in a manner ana.l.ogous to
trose in t:re previous section .
They are
given by H
Q(u,w)
m1.
= ·L0 L:0
Q(u,'W) =
>.=
r-
N
n.
L ~
r 0 p ' (u. ,w) 8 ~
PlO.II
. (u) (6-65)
r,~
-._.;... P..:Iu.:.::I)_ -!:!PII. II ('
0 5 P ' (u,-w)B
j=O s=O
5
(6-66)
j (w) '
and
P(u.OI
(6-67)
6-13 CooCLUS I(lll 'llle final two chapters have been an
introduction to scrre of the current rrethods for curve and surface descriptioo.s . l>\any other techniques have been
and used with varying
~s
suggested
of
This is a young and dynamic field with rte.J
h
SIXX)eSS.
ideas and techniques cx:nstanUy
Figure 6-14
Exterior and interior "blended" surfaces.
appearinq in the literature. To actually inplerrent these techniques in a production or desi gn environ-
rrent requires nany rrore considerations than can be discussed here. requ.irerrents will depend
\pln
the needs of specific applicat.J.oos.
specific Px001errs
such as jo.ininq surface patches, calculating cutter path offsets, determining intersections of ll.nes ard surfaces, designing data and file structures in
\oohi.ch to store and r.-anipulate the infollt'ation , writing analysis programs to calculate neoded paraneters, rEITOV:i.ng hi.ddE.>.n lines ard surfaces for proper
graphical dl.splay, etc . , are the types of requ.irerrents that will arise.
Sere
solutions to these problem5 can be fo\D'd in the current c::atpUter-aided design
liau-ature.
The reader of this book sOOuld be able to build upon the founda-
tion presented herein to create a capahil i ty to rreet his needs.
R£FERENCES 6-1
Bezier, P. E., E?tploi des Machines a Cunnande
1'.\Meri~,
r.u.sson et Cie,
Paris, France, 1970; Bezier, P. E. , "Exanple of an Existing System in the .t-btor !Mustry:
'lbe Un.isurf System," Proc. ~ -
soc.
(London) , Vol. A321,
pp. 207-218 , 1971. 6-2
Sabin, fot. A., "An Exi.sti.ng Systen in the Aircraft Industry.
'1he British
Aircraft CO.rporatioo NumetiC31 M3st.er Geatetry Systan," Proc. !by.
soc.
(Is:rdon), Vol. A321, pp. 197- 205, 1971.
6-3
Peau-s, G. J ., "Interactive Curputer GraJ;i'U.cs Applicatioo of the Bi-<::ubic
Pa.ranetric SUrface to Engineering Design Prob1ens," !-tD::nnel1 [):)uq1as Autatetion Conpany.
II
II
St. Ionis , Ml.ssouri, presented at Society of Indus-
trial and Applied Mathematics 1973 National l'leet.ing, HMpton, va., 18-21 June 1973. 6-4
COons, S. A. , "'Surfaces for Cooputer-Aided Design of Space Forrrs, " t-1. I. T . Project tW:, t-W:-TR-41, J une 1967.
(Also as MJ 663 504) .
6-5 Annit, A. P. , "Curputer systems for Interactive Design of Thrce-Dimmsiona1 Shapes," Ph. D. Thesis, Carrbridqe University, NoveTber 1970.
6-6
Lee, T. M. P. , "Analysis of an Efficient liCl'togeneoUS Tensor Representatioo
of Surfaces for Catputer Display,• in Advanced Carputer Graphics. by Parslo..r, R.
o. ,
Edited
and Green, R. E. , Plen\Jtl Press, New York, pp. lll9-ll41,
1971.
6-7
Ferguson, J. C. , "t-l.lltivariable CUrve Interpolation," J. Assoc. Cooprt:.
Mach., Vol. no. 2, pp. 221-2.2 8, April 1964. 6-8
Bezier, P., Numerical Control-fo'.athematics and Applications, (translated by A. R. fbrrest), John Wiley & Sons, Inc., Laden, 1972.
SURFJICE DESCRIPTTQI1 1\.''-m GEl!ERATICN 187
6-9
Armit , A. P. ,
"~ti.patch ;mel
f.Ultiobject Desiqn Systerrs," Ptoc.
~
Soc. (I.a1don) , 'A:>l. A32l, pp. 235-242, 1971.
6- 10 Bamh.il1, R. E. , and Riescnfeld, R. F. , "c:::onvuter Aldecl Gea1et:ric: Desiqn," Academic Press , New ¥ork, 1974. 6-11 Fo rrest, A. , "On CXlons and other M2thods for t;M Representation of CUrved Surfaces ," Colputer Grapucs and Image Processinq, Vol. 1, pp. 341-359, 1972.
APPENDIX A •
COMPUTER GRAPHICS SOFTWARE •
IP
lit
I~I
Ill
•
In developing the cx:n:::ept.s for a cr:rtprl:er graphics software system it is
ronvenient to diVJ.de the overall system into several parts .
'1\.o of the obvious
parts are the sq{X>rt CXJTpUter and the graphics device itself. harct~e.
taken.
These are t:oth
In considering these descriptions a not too nant:M vi ew sh::luld be
In general, the st.q?part catpUter is the system on \olhich the data base
exists and on which software manipulation of the data base occurs. ~re
However , in
syst:.ems this function rray be t:hysically split ootween tw::> or nore
systcns.
CXXJp..lter
'Ihe gra[i\ics device is the hara..mre which actually produces the picture.
A concoptual view of a c:x:JTpUt.er graphics software system is sho.om in Fig . A- 1. '1\.o sofb.ve systems produce an interface between the support oonput.er and
the graphics device.
'nle first is the data base manipulation software.
'llle
mathematical basis for this system is the topic of the main part of the book.
It consists of routines to translate, rotate, clip etc . ~ data base. Speci.fic algoritms to perform nany of these functions occur later in Appendix c . 'Ibis software system is mtpletely device-independent since it is only
noiTI\ll "drawing functions" and the hardware associated with a particulnr graphics device.
It is this system which causes the neoessnry codes (infomatioo in
t:.erns of binary bits) to be sent
to the grar:hics devioo to cause that
~ce to
CXMVl'ER GRAPHICS SOF'IWARE 189
draw lines, plot FOints, etc.
Although as outlined in Sees. A- 1 cUrl A-2 t.Alw,
this software system can be made minimally devi.oe-
Because of this Sees. A-1 and A-2 ally conocpt-
ually discuss this software system.
A-1 U:wuJtR
tMPHICS PRIMITIVES
As Feinted out aOOIIe, .....tlen designing cr ~a corputcr graphics
systan
it is convenient to define those areas ~e the 51J(JEX>rt cx:xrputer I!USt interl!ct with the grat:nics device. All graphics devirns, of course, reoeivc info:omtion fran the supp::>rt cx:xnputer as a data stream
cases cne or rrore bits tray be Sl.gnificant..
In partic-
However, except for very
sofhisticatcd users , the c:arposition of this data st.rerun is seldan of i.nt.&cst. It. is rrore convenient. to think of the fun.d.anental effoct.s that tho data stream has oo the graphics device. 'lhese fundamental effects are called graphic primitives . 'Ihcy generally fall into three areas: cursor control, device state or rrode control , and graphic input. Here cursor is used in a general sense to rrean the e.lec:..t:on beam on a
Q{}'
graphics device, the print head on a teletype,
high-speed printer , electrostatic dot matrix printer, or the pen oo a pen and
ink plotter, ~-
We Wl.ll briefly
discuss the grat:hic primitives associ-
IIRIT
ated with each of these areas.
Fran
these graphic primitives several carr-
pouro graphic cx:xnrands can OOA
~PI.lATIDl
::
i
be developed
which are of nore general interest. rumtl llJ'Mf.R '
These will be discussed in the next
secticn. 'lbe boO graphic primitives associ-
!£VI~
MlA &'~
ated with cursor control are t.nve the cursor TUrn the cursor en/of! 'lhe function of each of these primitives
.-1-.I C,_.,UCS tEVTCfS
is sel £-explanatory.
'nle graphic primitives associated W1th device state or nrx3c control are, for exanple:
Figure A-1 Q:r)ceptual vie.~ of a cx:xrputer graphics software system
Al(ilantm'eriC !rode (1)
Graphics m:x1e (2)
190 MM'HENATICAL
~
FOR CCK't1l'ER GRAPHICS
Standby rrode (3)
Absolute o:x>.rrlinate node (4) lelati~ (increrrental)
o:x>rdlnate ucde (5)
Frase the screen (6) Dash line rmdc (7)
01rVcd line nnie (8) I.oc:2l t.racki.ng a\ (9) I.ocal tracking off (10)
etc. For sare gra~cs dev:Loes , particular nodes ZMY not be appropriate and for
others additional
nojes
might be a color CRI' graphics device
also be regul.red .
For the latter
are required.
...mere
ease a
particular exanple
blue, red, and yel.l.o,..r ll'Odes
~uld
Ho.r.-ever, the above list represents those required for a
I
Before continuing, a ~d about the
I
reasooably sophisticated graphics device.
I~ II
function of each of these various m::rle.s or states is in order. Alphammaric and graphics mxles are generally associated with devices \oo'bich
can interpret a data stream as either alftl,anllroric characters or graphic elements. St.andhy rrnie generally is nsed to disable the terminal such that it ignores data sent to it and also data ge.'lerated by interactive devices such as key boards, light pens, etc. J;bsolute and relative cx::ordi.nate nodes detel:mine W'lether the o:x>rdinatcs specified in the graphic primitive positioo <:Xm'\\Uld are to be interpreted as absol ute or ro.lative o:x>rdinate data.
far stora
'!be erase node i s .ittp>rtant
Dashed line noie allows the generatiat
of dashed lines using the hard..are characteristics of the grapucs device. CUxved line node is used to activate hard.mre curve generatl..oo .
UX:a1 track.in9
node can be used to act.l.vate hardware, whlch allows a visible cursor at a CRI' ~cs
device to fol.l.ow the IrOveJTent of an interact!vc input device such as
a light pen or the pen at an analog table.
'lhe graphic primitive associated with graphic .input is read the cursor {X)Sitian .
positioo vector, i.e., either x,y-pai rs or x,y ,z-triplets.
All .interactive
co•puter systems and higher level languages have adequate faci 1 i ties for accepting alphamJroric character input, and hence character in?Jt is not considered
as a grafhic primitive .
Fran the above
see that there are essentially four g:raprlc primitives which CXIOCCptnally can be represented either as ccmrands within or subroutines of a hlgher level langu.l9e su:h as BASIC or FCRI'RAN. 'these four primitives are \o.e
cn-1Ptm:R GRAPHICS sc:FJW\RE 191 CHFP (control variable) 1-DVE (coordinate, coordinate, <.coordinate>) KlE (control variable)
oc ~
(coordinate, coordinate, )
< > is used to represent optional variables.
'1hc control variablP in CJIO'F
oo respectively.
'lhe control variable in f-OE could be an int.eqer assiyned a
specific nean.inq, as indicated above in parenthes.is to the right of each individual f\lJlCtion .
These four graphics pr.iJnitives can be cutbined into ~al
cntpound graphic elcrrents which are of nnre intel:'est to the general user.
A-2 CdwreR
GRAPHIC
Eumrrs
'1lle cxxtputer graE=hic primitives discussed in t:lx! previous section can be
cxxtbined into OCITpUter gra{ilic elerrents.
'lllese are generally c:crtp;:lUnd operations;
i.e., they generally involve rrore than ooe of the graphic primitives. 'lhe grapti.c elercnnts are of nore fundatrental interest to the user than the ~c primitives.
In fact , tOO
user
may not be aware of the graphic primitives .
h}a.in, as with the graphic primitives, the graphic elerents can be placed in three categories : drawing functions, term.ina.1 or picture control f\..B"'ct.ials, a.OO interactive device control func:tials. '1lle classification rrethod discussed in Sec. 1-6, as ~Nell as traditional
drawing rrethods, assist in dete.Imini.ng the fundarrental d.rawi.r¥;J elorents.
A user
will .....ant to be able to nove the cursor about without drawing, to draw a lule, and to plot a point.
It is convenient to be able to do this in either absolute
o r relative coordinates.
'nlese ideas lead to six gr~c elcroonts:
1-bve Absolute 1-blle Relative
Point Absolute Point Relative Draw Absolute Draw Relative
In addition the user will want to b! able to print textual nnterial at a speci-
fied location in the picture , say, Text Absolute Text Relatl.ve
il
ll
I~
II
In each e-lse it is <:nlvenient to oonsidcr t:hbt the action initiates fmn
the current
!X)Sit ion
of the cursor.
Functionally each o f these g:rcq:hlc elc:mmts
can be described in terms of grapuc primitives. algorithm might be
z.m£
For exanple 1 the t
1.
Set graphic rrode -
2. 3.
'1\lrn cursor off - Q.'D"F 0 Set absolute ooordinate m::xie - 1'100£ 4
4.
r-bve to the coordinates sped ficrl - MJJE (coer. I coer . 1 )
5.
Set alfila m:x:ie - MD£ l
2
•
•
The Point 1\bsolutc algoritlln might be
•
l.
Set graphic rrode - l-o:>E 2
2.
'!'urn cursor off - CNJFF 0
3.
Set absolute coordinate rrode - 1'100£ 4
4.
l1olle to the coordinates spec:i fi ed - MJJE (coer. 1 coor.
5.
'1\u:n cursor on - CNJfl' 1
6.
'1\u:n cursor off - CNJFF 0
7.
Set alpha node - M:DE l
1
)
'lhe Draw Absolute algoriilin might be 1.
Set gra{Xric m:x:ie - 1-o:>E 2
2.
'1\lrn cursor on - CNlFF 1
3.
Set absolute ccardi.nate nu de - MDE 4
4 • Move to the coerdinates specified - l-oJE (ooor. 1 coer. 1 ) 5. '1\lrn cursor off - CN)FF 0 6.
Set alpha rrode - ra>E l
a:x1 the Text .Absolute algorithn might be 1.
Set graphic JtPde -
2.
'1\u:n cursor off - CN:lFF 0
3. 4.
SeL absolute coordinate rrode - t-roE 4
l>bve to location - M:NE (ooor. 1 ooor. 1 )
5•
Set ctlpha rrode - l-DDE 1
6.
'1\lrn cursor on -
7.
Print characters
8.
'1\lrn cursor off - CN:lFF 0
M.DE 2
CN)FF
1
'The four relative coord..mate el.c:!rel\ts have the sare algoritl'ITsl except that the tlu.rcl oc::mnarrl sets relative nrxk• ~mE 5) . Tho above algorit:tm; set each required
'lhis is good i.n.surana::.
Hon1Cvcr 1 ilJl .increase
in
the efficiency of data transmissicn can be achie~-d at the expense of ~
additi
For exwplel the alqorithm for the Point J\bsolule cu:nmcl
1.
Is the devi~ in graphic rrode? node - r-a:£ 2 •
2.
Is the cursor off?
3.
Is the device in absolute coordinate node?
lf yes - CQ'lt..lnue I tf no - sci CJr•lrflic
If yes - CQ'ltinue, i f no - tum cursor off - c:N:E'F 0. If yes - CXlntinue, if no -
set absolute coordinate node - ln:E 4.
to the coordinate spocified -
4.
~bve
5.
'l\J.rn cursor on - OOOFF 1
~OVE
(coor. , coor. , ) •
'lhi.s schere requires that a small storage b.lffer \o.hich oootains infonnaHon
representing the "state of the device" at the end of the previous graph1c OJIIm:md exist.
~.
the potential exists for reducing the nlll't:>er o! data stroam
e1emmts which nust be camunicated to the graphic device.
Further, analysis
of the orlgin.:al algorit:lTn sl-oJs that if a rrethod can be devised to antic1patc
lo.hether the next camand will be a graphic c::cm'l\'md or a 110'\C]raphic (alphd) mand, then addi ticnal efficiencies result.
ca:~
Before cmt.inu..in<.J it should be noted
that although functionally identical, these alqorith!1\s might be structurod differently for different graphics devices .
Terminal or picture CCI'ltrol g:raphic elerents are associated w1.th initia 1iz-
ing the graphic device, erasing the screen for storage tube CRr graphics devices,
start.ing and ending frarres for refresh CRr graphic devices, and buffering the data stream and on CXl01'TIClOO sendiJ'lg it to the graphics device .
For exanple, the
graphic elerrents
Initialize the
~vice
Begin Fraire ~Buffer
could serve these purposes.
runcti
serves to set the default ccnditioos and zero the outpUt buffer; the Begin Fra!re
COII'and serves to erase the screen for storage tube CRl' grapuc devices , to indicate a new plot for pen and ink devices, am to rotify a refresh grapuc devioe that the picture is curplete and a new frmre nust be started. 'lbc 0\Ilp Buffer CXX'I'ITland serves to send the stored data stream to the graPrics device and
leave the device in alpha rrode. Fran the user ' s point of view the control of interactive devices rrust be
rrore explicit than a siiTple cx:rmand to read the cursor p:>sition .
in Sec. 1-5 there are several interactive grle.
Activat.i.ng an Analog Tablet
•
Activating a Jat Stick
llctivating a Light Pen
Acti.vatin
As an ex.anple, the algorithm for activating an analog table with
local tracking might be Set gratillcs rr00e - tOE 2
I
2.
Set absolute coordinate node - KX>E 4
3.
set
4.
lead the cursor positioo
5.
Read any table switch posit.i.a\s
6.
Turn off local tracking -raE 10
7.
Set a4ila node - MX1E 1
•
a
Reading the CUrsor Positi.oo
l.
Ill
As
mi.n.inun the input gnqiUc elesuents sOOuld include
~~ I~
As discussed
local tracking - t-IDE 9
'llle other graphic elenents would have simi..lar algorit:Nns.
Again ec:cnanies in
the data strean can be obtained by using sq:histicated chedti.ng procedures for the various nodes, etc .
A-3 CANoorCAL SPACe It is necessary to carplete our discusstoo of graphic syste.rrs by considering the various spaces or coordinate systems \o.tlich are o f interest to both the user and the system architect.
1here are three spaces that are of interest-user sp.x:c, canonical space,
am device spaoe.
user space represents the coordinates or uruts in \o.bich the
application is perfonmd, cancniCAl space is a pseWo space used to obtain device
, am device space represents the coordi.n.atc system in \o.hlch
the graphics device nust be
'lhese are illustrated in
Fig. A-2.
'Ihe cxu::ept of a caoonical or pseudo space is of interest.
By using a
standard cananical or pseudo space for all graphic devices, device-independent
CXJ•IPU'reR GAAPHICS SFlW\RE 195
graphic software CM rrore eas1.ly be
obtained.
In pa.rtio1lar, the picture
is
first trans!Onred fran user spoet or m-
ordinates to the st.andard canonical space
I
----
0 1'---
and then transfoDOOd fran the canonical
--~ space to the appropriate dPvioe space.
A s.i.!Tt>le concept for the canonical space assures lhat the addressable area is the unit square in the first quadrant;
I
• I ' - ----:-
0
n User space
6 211
i.e., the
addr~ssable
area of the psa;ido
graphic device represenl-ei by the caocn-
ical
~
is 0 -< Xa< 1 . 0, 0 -< ya< 1.0 • Although this aF{Cars to limit thc p1.cture ..r -
to a square format, this is not necessarily true.
'lhe 1.0 x 1.0 square fo.mat is
assl..llled for ·
arrong graphics
devices and is predicated on the basis that many plotters and uost storage bJbc CRr grafhics devices have a square ad-
-----
dressable area .
tiJWever, addressable and
viewable areas are not necessarily identiCA'll.
0
J
rectangle; e . g. , rrost storage tube CRl'
-
0'---- - -
1 graprics devices have a viewable area that is approx:i.Jrately 0.8 of the aMressable
b Canon•c al ip.acc
(\. )') r 1024
rhi~
area in one c:lim!nsioo , nost often the vertical dinen.sion . In this case , the viewable area is assured to be 0 ~ Xv ~ 1.0
area may not be '•cwablc
--
780
Frequently the vielriable area is a
o ~ Yv ~ o.a.
Further, llWlY ploLt.ers have the ability to extend one axis inand
definitely.
In this case the vi~'able
area is asS\.J'!9i to be 0 ~
Xv ~ a ,
where
a is SOli! nurber ...,hich is greater than
one.
Q1e
oould, of oourse, argue that the
addressable area slnl1.d then be rectangular. HQJ...-ever, unless extrete care was taken with scale factors , distortioo of gearet.ric 1024 (say)
Figure A-2
Grat:hlc spaces
figures ....ould occur; i.e., a circle "Ould be seen as an ellipse.
'lb.i.s is , of oourse,
unaccepta.ble for gearctric oc::rtpUter ~ ·
APPENDIX B •
MATRIX OPERATIONS •
I!
I
II
I
II
A few sirrple rules fran natrix algebra are given here for convenience.
matr-i:t is si.nply a rectangular array of nurbers \llh.ich are rules .
8-1
A
governed by these
T~Itn.OGY 'Ihe array of nll'lbers \o."hich make up a natrix are called its elements .
elements form rows and oolum.s within a natrix.
'Ihese
If the nl.ll'ber of rows and
ool ums are ogual, the matrix is cal l ed square. COnsider the 4 x 4 square matrix given by all ~2
8
13 ~4
~~2~3~4
a31 a32 a33 a34 a41 a42 a43 a44 'lbe first subscript refers to the natrix
l"CM
ard the second refers to the JTatr.ix
oolum.
Thus, a 34 is the element in the thixd reM and fourth oolum. A natrix of m l"CMS and n oohrm.s is cne of order m x n . In the square nstrix abo\.~ ,
'\!• ~ 2 ,
~ ,
3
44 are called its dia./'r:at cl.<- Pttc . 'lhe sun of the diagonal elements in a square natrix is called the trace .
m • nand the ele1ents
and a
'IATRlX OPEAATI<::I\'5 197
i
An
0 l 0
0 0 1 'l'No matrices are
qual only when each e1eroont in one is equal to the corresp :n:l-
ing elerent in the other; that is, they r.ust be i..de.ntical.
B-2
AooJTIOO AND SuBTRACTICtf If
b.Q
defined.
matrices are of the
sa~re
order, then additioo and subtracb.on are
To a&3 or substract tw:> matr.ia>s , the operation of addi t ioo o r sub-
tra ct.i.oo is applied to each o f the corres£Xlnding eletelts.
1 2 3
1 4 7
4 5 6 + 2 5 8
As an e.xanple :
2 6 10
=
6 10 14
7 8 9
3 6 9
10 14 18
1 2 3 4 5 6 7 8 9
1 4 7 2 5 8
0 - 2 -4
and
B-3
3 6 9
=
2 4
0 -2 2 0
t'u..TI PL ICATION t-latrix n'lll.tiplication is the nost useful matrix cp"ra tioo for cnrputer
graphics . n2 x
"'2·
Consider one matrix of order n1 x m1 and a semnd rmtrix of order For n'lll.tiplication of t:\<\0 matrices to be deflned, the value of "1 rrust
equal the value o f n • '111at is , the nllt'ber of co11.11V\S in the first natrix must 2 equal the nl.ll'b:!.r of rows in the second matrix. 'lhe rules for matrix nultiplicatioo are best described by the follCMi.ng
exanp1e.
If [A] is a matrix of order 4 x 3 and [B) is a square matrix of order
3 x 3 , then the matrix prodoct (A] (B) is defined by the following op?ratial .
•u •u •u . 21
"'ll
~1
bll
~)
.,1
b21 b2l 1:.2]
on 6 )]
l>ll bJl b)l
6
•n •u
010
2J
43
~~~•
~
• ~t'n ' 4 1~1 ~lbll. 'l.P22 • •lf>ll •ubu • •1/>n • 4 lf'JJ ·~1bu. • "2i'n • "2/'n -,lbU • a,_f7J -+ "J/>J~ •ubu • 'lt'll • •,fl33 ~l~l • •)-/>'ll • ~i>ll il11bu • •1i'22 • •rJ>u ~)lbll • 3 3ZD2J • 4 lJbJJ •c1~1 • •ct'7l • 0 ct>ll
6
clb17 • 0 cib22 • 4 cf>12 <~~c lbll • .ac:l'cJ • oc/>llj
Not.ice that a 4 x 3 matn.x In general,
'1.
a
!tl.ll tipl ied
by a 3 x 3 matrix produces a .; x 3 Matn.x.
f1. x ~
x 11'2 rratrix.
n1trix U."':£S a ~ x ~ matn.x, "'here l..s a nurerical exa:-ple , o::nsl.der
"'1 3- r1 21 4 2 L) 4j
-
1 1
4 +
6 8+ 8
1 +
3
G.
6 4 l3 2.,.
u, =
l+ 9 2 + 2 +
4
12 12 + 16J + 6 6 + 8
ttt = ~,
phcatlal 1.s izrpor...ant.
10 16 1 4 6 18 28
L 9 14_ 'Ihat is,
1.n
Thus, the order (sequence) of mllti-
Matrix operatior.s folla..r tt.e fust and second disc:i-
butive 18:'..'5; e.g., A(B + C)
ciative law
a
-10 141
The operatioo of matrix nultiphcation is ~ o::rrm.rt:ative.
general, [A) (B] is rot equal to [BJ (A).
produces
A(OC) '"' (AB}C
e
N3 + 1C. and (A+ B)C • AC +
a:.
Also,
t.~
asso-
applies .
'Ihc det.en:'J.nant of a square rat:rix (A]
l.S
denoted by
1
AI .
'lbe det.e.rrni.nant
For our ~ . it 1.s suffic1.ent to illustrate the nethod for find.inq the det.enninant of a 3 x 3 square rnatrix.
has rrany useful properties in rratrix theocy .
lf
M=
r all
en ~3
1
a21 a22 ~3 l a31 a32 a33J
~1 a12 ~3 4
21 ~2
8
23
a31 a32 a33 !
• ~ (~2a33- ~3a32) - a12 ra2la33 - a2~3l) + al3 (a211132 - ~2a31 )
nus gives a single m.rerical value for the dete.t:mmant.
of 3 9 4
6 1
81
12 5 3 'lhe result is
3(3 - 40) - 9(18 - 16) + 4(30 - 2}
= 3(- 37) - 9(2) + 4(28)
a
-17
Q::nsider the evall2ttm
1'11\TRIX
B-5
OPEAATia~
199
INVERSE OF A SGt..IARE f'1ATRI X There are many techniques for cal culat.i.nq the inverse of a squara trotrix.
t10St catp.lt.er languages have fmct.ions or subrouunes "ni.ch will perform U1e repetitive calculations necessary t:o evaluate the clCJI'Nlts of the inver$(. o f ' given matrix. In algebra, where single variabl es are ccnsidered , if ax:. Y 1 then x "' a- 1J',
...nere
a - 1 is s imply the reciprocal o f a, e . g ., a -
• 1/a.
In 1ratn.x al<)Cbr::ll,
aro
the reciprocal of a matrix does not exist. Ho..IL~r, the [X) = [A ) -l[Y), where (A) -l , is called the inverse of tre
divisioo is not defined
if (A)(X) = (Y] 1
1
square matrix fA} • A matrix inverse will exist i.f the matrix is square wid i f tho nntri.x is
noosingul.ar.
'Dlus, not every square IMtri.x has an inverse.
l:bNever, if the
determinant of the square matrix is nonzero and an inverse exists , then that
inverse is unique. 'Ihe inportant property o f a matrix uwerse that is used 1n thi s book is
that, for any square matrix [A), the equal.J.ty (A] (A]-l
= {I)
is valid, where
[I) is the irlentity rratri.x.
Coosider the matrix product o f the G.o folla.t.i.ng 3 x 3 tMtrioes :
1 2 3 1 3 3 2 4
6 -2 -3 1 0 0 -1 1 0 • 0 1 0 -1 0 0 0 1 1 If this p.roduct is represented by £1\1 (B] = fil, then (Bl is equal to (AJ- •
APPENDIX C DATA BASE MA IPULATION ALGORITHMS
•
II
II
I
The algorit:l"cns presented m t.l'ri.s sect.ron are based on the rmtheratica!
d' SC.JSsion presented in the l:x:xly of the te."«. ...Titten
l.1l
Oartrouth SJ.Xth Eiht.J.on Bi\SIC.
'they are given as subprogrc'li"S,
They can easily be converted to other
la."'9".1a9es . The algont:h:':'.s are rot intended to be tlY.. rrost efflcient posslhle. '!h..~ are mtended to be in.st:.n.r=tive. Eac.'ll of the transfomation a:go:·ithlts asst.meS that the data base is presented in ord:.nary coordinates. 'llle algorithm then develops the necessary hcn:x;eneous coordinate representat:.J.cn, performs the transfot:m:iticnl and returns to ordinary cxx:>rdinates.
The aw-1re.nt ineffic~c..-.cy
associated "'ith reconversion to ordina.ty coordinates has boon deliberately acx::epted since experience indicates that users norrrally W:J.ll perfonn or.e or o.o
transfon:ations, display tr.e result, \oohl.ch requires ordir.ary coord.lnatcs , and then c::ontwoie . If a&H tior.al efficiency is required the reader can either rrc:xti£y the algoritnr.s or develop his or her own based on t:re theory presented in the body of the text;.
algcnth:n for a."\ arbitrary translation uti Hzing han::qU'lE.'OUS coordl.J'lates is g!.ven bclo..~ il.S a BASIC !a.n3ua9e s...:bprogra:'n. It is based on sec. 2-15. An
2DTRANS 100 SUB" 2DTRAl5" : P, X( ) , Y() ,M, fJ ' 2-D 110 ' P:Ntl-:BER Of X,Y- PAIRS 120 ' X( ):ARRAY CONTAINit\G X-ax:JRDitU.'TES 130 • Y() : ARRAY CONTAlNir\G Y-COORDir~TES
'M:X TRANSLATION FACTOR
l 40 150 160 170 180 190 200 210 220
' N:. Y TRANSLATION FACI'OR
DIM U(100 , 3) , V(100,3)
U=ZER(P , 3) MAT V:ZER(P , 3) FOR 1:1 TO P LET U(l , 1}:X(I) LET U(l , 2):Y(I) LET U(l , 3)=1 NE:.XT I ~~T
230
240 250 260 270 280 290 300
310 320 330
TRANSLATION
MAT T:ZER(3 , 3) LET T(1 , 1):T(2 ,2}:T(3. 3)=1 LET T(3 , 1):M LET T(3 , 2):N ~1AT
V:U•T FOR I=1 TO P LET X(I):V{l , 1) LET Y(l):V(I , 2) NEXT 1 SUBEND
I
100 POSITICN VFrl'ORS
• R£DIMFt:SION
~v
FI LL WI1li ZC.Ros
SET UF H(l.l(Xjfljf:OOS • POSITION VECTORS I
REDIM.ElJSION T ~ILL ~lTH ZEROS I PLACE 1 I s ON LIA OF 'I MATH IX ' SET X TRANSLATION F/,CTOH 'SET Y TRANSLATIO!I FACTOR 'CALC, TRANSLAT~ POSITION VECTORS I
' RETURN TO PHYSICAL CWR .
A SC'Il ing algoritm is given bela;o~ as a 81\SIC language subprogram.
The
algorittm based on Sees . 2-11 and 2-15 provides both l ocal and overall scaling. 'lbus , distortion of a shape can be aa:aupl..ished.
It util izcs
l"otogetle()US
nates.
2DSCf..LE 100 SUB"2DSCALE": P, X() , Y() , A, O,S 110 ' P:MJHBER OF X, Y-PAIRS
120 130 1 ~0
150
' 2-D SCAlE
' X():ARRAY CGNTAINING X-COOJIDINATES ' Y{):AJIRAY CONTAINING X-COORDINATES ' A:X STRETCHING FACTOR A>1 STRETCHES A<1 CONTRACTS ' D:Y STRETCHING FACTOR D>l STRE'ICHES 0<1 CONTRACTS
160
' S:OVERALL SCALING S< 1 fl~LARGES S> 1 REDUCES
170 180 190
DIM S(100, 3} ,Q(100 , 3) II.AT S:ZER( P, 3) ~~~T O:ZER(P, 3) FOR 1:1 TO P LET S(I , l }:X(l) Lei S(I , ~):Y(I) LEi S(I , 3):1
2CO
210 220
230 240 250 26G 270 280
NEXT I HAT T:ZER(3, 3)
LF.T T(1 , 1):A LET T(2 ,2}:D LET T(3 , 3):S
'100 POSITION VECTORS ' REDIMENSION S&C FILL
WI~
ZEROS
' S£1' UP tn100ENEOUS I
POSITICtJ VECTORS
'RED!ME!\Siot~
1 FILL WlTr ZEROS
' SET X STRETCHING FACTOR ' SET Y STRETCHING FACTOR ' SE1' OVERALL SCAWKi
coordi-
If
II II
I!
202 l'tl\niEMM'ICAL 290 300 310 320 330
ELEMEl1I'S
FOR
a:l-lPlJTER
GRAPHICS
II.AT O=S*T FOR 1:1 TO P ~f X(I):Q(l , 1)/0(1 . 3) Ltl ¥(I):Q( I, 2)/Q(I,3) NEXT I
'CALC. TRANSFOJito1ED POltn'S 'CALC. PHYSICAL ODOR.
]llO SUBEND
•
C-3 A Tlo.O-DII'a4SIONAL Pm..ECTION
Ai..GOOJ~
A siJlt>le algorithn to reflect al:x:lut either the x- or y-axis is given below as a BASIC language subprogram. and uses l'xxtogeneous coordinates.
'Ihis routine depends on tro analysis of Sec. 2-5
This operation is also frequently referred
to as mirror itrage.
2DREFLT 100
110 120 130
1 ~0
150 160 170 180
190 200
210
SUB"2DREFLT": P, X() , ¥ () ,N ' 2- D REFl£CTION ' P:Ntt-!BER CF X,¥- PAIRS 1 X():ARRAY CONTAINING X-COORDINATES ' Y():ARRAY CONTAINING ¥-COORDINATES ' N=REFLECTION CODE 1=X-AXIS , 2:Y- AXIS Om S( 100 ,3) ,Q(100,3) '1 00 POSITION VECTORS 1 REDIHENSION S&Q FilL Wrni ZEROS HAT S:ZER( P, 3) HAT Q:ZER(P,3) 1SET UP l0100ENEOUS FOR I : 1 TO P LET S(I, 1)=X(I) ' POSITION VECTORS LET S( I,2):Y(I) LET S(I,3):1
220
~'EXT
230 2ll0
MAT T:ZER(3 ,3)
250 260
I
IF H:2 THEN 280 LET T( l , 1):T(3. 3)=1
300 310 320 330
1(2,2):-1 GO TO 300 LET T(2 , 2):T(3, 3)=1 LET 1( 1,1) : - 1 MAT O=S*T FOR I:1 TOP LET X(I):Q( I , 1) LET Y( l):Q(I , 2)
]110
NEXT I
270
280
290
' REDIMENSION T FlU. Wlnt ZEROS SET UP TRANSFORf'1ATION 11ATRIX
1
LET
' CALC. TRANSFOIIMED POINTS 'CALC. PHYSICAL COOR.
350 SUBEND
C-4 A G€NEAAL. TclJNENSIOOAL RoTATION An
~11H'1
algont:hn for o.o-dirnensiooal rotation
below as a Bol\SIC langu.'lge subroutine.
arout
an arbitrary point is g1ven
It utilizes the analysis of Sec. 2-16 .
2DROT 100 SUB"2DROT" :P, X() ,Y() ,T1,H,N '2-D ROTATION 110 ' P:NUMBER OF X,Y-PAlRS 120 ' X( ):ARRAY CONTAINING X-COOP.DINATFS 130 'Y():ARRAY CONTAINING Y-COORDINATES 140 'T1:ROTATl0N ANGLE IN D£n11EES 150 ' H:X TRANSLATION FACTOR 160 ' N:Y TRANSLATION fACTOR 170 DIM S( 100 , 3) ,0(100,3) I 100 POSITION VEX:TORS HAT S:ZER(P, 3) 180 'REDIMENSION S&O FILL 190 HAT Q:ZER(P , 3) 'WITH ZEROS 200 FOR I=1 TO P 'SET UP H~OGfliEOUS 210 LET S( I, 1):X(I) I POSITIOt; VECTORS 220 LET S(l , 2):Y(l) 230 LET S(l, 3):1 2110 NEXT I 250 LET T2:T1/57. 2957795 'CONVERT TO RADIANS 260 HAT T:ZER (3, 3) 'REDD1ENSION T FILL Wlnf ZEROS 270 LET T(l, 1):T(2,2): COS(T2) 'SET TRANSFORMATION MATRIX 280 LET T(1,2):SIN(T2) 290 LET T(2, 1 ) : ~IN(T2) 300 LET T(3, 1):-M• (COS{T2)-1 )+N•SIN(T2) 310 LET T(3 ,2):-M•SIN( T2)-N•( COS(T2) -1 ) 320 LET T(3, 3):1 330 HAT Q:S•T ' CALC. TRANSFORMED POINTS 340 FOR 1:1 TO P •CALC . PttYSICAL COC>R. 350 LET X(l):Q(I, 1) 360 LET Y(l) :Q(I, 2) 370 NEXT I 380 SUBEND
C-5 A T~-Dit-£NS ICJW. Sc.ALir«; AL.~Infl An alC}:)ritnn for t.hree-di.n'I!!TI.Sional scaling is given as a BASIC language sui~"'gtam.
It is based on Eq. (3-4).
3DSCALE
100 SUB11 3DSCALE" :P, X() ,Y() ,Z(},A,E,J,S '3-D SCALE 110 ' P:NUMBER OF X,Y, Z- TRIPLETS 120 ' X():ARRAY CONTAINING X-COORDINATE 130 'Y():ARRAY CONTAINING ¥-COORDINATE 140 ' Z():ARRAY CONTAINING Z-COORDlNATE 150 ' A:X STRETCHING FACTOR A)1 STRETCHES, A1 STRETCHES, El STRETCHES, J 1 ENLARGES S<1 REDUCES 1 100 POSITION VECTORS 190 DIM U( 100, 4) ,V( 100 ,4 ) 200 ~tAT U:ZER(P 11) 'REDIMENSION U&V Fill WITH ZEROS 210 ~!AT V:ZER(P, 11 ) 220 FOR I: 1 TO P ' SET UP HCMXJENEOUS 230 LET U(I , l):X(I ) ' POSITION VEC10RS 240 LET U(I, 2):Y(I) 250 LET U(l, 3):Z(l) 260 LET U(I , 4): 1 270
NEXT I
280 290 300
HAT T:ZER( 4, 11) lET T( 1, 1) :A LET T(2, 2)=E
•REDI.HENSION T FlU. WITH ZEROS ' SET X STRETCHING FACTOR •SET Y STRETCHING FACTOR
I~
204 t-IA'mEMI\TICAL
310
~
E'OR CO>IP\Il'ER GRAPHICS
LET T(3 , 3)=J LET T(4 , 4):1/S NAT V:U1 T FOR I: 1 TC P LET X(I):V(I , 1)/V(I , 4) LET Y(I ):V(l, 2)/V( I , 4) LET Z(I):V(I , 3)/V( I , 4)
320
330 340 350 360 370 380 tiEXT I 390 st.'BEND
C-6
•SET Z STRETCHING FACTOR ' SET OVERALL STRETCHll~ FACTOR ' CALC . TRANSFORMED POINTS ' CALC. PHYSICAL ODOR .
•
~ Al..GORITif-1 FOO THREE-DIMENSIONAL lbTATION
ABour
TI£ x-Ax is
"'t.e algorithm given belo..1 as a BASIC language subprogram alla..'S rotatjoo
of the position vectors of an object a1x>ut the x-axis . I t utilizes tatogenecus • coordinates.
3DXROT 100 SUB" 3DXR0'!": P, X( ) , Y() , Z() , Tl 110
120 130 140 150 160 17C 180 190 200
210 220
230 240 250 260 270 280 290 300 310 320 330 340 350 360 370
C-7
' P:~WM81:.R
OF X, Y. Z- TRIPLETS
' 3-D
ROTATION ABOUT
X- AXIS
' X():ARRAY CONTAI NING X-COORDINATES ' Y():ARRAY CONTAINIKG I -COORDINATES ' Z( ):ARRAY CONThlNit-.'G Z-COORDINATES 'Tl:ROTATION ANGLE IN ~GREES DIM U( 100 , 4) , V( 100 , 4) '1 00 POSITION VECTORS HAT t:ZER(P , 4) ' REDINENSlON U&V FILL WITH ZEROS MAT V:ZER( P,4 ) FOR I=l TO P ' SET UP lO'.oGENEOUS LET U( l , l ):X(l) ' POSITION VECTORS LET U( I , 2):Y(l) LET U(l , 3)=Z( l ) LET U( 1, 4): 1 t.EXT I
LET TZ:Tl/57 . 2957795 ~~T T=ZER(4 , 4) LET T(l , 1):1(4,4 ): 1 LET T(2 ,2) :T(3,3)=00S(T2) LET T(2 , 3): SIN(T2) LE~ T(3, 2):- T(2, 3) ~~T V:U1 T FOR 1=1 TO P LET X(I):V(I , 1) LET Y(I ):V{I, 2) LET Z(l):V(I , 3) NEXT I
' CONVERT TO RADIANS ' REDIMENSION T FILL WITH ZEROS ' SET UP TRANSFORMATION MATRIX
' CALC . TRANSFORMED POINTS ' CALC. PHYSICAL ODOR.
SUBE.~D
~ AI..GORITlf-t FOO T~EE Dit>ENSIOOAL RoTATION
Aaoor
n£ v-Axis
'llle algorittln given belo..1 as a MSIC language subprogxam alla..'S rotation
of the position vectors of an object alx>ut the y-ax.u;. coordinat:£!s.
It utilizes taJcxJeneons
3DYPOT
tOO SUB" 3DYROTW:P,X(} , Y(),Z(),T1 '3- D ROTATIC~ AEOui Y-hXIS 110 'P:NUMBER OF X,Y , Z-TRIF~Jr.S 120 ' X( )::ARRAY CONTAINING X-RDWATES 130 ' Y():ARRAY CGN'fAINitiG Y-COORDHJATES 140 ' Z():ARRAY C0~7AINING Z-COORDINATES 150 'T1::ROTATIOt\ At.CLE IN DECREES 160 DIM U(100 , 4),V{ 100 , 4) ·100 FOSITICN VEC"IORS 110 ~!AT U:ZER(P , 4) ' REDDiENSION U&V FlU. WITH ZEROS 180 ~!AT V:ZER{ P, 4) 190 FOR I: 1 TO p SET UP HCt-1CKiE.NFDUS 200 LE1 U( I , 1) :X( I) I POSITICtl Vf.CTCRS 21C LET U{I ,2):Y(I) 220 LET U(I , 3):Z(I) 230 LET U(I , 4): 1 21.;0 NEXT 1 250 LET T2:T1157 . 2957795 'CONVERT TO RADIANS 260 MAT T::ZER(4 , 4) ' REDIMENSION T FILL WITH ZEROS 270 LET T(1, 1}::T(3,3):COS(T2) 280 LET T(3, 1}:S!N(T2) 290 Lt.'T T( 1, 3) :-T(3, 1) 3CO LET T(2, 2)=T(4. 4):1 310 HAT V::U•T ' CALC. TRAl\SFORHED FOINTS 320 FOR 1::1 TOP ' CALC. PHYSICAL COOR. 330 LET X{I)::V(I, 1) 340 LET Y{I):V(l ,2) 350 LE: Z(I):V(I , 3) 360 NEXT I I
370 SUBENO
C-8
~ Al..GoR Jnt1 FOR Tt-REE DWENSIOOAL RorATIOO
ABouT
n£ Z-Axl s
The algorittln given below as a BASIC language sut:program p.rovides far three-
diJrensional rotation about the z-axis . 3DZROT
100 SUB"3DZHCT":P , X(),Y() , Z() ,T1 '3- D ROTATION AB0111' Z- AXIS 110 'P:NUMB~n OF X, Y, Z- TRIPLETS 120 ' X{)=ARRAY C(»lTAINI NG X-COORDINATES 130 'Y():ARRAY CONTAlttH.'G Y- COORDDJATES 1 Z():ARRAY OONTAINih'G Z- OOORDltlATES 140 150 'Tl:ROTATICN ANGLE IN Dt::GREES 160 DIM U(100, q} , V(100 , 4) ' 100 POSITION VECTORS 170 HAT U:ZER(P, II) ' RED~SION U&V FILL WITH ZEROS 180 11.AT V:ZER(P, 4) 190 FOR I: 1 TO P ' SET UP HOKCKifllEDUS 1 POSrTION VECTORS 200 LET U(l , l):X(l) 210 LET U(I , 2)=Y(l) 220 LET U(l , 3)=Z(I) 230 LET U(I , 4):1 2qO
UEXT I
250 260 270 280
LET T2:1 1/57.2957795 11AT T:ZER( 4, 4) LET T(1 , 1):T(2, 2):COS(T2) 1.1-.'T T( 1, t'):-SW(T2)
' CONVERT TO RADIANS 1 REDD'.SiSION T FILL W ITH ZEAOS
I~
I
II
I
I
'
290 3CO 310
LET T(2 , 1):-T{l , 2) LET T(3 , 3):T(~ . ~):1 ~~T V:U•T f"Ok I: 1 TO P LET X(l):V(I , 1} LET Y(I):V(I,2) LET Z(l)=V(I , 3) f,E.XT I
320
330 3~0
350 360
'CALC. TRANSfORMED POitrrs ' CALC. PHYSICAL COOn .
37C SUBEND
C-9
•
~ AlGCRint1 FOR THREE-DU'£NS IOOA1.. R£fLECTlct4S An algorithm which will perfonn three-dJ'iJTellSi.onal reflectialS th.rough the
xz, or yz is given bel~ as a BASIC language sutpr:ogxam. Reflections about other planes can be cbtained by a cacb.inatioo of rotation and reflection .
ooordi.nate plat'.es
~,
3DREFLT
100 SUb"3DREFLr' :P, X(} , Y() ,Z{) ,N
' 3~0
1110
'P:NUMBER OF X,Y , Z-TRIPLETS I X( ):ARRAY CONTllillt:G X-COORDINATES 'Y():ARRAY CONTAlfnNG Y-COORDINATES 'Z( ):ARRAY CONTAHIING Z-COOf!DINATES
150
I
110 120
130 160
170
180
190 20C
210 220 230 240 250
260 270 280
N:REFLECTION CODE: 1:)[{-Pl.ANE, 2:Yl-PLAflE, 3=XZ-PLA~t:
DIM U(100 ,4}, V( 100, 4} f"IAT U:ZER( P, 11)
·100 POSITION VECTORS
FOR 1:1 TO p LET U(l , 1 ) :X( I}
SET UP fD'.OOEl~S 1 POSITION VECTORS
~~T V:Z~R(P , ~)
LEi U(I , 2):Y(l)
flEXT 1 ~.AT T:ZER( 4 , L) If N:2 THEN 310 If N=3 THEN 3~0
T(1 . 1 ):T(2 , 2):T(~ , 4):1
290
~i
T{3 ,3):-l
300
GO TO 360
310 320 330 340 350
LET
310
38C
390
~00
~ 11 ~20
' REDIMENSION U&V FlU. WITH ZEROS I
LET U(l , 3):Z(I) LET U(I ,11):1
LET
360
REFLECTIOtiS
T (~ . 4):T{3 , 3):T(2 , 2) :!
LET T{1 , 1)=-1 GO TO 360 LET T{l, 1):T(3,3)=T(4 , 4):1 LET !(2 ,2) =-1 f".A T V: U•T FOR 1:1 TO P LET X(I):V(I , 1) LET Y{I):V(I , 2) LET Z(l):V(I,3) UEXT 1
SUBEND
' REDIMENSION T FILL WI'Ill ZEROS ' SET UP TRANSfORMATION ~lATRIX
CALC , TRANSFOR.'IED POHITS •CALC. PHYSICAL COOR. 1
DATA 81& f.'.ANIPUI.ATIQl ALOJRITI!I-ts 207
c-10
AA h.GORJTK'l FOO TlffE-Dit-ENSI~AL. TRANSlATIOO An
algorit:hn for translation in three dinlmsi.al!> is given below as a BASIC
l anguage
~rogram .
It mlkes usc
of~
100 SUB" 3DTRANS" :P , X(),Y(),Z() , L ,l'I,N 110 ' P:NUMBER Of X, Y, Z- TRlPLeiS
'3-D
coordinates.
TRANSUTJQN
120 130 140 150 160 170
' X():ARRAY CO~TAIN!NG X-COORDINATES I Y():ARRAY CONTAINING Y-COORDINATES •Z{):ARRAY CONTAHUNG Z-COORDINATFS ' L.:X TRANSLATION FACTOR ' N:Y TRANSLATION FACTOR I N:Z TRAl&SLATION f'ACl'OR
180 190 200
DIM U( 1 00 , 4) , V(100 , ~) MAT U:ZER(P , 4) MAT V: ZER(P , 4)
• RFDINENSION U&V f'ILL WITii ZEROS
2 10 220 230 240 250 260
FOR 1= 1 TO P LET U(l , l):X(I)
'SET UP HCI'lOOfllf..OUS 'POSITIOt~ VECTORS
LET U(l , 2):Y(I) LET U(I ,3)=Z(I)
LET U(I , ~)=1 NEXT I HAT T:ZER(4 , 4) LET- T{ ' , 1):!(2, 2):1 LET T{3,3)=T(4 , 4)=1 LETT(li,1)=L LET T(4,2):H LET T(4 , 3):N HAT V: U*T
270 280
290 300 310
320 330 3110 350 360
FOR I: 1 TO P
LET X(I):V(I , 1)
' 100 POSITION VECTORS
REDD-IENSION T FlU. WITH ZEROS • PLACE 1•s ON orAG . OF T-MATRIX I
'SET X TRANSLATION FACTOR ' SET Y TRAMSLATIOt~ FACTOR 'SET Z TRAUSLATION FACTOR 'CALC. TRAt~ POSITION VECTORS ' CALC. PHYSICAL COOR.
LE~
Y(l):V{l , 2) LET Z(l):V(l , 3) NEXT I
370
380
390 SUBEND
C-11
/'Jt4 Al.GORJlH"t FOO lt-REE-Dit-e4SIONAL ~TAT I ~
Ainrr Pt~v
ARBI'TAARY Axis IN
SPACE An algori t:hn which will produce a
ional rotat im about an
arbi trary axis in SP-loe is given below as a BASIC language subprogram.
based on Eqs. (3-17) ctnd (3-18).
3£X;ENROT
100 11 0
120 130 1110 150
SUE" 3DGEN~':P , X{) , Y() , Z() , Nl , N2 , N3,T1 ' P=~R OF X, Y, Z-TRIPLETS
' 3-D GENEFAL ROTATION
' X()=ARRAY CONTAllll~~ X-COORDINATES ' Y():ARRAY CONTAltllt~ Y-cooRDlNATFS ' Z():ARRAY COt.TA!Nlt;(; Z-OOORDlNATES ' N1:DIRECTION COSINE OF ROTATI~ AXIS W.R.T . X-DIRECTION
It is
t
II I II
16G 170 1BlJ 19~
2r 21~
nr 2Y 2~0 ~5r
;2:r-II.vc · c~ ,~ ~ ' lOG f":.>.'"IT!UN VECTOPS P~T L: ZE?(f , ~) ' FEDIY.E'lSlGtl ~'J FILL '.OITH ZEfiOS 1 1
P~T
FOr I= 1 ro P LET UCl 1 l):Y.(I) LF: UCI , 2):Y(I)
2£0
nr 2er
?jC
30"
;,r
~2G
3')f' 3140 "'SG 3~0
37r 3e('l
;r;,o llQt ~tr
1,2f' ~~G ~uo
I< 50 J~ f.O
V:ZEF.(r l 4)
~-r
U(I , 7 ):Z(I)
LE1
Ull,~J:l
t-."EYT 1 I'J 7 T:ZEP( I. , u) LI='T T2:T11!:7 .29':,7795
Lf.T l.E7 lEI' LE1' Lf.T LET LI='T
' ?EC:P.I.f1,!:IC'! 1 fill \HI'!" ZE.~S ' a;!NEF.l Tl TO RAtlANS 1(4,4):1 · ~r:; UP Tr·ANSFCF ~.ATlO~: ".ATPIY TC 1 1 • J :~;l • ',l+( ,-t.1 1 fol} 1 COSC12J 1( 1 ,2) :Nl • t;2 1 ( l-COS(T2) J•'f» !:;IN(T2) 1( 1, 3J=~i1 1 1·3 1 C 1- COHT2) )- 1~~ "'1!.(1'2 ) 1(2 1 1):•,1 1 !.2 1 ( 1 -CGS(T2)H\3- :.It~(T2l TC 2 ,2) :'.2 1 1.2+( 1-t.~• t;2) 1 COSCT2) T(2, ;):t.c•:;··•( 1 -CUSCT?J> + ~:t • Sll.(T2)
LET rc3, '>=~l ' N!' =~f'IN:~ · c t-ccs
ti.T TO, ~):~;~ • •(;( 1-!.!1 'l!J 1 CVSCT2 ) W,T V:U 11 ' ':I.LC. TFM;:FOf"EC FC!t\TS FOr 1:1 T~ r LET Y( 1 ):V( I I 1)
' C~LC. fP.Y~IC~L coo~.
LET l(l):V(l , 2)
LET
Z(I):~(I,!)
t.'EY.T 1
suwm
An ll9Qritl"rl ft;_,r a s.wq::le c~.J.c proJectl.oo c.nto a ~ro plane per-
IY-"ldlcular w MY of the tl'.ree
o~nal
axes is
g1~ l:x:lo.~
e.s a BASIC
S'~Og't'al"'.
PFGJ lif. ..UE:"HCJ" : P, Y.(), Y( J , Z{) 1 t1 ' I'FOJE.CTIOtl 11C •r-=r:uveEP or x, YI z- TRIPL'!S 12C ' Y( ):AP~AY CGtm,H\HC Y-COORDINATES 1~" ' Y( ):APPAY CCI;TAININC Y- COJFCINATES lt:C ' l( ):APf
t'~T 'J=ZE~(P,4)
2r'
FCF I= 1 Th P L£: UC: , l):X(J) 1..£7" uc , l' J='f CI J
21~
nr
2~0
' SET UP
~~E.'~ctiS
' ?r~!TIO\ VEC7C~S
LET U(1 , 3}=ZC!)
2~r
LE7
2Sr
Pt'E>'T I
?'f
111\l T:ll~(IJ,t)
Uf! , ~):l
· ~EDl"~PlSICti
T Fill IOI'ni ZETtOS
la.'lqllage
270 280 290 300 310 320
330 3110
350 360 370 380 390 QOO 410 1120
LET T(1,1):T(2 ,2)=1 LET T(3,3):T(~ , Q):1 IF N= 3 THEN 350 IF N:2 THEtl 330 LET T(1 ,1 ):0 00 TO 360 LET T(2,2):0 rn TO 360 LET T(3 , 3): 0 ..-.AT V=U•T FOR 1:1 TO P LET X(I):V(l,l) LET Y(l):V(I, 2) LET Z(I):V(1,3) NEXT I SUBENO
' PUT
1' S 00 AlL Ot A ELF-Y.ENTS
' SET APPOOPRIATE
DIAOOt~AL
TERt1:0
' CALC. TPAt:SFOFME'D POI NTS ' CALC. P~YS!CAL ODOR .
C-13 A DIMETRIC PROJECTIVE AlGORITif'l An algoritlm which i.nplercents the specific dirretric transfornatioo given
in Eq. (3-26) and ass\11\ing projection ooto the z = 0 plane is giwn belo.~ as a
BASIC language subp.rogmrn.
Dlt1E1RIC 100 SUB"Dlt-IETRIC": P, X(), Y() ,Z() ' Dlt-IF1'RIC Pnc)JFCTION 110 ' P:NUMBER Of X, Y,Z-TRIPLETS 120 ' X():ARRAY ~iAINI~G X-COORDINATES 130 ' Y():ARRAY CONTAINING Y-COORDllJATES 140 ' Z():ARRAY CONTAINING Z-COOPD~ATES 150 DIH U( 100 ,4) ,V(100,4) ' 100 POSITIO~ VECTORS 160 MAT U:ZER(P , 4) ' REDIY.EriSION UbV FILL WITH ZFROS 170 MAT V:ZER(P , 4) 180 FOR I= 1 TO P ' SET UP OOMOOENIDUS 190 LET U(I,l):X(I) 'POSITION VECTORS 200 LET U(l , 2):Y(l) 210 LET U(l , 3):Z(I) 220 LET U(l ,4)= 1 230 flEXT I 240 MAT T:ZER( 4, 4) 'REDIMENSION T FILL WIW ZEfiOS 250 LET T(1,1):0.925820 ' SET UP TRA NSFOR~ATION ~~TRIX 260 LET T(1 ,2):0.133631 270 LET T(2 ,3): 0.353553 280 LET T(1,3):-0. 353553 290 LET T(2 ,2):0.935414 300 LET T(3,1):0. 37796Q 310 LET T() ,?):-0. 327327 320 LET T(3 , 3):0.866025 330 LET T(!l ,4):1 340 t'AT V:U•T ' CALC. TRAfJSFORfiATlON 350 FOR 1:1 TOP ' CALC. PPYSlCAL ODOR. 360 LET X(I):V(l ,1 ) 370 LET Y(I):V(l,2) 380 LET Z(I):V(I , 3) 390 NEXT I 400 CALL "PROJ": P,X() ,Y(),Z() ,3 'PROJECT ON TO Z=O PLA~f 410 SUEEIJD
It
C-14
fJ/~ lsa-£TR1C PRoJECTIVE /\.GOOint1 An
algonthn \ofu.ch ir.ple-ents the
Sec. 3-!.o (1 .e.,
c.q. (3-26)
wit.'"t e
i
sa etric tranSfon ation disc:ussec! in
=35.264°
on the z • 0 plane is given bel0oo1 as a BASIC
CiOO ~ = 45°] aro assu-.J.ng proJect:..ioo 1~
sd:program.
1CC SUE"lSOt'..ET":F,X(),Y(),Z() ' ~!C PPJJex:Timl 110 'P:t~EP OF X,Y,Z-!RIPL~ 120 ' X():APPAY COh'TAII~I.ll; X.COOPDll\A!ES 130 ' 't():APPAY CC:;tAINn~ Y-cooPDINATES 11;0 ' Z{) :t.PP.AY CONTAINL~ Z...COOPCI~iATES 150
160 170 160 190
200 210 220 230
zuo
DIY U(100 ,u ) ,V(100 ,JJ ) f'I.AT U:ZE.Il(P,1.)
f'I.AT V:ZEP(P,~) f"'F I= 1 TO P
•SET
U(I ,l ):X(l) LET U(I ,2): Y(I) LET U(I ,3): Z(I) LET U(I ,~ } =1 LET
t,"EX7 I ~.AT T:ZEP(4 , U)
280 290
LET 1(1 , 1) : 7(3,1): 0.707107 LET 7(1,2):0.4082~8 LET T(? ,2):-C. C82~8 LET 1(1,3): -0.577353 LE: !(!, 3):0. 577353
300
LET
310 320 330 340 350 360 370 380
LE7 !(2,3)=0.577345 LET T(1.,4):1
250 260 210
I"J.T V: U1 T
FOR I: 1 TO P LET X(I):V(I ,1 ) LET Y(1):V(I ,2) LET Z{I):V(I,3)
ZEPOS
HJKXjE!."EOUS ' POSITICN VECTOPS (p)
' FED!~!:~ T FILL ~!TH ZEPCS ' SE! L? !PA~~fCP~ATICS ~~!PIX
' CAU:: . TRAl~FQFI.#.ATIO~ 'CAU::. PHYSICAL CXJP .
NEXT I 'PRCJECT Q:rG Z: O PtA1:E
Pi~ fu;oo tnf'l FOO PeRsPECTtVE TRANs~ATlOOS An alqontlrl ~o.hl.ch
as a
~ITr
1(2 , 2):0.816~97
390 CALL "PPOJ":P,X() ,'t() ,Z () , 3 1100 SIPI....£ND
C-15
' 100 POSITIOt: VECTC?S ' fEDI'"ENSIOt: ~V Fn.L
Bl-SIC
will generate a
~
perspective
V1ewi
1s given
lan
l UJ SUC." FC.r~FEC": f 1 ,X( J , Y() , Z() , F, C, F. • fE.f.SPECT 1\'E. !f..:J...Fvf.YJ.UC!i 11 C: ' P 1:MJ.'o.!:Ef' Vf X, Y, Z- :?IPLETS 1eO 'Y (): A"FA Y CG.,1/.l1.lt.G >:- CCA..FIL:tJ.rt..S
b?lcy
130 140 150 160 170 180
190 200 210 220
230 240 250 260 270 260
290 300 310 320 330 340 350 360 370
380 390
'Y():ARRAY-CO~TAltcll\G
Y-COOfiDINATES
' Z():ARRAY CO~TAI~1NG Z-COO~Dl~ATES 1 P:X- AXIS POINT (;f PROJt.CTlOI• 'C:Y-AXIS f'Ol~T Of PROJECTIOf, ' R:Z-AXIS POINT OF PROJECT!~ DIH U( 100,4),V(100,4) I 100 f'CSITIOJ'oc VE.C'I\JhS MAT U:ZER(P1, 4) I REDlMEI..SlU. ~ v FlU. ~11li ZE.hGS MAT V:ZER(P1 , 4) FOR 1:1 1U P1 ' SET UP IUIC(jEM:OUS2 10 LET U( l ,1):X(l) I POSITION VEX:TORS Lt.T U(l , 2):Y(l) LET U(l, 3):Z(l) LiT 0(1,4):1 HI:.XT I NAT-T:ZER( 4,11) REI.ilMOOlON T- FILL ~ITh ZEhOS LET T( 1, 1):T(2,2):1 I PL.ACf. 1' s 0~ D lAG. Gf T I'IATfHX LeT 1(3, 3):T(4, 4):1 Ll:.-r T(1,4): P ' SET X PROJbCTiotJ Pvl~~ U-1' T( 2,4 ):0 I SE.T y Pf\GJECUON POltlf LET T(3,4):fi I SET z PftbJECTION PU 11.1 MAT V=U•T ' CALC . TfiANSfORMA1'lON FOR I:1 10 Pl ' CALC. PHYSICAL COOn. Lli X(l}:V(1 ,1 )/V(l ,4) LI:.T Y(l):V(I ,2)/V(l , 4) LiT Z(l):V(I,3)/V(I , 4) NeXT 1 SUbE.t
C-16
I
Ttte-DWENSJONAL RECONSTROCTJON Au~rm'IS
tive project.icns, say, tw:> fh:::>togxafi\5, \orilen the natrix transfOll'IVltion e.l.elrents
are known is given
bela.~
as a BASIC language sutprogxam.
It is based
(3-48) .
3DRECON1 100 SUB"3DRECON 1":T( , ) ,S( , ),X1 ,Y1,X2 ,Y2,X(, ) ' 3-D RECONSTRUCTION-I 110 ' T(, ):4• 1l TRANSFORMATION ~lATRIX FOR THE FIRST VIEW 120 ' S( , ):tl1 4 TRANSFO~ATION mTRIX FOR THE SEX:OND VIEW 130 'X1 , Y1:000RD1NATES OF POINT IN FIRST VIEW 140 'X2,Y2=000RDINATES OF POINT IN SECOND VIEW 150 DIM A( 4, 3) ,B(4 ,t ) ,C( 3, 4) , D(3 , 3) , E(3,3) , Y(3 , 1) 160 ~AT X=ZER( 3,1 ) ' FILL X-MATRIX-WITH ZEROS 170
180 190
200 2 10 220 230 240 250 260 270
280 290
FOR J: 1 TO
3
FOR I: 1 TO tl
' FIND
POOJEX:'TIOt~
PLANE
' IN T-t".ATRlX
IF' T(l,J)<>O niEN 220
NEXT I 00 TO 250 NEXT J PRINT "TRANS T IS rm A PROJECTION" 00 TO 660 ON J 00 TO 260 , 290 , 320 'ELIMl~ATE 0-COL~J LET I 1=2 I FR(Xo1 EOOA TIONS LET 12=3 GO TO 340 LET 11:1
Q'l Elg.
300
310 320 330
3il0 350 360 370 380 390 400 1;10 420 430 IIIJO 450 IJ6C
470 460 490 500 I
510 520 530 540 550 560 570 580 590 600 610
LET I2=3
ac m
3110
LE1' 11:1 LET 12:2
FOR J=l TO 3 'F'IND PAOJE'CTION PLAr.E FOR I: 1 TO ll ' IN s-f".ATRIX IF S(I , J)<>O TPEN 390 NEXT I GO TO 420 • NEXT J PR!tii "TFANS S IS NOT A PROJECTION" ro ro 660 Oll J GO TO II 30, 460, 49C' ELit'IMTE D-COLl11 N LET J1:2 ' f'PO~l EctJATIONS LET J2=3 GO TC 510 LET Jl: 1 • l-...t'T J2::3 CO TO 510 LET J1:1 LET J2=2 FOR 1:1 TO 3 ' SET UP A-fo'ATRlX 1 LET A(1,I):T(I,I1) - T(I,4) X1 LET A(2,I):T(I,I2)-T(I,4) 1 Y1 l-~ A(3,I)~S(I,Jl)-5(I,4) • X2
LET A(4,l):S(I,J2) -S( I, 4) 1 Y2
I l-~ E(1,1):T(IJ,4)'X1-T(4 , 11) LET E(2,1):T(Il,li) 1 Y1-T(IJ , I2) LET e(3,1):S(4 ,4 )1 X2-S(4,J1) LET 8(11,1):S(4,1J) ' Y2-S(4 ,J2)
h"EXT
~AT C::TRN(A) ~:ATRIX D::C1 A
620 610 fJ.AT E:li\V(C) • 640 fo'AT Y:C1 E 650 ~:AT Y.=E'Y 660 SUEElJO
' SET UP P-1-'.ATfiiX
'TRANSPOSE' A ' fORM SC ('IATfllX FRO~ A & lT' S ' INVE~T SCOARE VATRIX ' FORM A TRANSPOSE • E '08TA IN SOLUT!Ot.
~ANSPCSE
to reconstruct t.~ transformation used to obtain a perspective projection fran the )o1or..n location of six p::>i.nts in the I=bYsical and projectiO'\ An algoritlln
spaces is given below as a BASIC language subpt'Ogram.
It is based on F.q . (J- 50)
in the oor.nalized fonn.
3DF£CON2 1C<> SlJ8"30REX:ON2" :X(,) , ll( , ) , TC , ) '3-D RECONSTRUCT!Oti- 2 110 'X( , ):61 3 fo'ATIHX COIITA~lt.G n!E LOCATIO.\S Of' nzE SIX POif.liS 120 ' lN PHYSICAL OOORDI~ATES I~ ORDER 130 ' U( , ):6• 2 ~ATRIX COUTAINING WE LCCATIOt,S Of' TilE SA:-'E SIX 140 ' POH:I'S IN 1liE PEPSPECTI\'E VIEW I~ nzE SAt--E OROEP 150 'T( , ) :TJJE ll• 4 TPAt~F'OR."'.ATlON HATfllX 160 DDI. A( 12 , 12) , 8 ( 11 , 1l ) , C( 11 , 1) , D( 11, 1) , E( 11 , 11 ) 170 MAT T::ZER(4 ,ll ) ' FILL T&A ~ATRICES ~ ZEPOS 180 MAT A=ZER( 12,12) 190 FOR 1=1 TO 12 STEP 2 ' SET UP A~.ATPIX 200 LET J=J+l
210 220 230 240 250 260
LET A(I , 1):A{l+1, ?):X(J , 1) LET A(I,4):A{I+1 , 5):X(J, 2) LET A(I , 7):A(I+1,8):X(J,3) LET A(I , 10):A(l+1,11 ):1 LET A(l , 12):U(J,1 ) LET A(I+1 , 12):U(J , 2) LET A(l , 3)=-X(J, 1) 1 U(J ,1 ) LET A(l+1 , 3)=-X(J , 1) 1 U(J , 2) LET A(I , 6) : - X(J, 2) 1 U(J ,1 ) LET A(I+1,6):- X(J,2) 1 U(J,2) LET A(I , 9):-X(J, 3) 1 U(J , 1) LET A(l+1 1 9):-X(J , 3) 1 U(J,2) NEXT I FOR 1=1 TO 11 FOR J:l TO 11 LET B(l ,J ):A(I,J)
210
280 290 300 )10
320 330 340 350 360 370 h"EXT J 380 LET C(I , 1):A(l,12) 390 NEXT I 400 MAT E:lNV(8) 410 MAT D=E*C 420 LET T(1 , 1):D(1 , 1) 430 LET T(1 , 2):D(2 , 1) 440 LET T(1,4):D(3,1) 450 LET T(2, 1):0(4 , 1) 460 LET T(2 , 2): D(5 ,1 ) 470 LET T(2, 4):D(6,1 ) 480 LET T(3,1): 0(7 , 1) 490 LET T(3 , 2):D(8,1) 500 LET T(3 , 4):0(9,1 ) 510 LET T(4 , 1) : D(10 , 1) 520 LET T(4 , 2):D(11,1 ) 530 LET T(4 , 4):1 540 SUBEND
C-17 A STEREO
' SET UP P.&C t'.ATRICFS ' FOR~ e-~ATP.IX
' fORM
C~ATRIX
'CALC. TRANS.
~~TRIX
~Illfl
to generate a stereo pair for an abject ....nich has been positiooed and rotated to fonn the desired view and perspective is given bel.c7.ol as a 81\SIC language subprogram. An algorithm
STEREO
100 SUB" STERfl>" : P1 , X( ) , Y( ) , 1 () , K, E' , A( , ) , B( , ) 110 ' P1:tlUM!ER CF POINTS USED 10 DEFitiE OBJECT
120 130 140 150
160 170 180 190
200 210 220 230
' X0=ARRAY OOtri'AINI~G X-COORDD;ATES ' Y():ARRAY OOtri'AI NING Y-GOOPDINATES ' Z( ):ARPAY rotri'AINING Z-cODRDINATES ' DIM X(Pl) , Y{P l), Z(Pl) It-; t-'Alt\ PflOORAt-1
' K=FOCAL LENGTH OF STFRW VJE'fo'ER ' E:DISTAt\CE EETWEEti EYES Of OBSERVE? (RELATIVE' 'IO OUTPUT g:ALE) ' A( , ):RESULTUlJ X1 -, Y1 -COORDINATES FOR LEFT EYF
' DIM A(P1, 2) IN MAIN PROGRAt-1 ' B( , ):RESULTING X 1 -,Y 1 -COO~DINATES FOR RIGHT EYE 'DIN B(P1 ,2) IN t'.Alll PROGf!AtJ Dlfo! U( 100, 4) DUI l( 100 , b)
214
~CAL
ElEMENTS FOR a:t-1Plll'ER GRAPHICS
240 DIM R(100 ,4) 250 DIH P(100 ,4) 260 DIM 0(100 , 4) no FOR I:l TO Pl ' SET UP ~"X3 THEN 520 510 LET X3:X1 520 IF X1X7 THEN 560 550 LET X7=X5 560 IF X5
590 600 610 620 630 640 650 660 670 680 690 700 710 720 730 740
NEXT I LET O:(X6+X7)/2- (X3+X2)/2 LET
t-'.AT
LET LET ti.AT LET
' F~D
EYE SEPARATION
DISTA~CE
L:D+(E-D)/2 T: ZER(ll , 4) 'TRANSLATE PICHT- EYE VIEW T(1 ,1 ):T(2 ,2):T(3 ,!)=T(4 , 4):1 T(4 ,1 )=L R:P1 T 1(4 ,1 ):- L 'TRANSLATE LEFT-EYE VIEW
t-'.AT L=01 T
MAT A=ZF.R(P1,2) MAT S:ZER(P1 , 2) FOR 1: 1 TO P1 LET A(1 ,1 )=L(I ,1 ) LET A(I ,2): L(I , 2) LET B( I, 1):R(I , 1) LET E(l ,2):R(I , 2) ~EXT
' SET UP RETURN
~ATRIX
1
750 SUE!EtJD
C-18
~ Al.GOO lnt\
FOR
A CbwARN"ETRIC CIRCLE
'Ille Al.goritnn given below as a BASIC l..anguage subprogram generates a circle based on a nonpa.rairetric representatioo.
4-5.
The Algorithm
1.n
It is based on the discussi oo of Sec.
the next section is reo:xntended for circle generation .
t-.PCIRCU 100 SUB"hPCIRCLe" : H, K, R, t-. , X() , Y()
110 120 130 140 145 150 160 170 180
190 200 210 230 240 250
260 270 280 290 300 31o 320
330 340 350 360
37C
380
390 400 410 1120
430 440 450 460 ll70 480 490
500 510 520
530 5~0
550 560 570 560 590 600 6 10
' ' ' ' ' ' '
H, K:X&Y COOR . OF CEIITER R: RADI US OF CIRCLE
' f,(ltiPARAMITRIC CHiCLE
t-.: NlR-lbER Of POIf\ T!:> L'£.SII1ED X( ):ARitt.Y COUTAit>lhG TI-E X-cooROIMTES Cf CIRCU: Y( ):ARRAY CONTAlhlt.'G lliE Y-OX)RDlkATFS OF ClkCLE NOTE :t\ HliST BE DIVISIBLE BY EIGHT . If NOT PPOGkAM WILL USE ~EXT HIGHEJ\ NUMBER Dl VISIBLE BY EIGhT LE.T A::lloT(~/8) ' DE1'E.I
HAT
u..,-
Y: ZER(~ 1+1 )
D:R1 ( 1-1 /SCR(2))/(N1/8)
LiT X( l) : R
' CALC. INCREl'IF.NT ' CALC. INI1IAL POIN1
LET Y(l ):O FOR 1:1 TO N1 I F 1: 1 TIJE.tl 370
' CALC . POINTS AROUND ORIGiN ' DETERMINE SECT. FOR CALC .
I F A&S(X(I))<0. 0005 THEN 320
GO ro 340 lF Y(l )>O THEtl 440 GO TO 590 IF Y(l)<0. 0005 THEt~ 500 IF Y( 1)<0. 0005 11-lEN 430 1F Y(l )>: R/SCR(2) - 0.0005 Thlll 400 U.T X( I+l ): X(I) - 0 ' CALC. PO~TS LET Y(l+1):SQR(ABS(R• R- X(I+1 ) 1 X(I+1 ))) au 'IV 640 LlT Y( l+1):Y(l) +D ' CALC. POit.TS LET X( l+ 1):SCR(A6S(R*R- Y(l+1)• Y( l +1))) GO ro 640 IF Y( l )<:R!SCR(2)+0. 0005 Tr~ 470 Lli Y( l +1):Y(I)- D •CALC. POlt.TS LlT X( l +1):-SOR(A&S(R*R-Y(l+1 ) 1 Y(l+1 ))) GV TO 640 LE.T X(l+1): X(I) - D ' CALC . PO:U.TS LeT Y(l+1):SCR(AbS(R*R- X(I+1 )*X(l+1 ))) GO TV 640 IF X( l)>- 0.0005 THEN 580 IF -Y{l) >:R/SQR( 2) - 0. C005 THEN 550 LET X{ I +l) :X(l)+D ' CALC. POINTS Le7 Y( l +l) : - SUR(AbS{n*R- X( l+l )• X(l+l ))) Gu TO 6UO LET Y( l +l):Y( I)-0 ' CALC. POINTS LET X( l +1):-SCR(AfS(k*R- Y(l+l )•Y(l+l ))) GC TO 640 l f - Y(l)<:R/SCR(2)+0. C005 THEf4 620 LeT Y{l +l):Y( l)+D ' CALC . POINTS LET X(l+l ):SCR(ABS(R• R-Y{1+1 ) 1 Y(I+1 ))) cv ro 640 Lei X( l +l) ;X(l)+O ' CALC . POINTS
0-~5
DEG.
45- 90 DEG.
90-135 DEC. 135- 180 Df..G .
18D-225 DEG . 225- 270 DEG .
270-315 DEG.
620 315-360 CEC. 630 llT Y( 1+ 1 ):- SQR(ASS( R*R- X( 1+ l) I X( 1+ 1))) 6fJO NEXT I 650 FOh 1: 1 TO ~ 1+ 1 ' T'RAHSLA'I£ ClRCl.E 660 LlT X{l):X(I)+H 670 llT Y(l):Y(l)+K 680 NEXT I 730 SllBENO
II
216 MAmEl-MICAL EUMNl'S FOR CXMroi'ER GRAPHICS
C-19 P« At..GORITK'i
~A PAJW£mtC
CIRCLE
An algontiln for a pararootric representati.al of a circl e is given as a
BASIC language subprogxam.
I t is based oo f):}s . (4-28) and (4-29) of Sec. 4-6 .
CIRCU:. 100 SLb"ClitCLE":t1 , }, , P, /. , X() , Y() ' PAF-AI'iETFIC CIRCLE 110 'h:X- CUJ110I;,ATE. OF CE.hTE.h OF THE CIRCLE 120 ·~=Y-COORDihATE Of CENTEfi OF tr.E CIFClE 13C ' .h:F.ADIUS 1l40 ' N:NUMbE.H Of POIM'S Ql', CIRCLE 150 ' X() ARkAY COI~TAirJING THE X-COORDINATES Of FOWTS Of~ CinCLE 160 ' Y() ARRAY CONTAlllUIG Tr.E Y-COOFDI!,ATES OF POINTS Qt. CIRCLE 170 LeT P:2*3.111156/(tl- 1) ' CALC . INCREMEliT Th THETA 180 LET C1:CUS( P) ' CALC. CuNSTANiS
190
LET Sl:SIN(P)
200
LE.T X(C):H+R
210
~T
Y(O):K
' CALC . INITIAL POit.T
FOR H: 1 TO N ' IM~ER LOOP LET X(M):H+(X(M-1) - H) 1 C1- (Y(K- 1) - K) ' S1
220
230 2110 250
LET Y(H):K+(X(M- 1) -H) •S1 +(Y(M-1) - K) ' C1 NEXT H
~
.<\llf,ff.JO
C-20
PARN~ETRIC Eu.IPSE Al..GOOmt\ 1he alC}Clrit:hm given below as a BASI C l.angu8
ellipse.
It is based oo Eqs. (4-38) and (4-39) of Sec. 4-8.
E.LUl'SEl lCO Sl.c"ELUPSC.l" :H,K ,A,B,I,ri, X(} , Y(} 110 ' r:X-COOFDI~ATE Of CENTER Cf THE ELLIPSE 120 130 1~0 150
160 170 180 1<;0
2CO 210 220 230
21l0 250 2£0
270 280 29C
' 1\:Y-coofiDINATE Of CEtiTER Of THE ELLIPSE ' A:LEt.GTH Of SEMI- MAJOR AXIS ' b: LEl:CTH Gf Sffil - t-:IIJOR AXIS 'I:INCL~ATIU• A~GLE OF MAJOR AXIS IN ~cGREES ' N: MJMBER Of POlt1TS ON ELLIPSE ' X() :ARfiAY CXitiTAThmG THE X-CGOROI.NATES Cf POit>TS ON ELLIPSE ' Y(}:ARHAY ca~TAl~ING TP£ Y-COORDlNATES Of POI NTS ON ELLIPSE 1 CALC . THE INCREMEliT IN TilE PARAMETER LE.T P:2• 3.1.1!156/(t-.-1 } LET 11: 1/57 .2957795 ' CONVERT I TO RADIANS
LET C1:CGS(I1) LET S 1:SIM 11) U.T C2:COS( P) U.T S2:Sit.( P) Ll1 C3:1 LET S3:0 FOR M: 1 TG t. LET X1:A1 C3 LET Y1:61 S3
' CALC . THE COSINE AND ' Sll~E
CR Til£ l t.CLlNATION At-GEL ' CALC . TilE lt-.CFIEMEh'TS HI COSINE ' A~O SINE OF THE PAPAMETER u;cREME~;r •INITIALIZE THE ACCU~ULATIO~ ' VARIABLES ' BEllIt-. INNER LOOP ' CALC. Ir.CRE11EUTS It. X AND Y
3CO 310
LET X(~)=H+X1 1 C1 -Y1 1 S 1 lti Y(I1 )=1<+Xl ' Sl+Yl 1 C1
' CALC .
320
LET T1:C3'C2-S~' S2 LET S1: S31 C2+C31 S2
•CALC . NFJoi At.GLE fOWLLAS 'DOUblE ANGLE. F'Ofi SI,_E ' OOUE!LE. AIJGLE FGk COSINE
33~ 3~0
350
360
LEi Cj:Tl hEY.T M SUbEtrD
~~~
X At.O Y
C-21
~ Al.G001nt1 ~ A PARN£rRIC PARA&U 'Ihe al
is based oo Eqs. (4-54) and (4-55) of Sec. 4-9.
subprogram
PAf\A80L.A
100 SUE" PAIW30I..An :A, PI, N,X () , Y(} ' PARA."ETRlC PARAOOU. 110 ' A:VlSTAtiCE FRCJ1 FOCUS TO VERTEX OF P~ABQLf, 120 ' Pl:I~.AXll-lUM ANGLE (DEGREES) (CF. EO. ~-45 OR 11-116) 130 ' N: Mlt'.SER Of POINTS ON PARABOLA 140 ' X():ARRAY CONTAlNII."C WE X-COORDINATES CF POINTS Cf: PAFAfCLA 150 ' Y():ARRAY COt-'TAlNlt.:c TilE Y-COORDINATES OF f>ClriTS Ct, PARAECLA 160 LET P:P1/((N-1 ) 1 57 . 295)7795) ' CALC. l~CREY.E~T IN THF.TA 170 LET X( 1): 0 ' INITIALIZE 180 LET Y( l )=O 190 LET Al:A*P*P 200 LC~ B1:2*A*P 210 FOR ~·=z TO N ' ll!t!ER LOOP 220 LET X(~I):Al+X(M-l)+P*Y(N-1) 230 240
LET Y(M):Bl+Y(I'.-1 ) NEYT M 250 SUBEND
c-22
~ ITWIS
Foo
PAIWETRIC
HvPEROOLAS
An algoritbn which generates a pa.ranetric hyperbolll is given belCM as a
BASIC snbpxogcan.
'lhe al
sec.
4-10.
l1YPE.I1b 1
100 SUB"HYPEfiB l" :A , b , N, X{),Y() 'PAAAH.ETRIC HYPERBOLA 110 ' A:DISTA~CE FllOI-' Cl:.mER Of HYPERBOLA TO VERTEX 120 ' B:DETEfiHH4ES SLOPI:.. FO ASYP.PlGTES:+-6/~ 130 't.:tMlEE11 OF POI~TS Otl HYPERBOLA 140 ' X():ARRAY CONTAlt.U~ X- COORDINATES OF POINTS ON HYPERBOLA 150 ' Y( }:ARRAY CONTAH INC Y- COORDINATES OF POit.TS ON HYPERBOLA 200 LET P:) .l 41592654/(2*( N- 1)) ' CALC . PARAt'.ETEk INCREJ.~E.NT 210 Lf.,T C2:CGS( P) ' CALC. COSlNE ,SlNE , TAMiENT 220 LE.T S2:SW( P) ' OF PARAfJETER I~CRENENT 230 LET 12=T~(P) 240 250 ~60
270 280 290
300 310
LCT Q9:b1 T2 ~1 S9:b1 C2 LET X(l):A
'INITIALIZE
LET Y(1):0 FOF< H:2 TO N 'U,NER LCOP LET X(M):(S• X(H-1))/(S9-Y(N-l)•S2)
LET t\E.XT
320 SIJSEND
Y(H):(b' (Y(~-1)+09))/(8-Y(M- 1) 1 12)
~
Another alla is given below as a BASIC subprogram.
'1his al
,
218 MMl!E:'.ATICAL EILMENl'S FOR a::MPUlER GRAPHICS
Sec. 4-10.
'1his alcprithn yields the port.ial of the hypcrlx>la in the first 'lbe port:.i.als in the other quadrants or nonorigin-oentered hyperlx»as
quadrant.
can be obtained by appropriate .rotation, reflect..i.oo, and translatioo operations .
HYPERE.2 100 Sub"hYPERB2" :A , B,C,N,X(), Y() 110 'A::DISTMICE fRGI CE~'TER OF
120 130 140 150 160 170 180 190 200 210
220 230
• PAJWIETI\IC HYPEf\BOLA
HYPERECLA TO VERTEX
•&:DETtRHI~ES
SLOPE fO ASYM~ES=+-B/A ' C::LltUT OF X-COOi1D:U ATE ' lY::Ml-lEEk Of POlhTS CU HYFERBOLA
' X():ARRAY
CON1AINI~G
X-COORDINATES OF POINTS ON HYPEI\BOLA ' Y():ARkAY CONTAlNH.C Y-CC
' lNITlALIZE
I
2110
FOn M=2 TO t. ' lJ:t,EJi LOOP U.T X(M):Ci_>aX(M-1)+(A/E)*S2*Y(N- 1)
I
250 260
LET Y(M):(b/A) *S2*X{M-l )+C2*Y (~~1) t.£XT M
I(
It li
If
II
270 SUBE!\L
C-23
~ Al.OORillfl ~
ACIRCLE TtmG! T~ PoiNTS
'!he alcprithm given below as a BASIC Stq>rogra.'ll detelll\i.nes the circular arc passing through three points.
It is based oo the discussioo of Sec. 4-ll.
3PClhAkC 100 SUE"3PCIRARC" : X(),Y() , t- , H,K, R,S(),T() 'ARC ntRU THREE POINTS 110 I X(). Y( ):COORDINATES Cf THfiEi POll\'TS 120 ' N::NW.BER OF STRAlChT UNES TO REPRESENT ARC 130 I H' K::); . ¥- COORDINATES Gf CEt'fTER ( RETURf,ED) 140 ' fi:RADIUS (RETURNED) 150 ' S():ARRAY CON1AIN1NC X-OOORDI1~ATES OF CIRCLE 1 160 T():ARRAY CONTAINING ¥-COORDINATES OF CIRCLE 170 ~~T A:ZER(3) •DIM & SET EQUAL TO ZERO 160 MAT B=ZER( 3) 190 FOR 1: 1 TO 3 ' CGPY X() & Y() ~~~ h() & B() 200 LET A(l):X(I)
210 220 230 240
2SO
260 270 260 290
300 310
320 330
LET B(l):Y(I)
!~EXT
I
CALL"21YfRANS.":3,A(),B() ,-A(1),-8(1) ' TRANS. 1ST POINT TO ORIClt\ CALL"SOLVE" :A() , 8() , H1 ,K1 ' SOLVE fOR CENTER Of TRANS. ARC
LET H:X( 1)+H 1 LET K:Y(1)+K1
' CALC.
LET R:SOR(H1*H1+Kl ' K1)
' CALC. RADIUS ' TnAfiS. CIRCLE TO ORICD; ' CALC. BECit,NlM; & ENDING At.GLES
CALL"2DTRAt~.":3,A(),8() ,-h1 ,-K1
CALL" ATAt," : A( 1) , 8( 1), T1
CALL"ATAh'':A(2) , b(2) ,T2 CALL"ATAtt" :A( 3) ,8(3),T)
11:11*57 .2957795 1 LET T2=T21 57.29577951
LET
C~TER-COOJ\OINATES
' CHANGE RADlAhS TO DEGREES
~0
350 360
310 380 390 400
410 420 430 440 450 460 470 1180 490
500
5 10 520 530 540 550
560
570
sao 590 600
LET T3:T31 57 . 29577951 IF T1T3 1hEl• 470 'Tl >T2>T3 l.I:.T t3:T3-360 ' T1>12T213 CASE 1 lF ABS(T3- T1 )<360 HltJ, ll70 ' TEST If CI.SE 1 CCR!lECT LlT T1:T1+360 'T1T3 ~ASE 2 LET T3:T3- 360 CAl.L"AkCA":H.K,fi,T1,T3,N, S() , T{) ' IlM\<1 ARC TifHIJ Tift 11iftE.E Ptilt.TS SUBEND SlJB11 SOl.VE11 : A() , B() , H1, J\1 ' SOLVE FLR COOlfi at- ClhCI.£ 1 A() , B():COOR OF 3 POINTS WHEn£ A( 1):8(1):0 ' Hl , Kl:X , Y-COOR OF CEhTER (RE1'URt.Et} MAT C:ZER(2,2) t1AT D:ZER( 2 , 1) f-1AT E:ZER( 2 ,2) JI.AT F':ZER( 2. 1) FOR 1:2 TO 3 I SET UP MATRIX LET C( l-1, l ):~A(I) LLT c< I-1, 2>=~~
l.lT D( l-1, 1):A(I) 1 A(l )+B(l) 1 8(1) NEXT I MAT E:~~V(C)
610 ' CALC. CE~TE11 620 HAT F:E'D 630 LET H1:F(1 , 1) ' SET CEliTER ECUAL TO H1 , K1 640 LLT K1:F(2 , 1) 650 SUBEI\D 660 SUB" ATAt.'' :X , 'i , A ' TAl\ G- (YX) 670 I 'i:PROGftAH Flt.:DS TAf.GI:.liT Of ('i/)1) 680 ' A:TAtiC~'T Of ( Y/X) It• RADIAl-S ( RETURUE.C) 690 IF Y
x.
85o st..w.o
86C S{j8"ARCA" :X,Y , R,Tl.T2 . l•.X() , Y{) ' PARA!-'E'Tn:C CifiCULAk AliC 670 ' X: X-CUJfd.at~A TE CR CEfiTEf.
900 9 10 920 930
' T1 , T2:bi:.GlliNH,G AND Etl!il"G AtiGlFS II~ r.t:.GfiEES 'f•:NlJJ-.bE.h
220 MMREl-'J\l'ICAL El..EMENTS FOR ClH'U'I'ER GRAPHICS
guo
LiT T3:T1/57 . 395b LET P:(12-T1)/()7 .295b •(~-1 )) LET C1:CGS(P)
95C 960 970
U:."T Sl : Slr~(P) l£'1 X( 1):X+R1 COS(T3) LeT Y(1):Y+h 1 Slh(T)) Fu£4 1'.: 2 TO N
98C 99C
1COO
' CHA~E
OtCREES TO
RADIA~
' CALC. lNCkEr-1E(ti-1)T IN RADlA(h-l )S ' CALC . COS & Sltl Cf' lliCREJ-'.EtlT ' ll.lTIIILIZE TilE ACCI.i1Ut..ATIOtl 'VARABLE
' CALC . X, Y R .
1010 LET X(N):X+( X(l'l-1 ) - X)•C 1-( Y(N-1 )- Y)*S 1 1020 LET Y(I'I):Y+(X(~l-1)-X) * S1+(Y(t-1-1}-Y}•C1 1030 fJUT ~ 1040 SUBErlu
C-24
~ Au:.ootnt1
Foo
C£NERAT1NG
Cusrc SPurt:s
II
to n lax:Jwn data r;oints with various end bounda.ry cxnlitioos is given belCM as a 81\SIC language subpX'OgUim. '1be algorithm calculates the paran-eter range based upon chord di.stance between data r;oints rather than usi.nq noonalized splines .
I
SPUhE
An algorithn for generatin
100 SUB"SPLit.E" :S ,t. ,P( , ) , Cl ,C2 , N( ,) ,B(,) ,L(), Z, U(,) ,C(,) 110 ' S:CONTROL VARIA8LE S=2 FOP 2-D CURVE, S=3 FOR 3- D CUIWF. 120 • tl=t:tJPEER OF DATA rorrrrs 130 ' P( , ):ARRAY COt.'TA lt.l}.{; DATA POINTS 1!10 ' Cl:WITIIil END COt\DITIOr, 150 ' 1: RELAXED 160 ' 2:CLAt·IPED 170 I 3=CYCLIC 180 'll:AJITICYCI..IC 190 ' C2:Fit.AL E~D OONDITION 200 '!:RELAXED 210 ' 2:CLAMPED 220 · ~mE: IF A CYCLIC OR AA'TICYCUC TIIITIAL EllD CONDITION 230 ' IS SPECIFIED TI!F FINAL EJJD CONDITION IS ALSO SPECIFIED 21l0 'N(,):MltiZEliO El.FI~ ENTS OF M-t-'..ATRIX, SEE EX). (5-33) 250 'f~ MUST HAVE Dlf'ltflSSIONS OF ( N, 2) OR 01 , 3) 260 ' 8( , ):ELEMENTS OF B-11ATRIX , Sfl: EO. (5-33) 270 ' B ~lUST HAVE Dll'l.ENSIONS OF (2 ,N) OR (3 ,N ) 2b0 ' l():SPAN CHORD LENGTiiS 290 ' L ~11JST HAVE DlJffiNSIORS OF ( N- 1) 300 ' Z.:tru'~BER CF IrrTEf\f1.EDIATE FQHITS PER SPAN 310 ' U( , );;ELEI".EtJTS IN TANGENT VECTOR MATRIX , SEE EO. (5-33) 320 'U MUST HAVE D~NSIOtlS OF (2 ,N ) FOR 2-D AND (3 , N) FOR 3- D 330 'U( 1, ):X-COMPQt,J:lrr 31.0 'U(2, ):Y-COOPOt~Etrr 350 'U(3 , ):Z-COMPOl f.l,T IF APPROPRIATE 360 'f.oTE: IF CLAMPED El\0 CONDITIOt-JS ARE SPECIFIED 370 ' 1FFJ~ 11fF VALUES FOR TilE TANGFUT VJ:X:TORS FOR 380 'lliE fiRST ao Cl>tiDITION tiUST EE IN U( 1 , 1), U( 2 , 1), U( 3 , 1) 390 ' AIJD FOR WE LAST El~D COliDITiotJ IN U( 1,tl), U(2 , N}, U('3 , N) !IOO ' C( , ):ARRAY rorrrAINnx; GEltERATED FQlllTS Al..Of
ll50
LET Z=Z+1 'ADD a\E TO 00. INTE!lMEDIATE POINTS IF C1 >2 THEN 560 ' GENERATE OiORD L.ENGn1S AND El.rnEt-.TS IN B-,u-, At..n 1"-t'ATRICES ' \t.'HIOI DEPEt.D l1POl\ Th"E SPa:IFIED El>L rotUHTIOt~S.
ll60
ll70 l!8o ll90
CALL" EhDP~T": S ,N,P ( , ) ,N(,), H( , ) , L(),Z,U(,),C1 , C2 'D~.PLEME~i
GAUSSIAN EllMINATION TO S)LVE FOR WE ' U"1
500
510 520 530
co ro sao
51l0
'INVERT ~:ATIUX M AltO SOLVE FOR m;K~tl TANGEUT \lECTOR ' (CYCLIC AND ANTICYCLIC SPLThES ONLY) CALL''MATINV": S , N, P( , ) ,N(,), I1(,),L() , U(,),C1,Z ' GENERATE ll!E POl US Clf TliE SPUUE CURVE CALL"CURGEN":S , .. ,P( 1 ) ,L() , Z, U( , ) , C( , )
550 560 570 580
590 SUBEND 600 SUB"Et.DPNT'' :S, N, P( I ) ~ ~< 6 10 620 630
~IATF!IY
U.
' ) , B( ' ) ,I..() IZ , U( I) ,C1 ,C2
lF C1: 1 THEN 680
LET N(1,2):1 LET N( 1,3):0
61lO
FOR K:l TO S LET B(K 1 1):U(K ,1 ) NEXT K GO TO 700
650 660 670 680
' FIRST POW OF
t'-~ATRIX
' FOR CLAMPED INITIAL fllD COIJDlTlotl ' SET
B(K, l):U(K,l) FOF CL»1PF.D END
LET M1 , 2) =1 'FIRST ROW Of t-1-MATRIX LET tl( 11 3) : . 5 ' FOR PELAXEC D;D CO"DITIOI~ FOR J:l TO N-1 lF C):3 THEt 760 ' CALCULATE TliE 2- D SPAN CEORD l..EttcniS
690 700 710
720 730 71lO
LET L(J ):SCR((P( 1,J+1)- P( 1,J)).2+(P(2,J+1 )- P(2,J))-2) GO TO 770 ' CALCULATE iliE 3- D SPAN CHORD LEtm-P.S
750
760 770
8oo
LET L(J)=SCR((P( 1,J+1 )- P( 1,J))-2+(P(2,J+1 )-P(2,J)).2+(P(3 ,J+ 1)- P(3 ,J))-2) NEXT J IF C1:2 THEN 820 FOR K:!l TO S ' SET B( K, 1) FOR RELAXED END LET B(K , 1):(3/(21 L(1)))• (P(K ,2)- P(K,1))
8 10
NEXT K
fl20
IF C2: t
830 840 850 860 870
LET ~(N , 1):0
7&>
790
880
890 900
910 920 930
940
TH~ 890
LET N(N, 2): 1
' LAST RO',J OF t-1-MATRU ' FOF CLAl-lPED FINAL DJD CONDITION
FOR K:l TO S LET S(K , t~):U(K , t-t) ' SET E(K,N):U(K , N) FOR CL.Afo'PED EJ.ID NEXT K GO TO 9ijO LET N(N ,1):2 ' LAST ROW OF 11-MATJU)( LET N(N 1 2):1l ' FOR RELAXFD END CONDITION FOR K:1 TO S ' SET B(K ,N) FOR RELAXED END LET B(K , N):(6/l.(t\-1)) 1 (P(K , N)-P(K , N-1}) NEXT K SUBEND
950 SUB"CAUSS": S, t, , P ( , ) , ~ ( , ) , f( , ) , L() , U( , )
960 970 980
990 1000
1010 1020 1030 l OllO
FOR
J:2 TO N-1
LET N(J,l}:L(J) ' CREATE t.()t\ZERO VALUES FOR L.ET N( J , 2):2i'( L(J)+L(J- l)) 'ltiTERNAL RO\t.'S CF ~1-!·'ATRIX LET ti(J , 3):L(J- 1) FOil K= 1 TO S ' CREATE flOWS 2 nillOUGH r.- 1 OF P41ATJHX LET B(K , J)=3• (L(J-1)~2' (P(K,J+1) -P(K,J)) +L (J)-?'(P(K , J)-P(K , J- 1 ))) LET B(K ,J):B(K ,J)/(L(J)*L(J- 1)) ~~T K NFXT J
1050 1060 107C
' THE FOU.
FOR 1:2 TO ~
IF ft( 1 , 1) :0 TrEN 118C l..E'T ():?~(1- 1 , 2)/1;(1 , 1)
1o.80
FOR K: 1 TO 3 LET MI,I<):N(I , IO*C-N(l- 1,K+1)
1090
1100 1110
L-~ B(K ,I):B( K , I) • C-8(~,1- 1)
t~XT
1120 1130 1140
1180 1190 1200 1210
1220
1230
' RE'DUCE
K •
LET C:N(l,2} FOR K: 1 TO 3 LET
1150 1160
1170
' ll>PHALIZ£'
~{I,K}:N(l , K}/C
' t.OR~1ALIZF
LET B(K , l):B(K,l}/C NEXT K tiEXT I FOR K: 1 iO S
FOR J:O TO N- 1 LET
U(I< , ~-J):{B( K, t•-
NEXT J NEXT K
' SOL\'f FOR UNKtl>\iN
TA~ENT
VECTORS
J)- N( ~-J , 3) 1 U(K , Jl+ 1- J) )/N(N- J , 2)
1240 SUBEND
1250
SUB"CUflCEN":S,N , P(,) , L() , Z, U( , ) , C( , )
1260
LET 1:1
1270 1280 1290
FOP J:1 TO N- 1
' SOLVE FOR n'E FOUR CUEIC SPLD;t: COEFF .
FOR K: 1 TO S
LET F{1 1 K):P(K,J)
1300
LET F(2,K):U(K , J)
131 0
LET F(3 ,K): (3/L(J)A2) 1 (P(K , J+1)- P(K , J})- ( 1/L(J)} • (U(K , J+ 1)+2*u(K , J)) LET F(4 , K}:( - 2/L(J}-3)*(P(K,J+1)-P(K 1 J))+( l/L(J)-2) *(G( K1 J+1) +U( K1 J))
1320 1330 1340
t.EXT K FOR
1350
1360
T:O TO L(J) STEP L(J)/Z IF J: 1 THEN 1380
IF T<>O THEN 1380 GO TO 1490 FOP K: 1 TO S ' CALC. POI.trTS ALOI~ COEIC SPlliE LET R(K):F{1,K)+F(2 , K) 1i+F(3 , K) ~ (T-2) +F(4 , K) ' (T-3) NEXT K IF S=3 THE~ 1450 LET C( l,I}:R( 1) ' CREATE C( , } MATRIX OF RESULTS
1370 1380 1390 1400 1410
1420
1L30
LET C(2 , I):R{2) GO TO 1480
1illl0
LET C(1 1 I):R(1)
1450 1460 1470
LET C(2,l):P(2)
111~0
LET 1=1+1
1490
LET C(3,I):R(3) h!XT T
1500 NEXT J 15 10 SUBEJ.'D 1520 SUB"!oC.ATit•V":S , N,PC , >IN<,> ,o< I>,L( >,u( I> ,c1 , z 1530 Dltl. t{(70 ,70) ,C(70) , V(70 , 70) ,W(70 ) ' DIYENSION WTE11NAL JJ.ATRICFS
1540 1550
1560 1570 1560 1590
1600 1610
1620 1630 1640
ti.AT f-': ZER(N- 11 tl- 1) MAT C:ZER(N-1) MAT \or:ZER(f,- 1}
' l1UTIALIZE
AND ~EDIHHiS!Otl ' EACH INTERNAL t'.ATRIY
HAT V:ZERO.-l , tl- 1)
J:l TO ~1 ' CALC. SPAt; CHORD W~TPS IF S=3 THE'N 1620 LE·r L(J)=SCIR( ( £-{ 1 ,J + 1) - P( 1 1 J}) -2+( P( 2 ,J• 1)- P{2 , J )) - 2) OG TO 1630
FOF
LfT L(J):SCR((F(1 1 J+l) -P( t.'EXT J LET S):L(N- 1)/L( 1)
1, J))-2+(P(2 , J+ l} -P(2 , J })-2+(P( 3 , J+1) - P(~ , J ))-2 )
If' Cl : 4 TBEN 1730 U:T f" ( l , 1):2+?*S3 ' FIRST RO" Of fi-~ATRU LET H( I , 2):S3 ' FOR CYCLIC SPLINE L...!:i 1·:( I , t~ l): 1 FOR I<: l TO S ' SET B(f , 1) f'U~ CYrUC Sf-Llt.F. LET S(K, 1) :(3/L(l)) * (S3 1 (P(K,2)-P(K,1))+(1/S~)t(p(r,,N)-P(Y. N-1))) NEXT I< ' GO TO 1800 LET P( 1, 1)=2+2*S3 ' fifiST POll OF t'~IATfflY l.FT ~( 1, 2):53 ' FOR AUTICYCLIC Sf'LltlF. LET M( 1 , t,-1 ):-1 FOR K=1 TO S ' SET E{K ,l) FOR ANTICYCLIC SPLH•f
1650 1660 1670 1680 1690
1700 1710
1720 1730 1740 1750 1760 1770
LET B(K , 1 ):(3/L(1)) 1 (S3 1 (P(K , 2)-P(K,1))-(1/S3)'(P(K,N)-P(~,N-1)))
1780
NeXT K
1790
I
1800
' SET UP I NTERIOR 1-'.ATPIX AND mVERT
1610
FOR
J:2 TO N1 LET H(J ,J-1)=L(J) LET M(J ,J ):2*(L( J )TL(J-l))
1820
1830 1840 1b50
FOR K: 1 TO S
LET
B(K,J)=31
' C~F.ATE RO~!S Of' 8-t-I.ATRIX
(L(J- 1)-21 (P(K , J+1)-P(r , J))+L(J)-2*(P(K ,J)- P{Y. ,J-l))) LET B(K , J):B{K ,J)/(L(J) •L(J- 1))
1860 1870 1880 1890
t-EXT K hEXT J
~~T V:INV(M) FOR K:1 TO S
1900
FOR
19 10
1920
n- 1 LET C(J):B(K ,J)
' INVERT M-f".ATP.IX
J: l TO
' CALC. SINGLE llOI C-11.ATPIX
NEXT J
1930 1940 1950
1960 1970 1980 1990
2000
MAT H:V 1 C FOR J: 1 TO N- 1
' CALC . TA"GEliT VECTOR VALUFS
LET U(K ,J):l\(J) I•EXT J NEXT K IF C1=4 THElJ 2040 FOR K= 1 TO S LET U(K ,IH:U(K, 1)
' CREATE TAt.GElli' VECroP 1-'.ATRIY
2010 2020 NEXT K 2030 GO TO 2070 2040 FOR K: 1 TO S 2050 LET U(K , ti):- U(K , 1) 2060 NEXT K 2070 SUBEND
C-25
' CRF:ATF t.'O~ZEhO VALUE"S FOR 'lh1'ERNAL FOWS OF 11-t-IATFIX
' SET FINAL TANGE:toT:lliiTIAL TANGM
' SET fiNAL TANGEtiT:- IIHTIAL TANGENT
~ t\Lc;oo(llf1 FOR PARABOLIC &..ENDING An algoritlm which iJTplenents the parabolic blending t.ec::hni.q\e described
in Sec. 5-6 is given belCM as a BASIC language S\i'p1 o.:~ram. PARBLEND
100 SUE"PAP£·l£fRDit.ATES Of FOUR POltiTS 120
130
' R( 1, ):X-COOnDlNATES ' R(2 , ):Y-OOORDINATES
224 1-!M'HI:MM'ICAL EI»!ENIS FOR CGlPUI'ER GRAPHICS
140 150 160
' R(3 , ):Z-CCORDUJATfS ' N:NUHBER (F POINTS ~ CURVE 'C( , ):ARRAY roNTAir.ING POUlTS Ot• ELEt:DED CURVE
170 180 190
' C( 1, ): X-CCORDINATFS ' C(2, ):Y-COORDINATES ' C(3, ):Z-ccORDINATES
200
I
21C 220
230 2110 250 260
270 280 290 300 310
320 330 3140 350 360 370
t.OTE :
c 1-1l.ST
HAVE DD'!EliSIONS (F (3 ' t!)
01._, P(3 , 611) ,0(3,61;) ~'AT ~AT
S:ZfR(3, 3) T:lER(3 , 3) HAT F=ZEfl(3 ,N ) HAT C=ZER(3 , tl)
' Al.LO\\S UP 1U 6!1 SEG'tliTS ' IrJI1IALIZE AtiD Dl~EllSIOl>'
LET TO:SQR((~(1 , 3) -R(1 , 2))-2+ (R(2 , 3) - R(2 , 2))-2+(R(3 , 3) -R(3 , 2))-2)
FOR J : 1 TO 3 FOR 1:1 TO 3 L£T S(l ,J):R(I , J) LET T(I,J):R(I,J+1)
NEXT I
' CALC . TO
' SET UP TO GEtiERATf PARAEOLAS
THRU FIRST WREF.• POHrJ'S ' THRU LAST 'JllRE£ POIN1S
1
NEXT J
CALL"GEtl" : S( , ) , 1, TO, N, P( , ) CALL"GEN" : T( , ) , 2 , TO , N,O(,)
' GENERATE PARAFOLA 'JllRU FIRST THR~E POINTS ' GEtlERAT£ PARABOLA WJ!U LAST TI1REE POilns
K:O FOR 1:0 TO
' GElJERATE IlLEt'IDED aJRVE
LET
1 LFT K=K•1
38o 390 LIOO
STEP ( 1/(N-1 ))
FOR 1:1 TO 3 LET C(l,K):( t-T) 1 P{ l,K)+T*O(I , K) t.'EXT I
1110
f;EXT T 420 SUEEtiD
430 1140
450 460 470 1180 490
500 510 520 530 5110 550 560 570 580 590 600 610 620
630 640 650 660
SUB"GEJ~":P(
, ) ,St , TO , N, X( , )
'P( , ):ARRAY CONTAihlt.x:i ntf;E'E POlh-iS FOR PARABOLA ' P( 1 , ) :X CCt-'POtiENT ' P(2 , ):Y COMPONE~T ' P(3 , ):Z COf.1l'Ot,ENT
' S1:C0tn'ROL VARIABLE l :FIJ!ST PARMlOLA , 2:SECOND PARABOLA ' N:tml-1I!ER CF POINTS 01~ PARAE
' X(2 , )=Y- CC+IFONEI!'T I
X( 3 • ) =Z-COI'POf~fl;T
~~T T:ZER(3)
FOR I= 1 TO 3 LET T(I):P(I , 2)- P(l , 1) LET S(I):P{I ,3)- P(l , 1) LET I~( I):P(l,3)-P(I , 2) J,f.XT I NAT U:S*S
' SET UP ' P(SUBII)- P(SUB3) ' P(SUD5)- P(SUB3)
HAT V:T*S
' CALC. n'E 001' PflOOOCT (CF EC. 5-Sll) ' CALC . X (CF' EO. 5-54) ' CALC. ALPHA (CF 8Q. 5- 57)
LET D:SOR(U)
670 680 690 700
fiAT W:fi*S LET T1 :\,/(T01 D)
710
0010 730 LET T1:V/(TO*D) lf1' K:O
730
DII~EliSIO
HAT S:Z£R(3) liAT IJ:ZER( 3)
I..E'T X:V/U LET A:1/(U•X• ( t-57X)) If 51:2 THE~ 720
720
'INITIALIZE &
' P(SUB5) - P(SUBII)
• ( CF' EO. 5-Sll)
' CALC . D-2 ' CALC. 0
' CALC . COS(nlFTA) (CF' EO . '.)-5Q) ' CALC. COS(TI'f'TA) (Cf
ro.
'i-SQ)
F'OR T2=0 TO 1 STEP(1/(~-1 )) LET K=K+1 LET T=i0 1 T2 LET R=T*Tl IF S 1=2 TH£1'.; 800 LET R=R+X1 D FOR J=l iO 3
740 750 760
110
' CALC. POUlTS ctl PAPAKILA ' CALC. T
780 ' 'If.ST FCR FlPSl OR SECOND P ARAf~ A 790 800 810 LET X(J 1 K):P(J 1 1)+(RID} 1 S(J)+A1 P1 (D-R)I(T( J)-Y ' S(J ) ) 820 NEXTJ 830 t;EXT T2 840 SUSf.tiD
c-26
A Bezi ER CtJM
Al.GOOITif'\
lin alqoritl'ln which Will
a BASIC language subprogram.
generate BezJ.er curve segronts is given belo.t as It is based oo the discussioo of Sec. 5-7 .
BEZIER 100 SUB"BEZIER": N1,,S,X(,),Y(, ) 1 Z(,) , P,R(,) ' EEZIER CURVE 11 0 ' Ul:NlJI.BER OF VERTICES IN eEZIEJl POLYGON 120 ' S:COtiTROL VARIABLE 2=PLAtiE CURVE , ) : SPACE CUPVE 130 ' X( , !):ARRAY CONTAINit,G lliE X- CCfo!PQt,Et-,T
LET Y:Y1 X
330 31:0 350 360 370 380
LET X=X-1 IF X:O THEt\ 380
roro
32o
LET fllf =1 0010 390 LET fliF: Y
390
Pt;a.o
400
LET K: 1
410 420 430 ll!IO 450
f'OR T:O TO 1 STEP 11 ( P-64 1) ' GF.tlEfiATE EASlS FUt!CTiot, ( CF EO. 5-64) FOR I= 1 TO t. LET J( 1I I+ 1):( FNF(~)/{F11F( l)*FtlF(N- I)) ) ' TAl* ( 1-T)- ( t.- I) r.EXT I t".AT C=J'X ' GEUERATE POI~'TS AWiC A 1 2- D OTi 3- D BEZIER UJRVE (CF EO . 5-66) P.IAT D=J'Y
1160
226 MJ\1111:M\'riCAL lill.1'U:N'l'S FOR CCl1PlT.IER GAAPHICS
ll70 ll80
LF.T h( l , K):C( 1, 1) LFT R(?, K):0(1 ,1 )
lJ90 500
IF S:2 THEN 520 ~AT E:J• z
5 10 5?0
LEi
510
IIEXT T
' CREATE POUlT RESULTS
R(3 , K):f(1 ,1 ) LET K=K+1
5ll0 SUB€ND •
C-27
8-SPLINE ALGORITHM Algorithms "'hlch will generate the required B-splinc basis knot vectors
and &-spline curves are given
bela.~
as BASIC l.ar¥Juage s\bcrograms .
Msed oo tOO di.scussion of Sec. S-8 .
1lley are
Note that tha subprogram B-spl 1ne CAlls
'nlc 5\.bprog:ram a-spline uses the Cox and de Boor algorithm to generate
1<1\'01'.
the a-spl ine curves o r various order.
PSPLINE 100 SUb "BSPLli~E" : A, C, V( , ) , S , P , R( , ) 11 0 'A:Nl.I-'BER CF POLYGOt. VEfiTICES ~n:us at-E
120 110 140 150 160
170 180 190
700 210 220 230 ? liO 250 260
270 280
' C: O"rfR Of 8-SPLlNE bASIS ' V( , ):ARRAY CONTAINit.'G DEFINING POLYGON VEPTlCFS ' V( 1, ) COti'TAI NS X-COORDWATES ' V(2, ) CONTAIKS Y-COORDUJATES ' V( 3, ) COtrfAlt.S Z-COORDINATES ' S:CONTROL VARIABL.E S:2 FOR 2- 0 CUflVE , S: 3 FOR 3- 0 CURVE ' P:THE UU}1JER Of POit;JS GEJ•ERATED ALOfXi ntE CURVF ' h(, ):ARRAY COtiTAlNII-.G 1lfE F£SULTit.G B-SPLINE CURVE I k( 1. ) CONTAI~ X-COORDINATFS ' R(2, ) CONTAINS Y-GCORDINATES I R( '3 , ) CONTAINS Z-COORDitlATES
' N( ,):Io.ElGHTlNG
fUt.CiiON (Cf EO. 5- 78 )
DlM N(25 , ?5) , X(100) !~.AT 1\:ZER((A+C) , (A+C)) LET B:A-C+2 CAU.." KNaf":V( , ) , B, C, X() , S
300
FOR w:C- 1 TO C.B FOFI 1=0 TO B+( C-1) • 2-1 IF I<>H THE~ 350
320
LfT Ml , 1): 1
330
GO TO
290 ~10
3~0
35C 3N' 370
3b0
390
liDO 4 10 420
430 440
' REDlMEt.S lCti N & FlU. Win! ZF'POS
IF X(l):X(l+l ) THEN
350
LET N(I , 1):0
3~0
' INCREMENT KOOT VFC'TOR SUBSCRIPT ' CHECK FOR A GF.OMFTRIC KOOT ' CALC. VALUF.S ' fOR N(l , l) i U~ES 260- 320
NEX1 I F'CJR T:X(\.1) TO X(W+-1)-. 0S STEP . OS L£T L=L+ I
K:2 TO C fOh 1:0 TO A
FOR
'INCFI&.ENT PARAtAfTER i
' CALC. VALUES
OF N(I , K) IN
' LINES 3~0-4 lJO If N(l , K-t )<>O 1HEIJ 1J30 LET 0:0 GO TO 11110 LEi D:((T-X(I))• tHI , I<- 1})/(X(I•I<- 1)-lC(I)) If fl(l+1,Y - 1) <>O THEN 470
' fiRST TERH , ' FC'. ( 5- 78)
450 460
LfT E=O GO 10 480 LET f::((X(l+K)-T) 'tl(I+l , K-1) ) /(X{ +1!'}-Y(l+l)) ' SECGW JF~•· , 1 E(.l. ( 5-/r ) LET N(l ,K)=D+E
470
!lliO 490 510 520 530 540
LET G:V(1 , I) • U(l,K)..C ' X-COY.POt,fJ;T OF P{T) lEI' I!=V(2 ,I) l t-;(I , K)+H If 5::2 THEN 5 30 LET 7.:V(3, I) •N(J,K)+Z NEXT 1 IF K=C ~E~ 590
550
LET G:O
SM 570
LET H:O
500
LET Z:O
500
t1EX1 K
590 600
LET R(L, 1):G LET R(L ,2):H
610
If S:2 THEN 630
620 LET R(L, 3)=Z 630 640 650
660 670 680 690 100 710 720
730
LET G:O
LET H:O
' RESET INITIAL X, Y, Z VALUES
LET Z=O NEXT T
NEXT W LET L=L+ 1
LET R(L, l):V(l , A) LET R(L , 2):V(2 , A) IF S:2 THEN 730
LET R\L ,3):V(3 , 1) SUEEND
740 SUB"KNGr": V( , ) , B,C, X() , S 750 ' V( , ): ARRAY CONTAINII«:i DEFINH!G POLYGON VERTICES 760 I B:l-ii\XIKUI~ T VALUE=A-C+2 770 ' C:ORDER OF B-SPLIUE BASIS 780 ' X( ):ARRAY OONTAINlt\G YtJCYf VECTORS GEIJERATED W TIHS SUPRO\fl'ltiE' 790 ' S:O:JNTROL VARIABLE S=2 PLANE' S:~ SPACE CURVE 800
8 10 820
830 840 850 860 870 880 890 900 9 10
920 930 940
950
960 970
FOR I=O TO B+(C-1) • 2
IF I>C-1 THEN 840 ' ASSURE MULTIPLICITY Cf' DEGRE'F' C LET >:(1):0 ' I.S..<;IGN ~IULTIPLE El4t KmT VELIORS GO 10 960 IF IV( l, I-c+l) Tf-1Efl950 1CHfC(( FOR REPEATIIOCI VERTICES IF V(2 ,I-C)<>V(2 , I-C.1) THEtl 950 'C~ECK FOR REPEATltol; VF'RTICFS GO TO 650 IF V(l , I - C}<>V(l , l-C+l) 111Eil 950 IF V(2 , I - C)<>V( 2 , I-C+1) 11-lHJ 950 IF V(3 ,1-C)<>V(3 , I-C+1) THEt. 950 GO TO 850 LET X(I):X(I- 1) +1 ' ASS IGf~ SUCCFS.<;IVF IN'Jl:RtfAl. Vtx:lORS tlF.XT I SUBEND
________________________________________.....
C-28 fw
AlroRlnt'l FOR A Blllt£AR ~ACE PATOi
An alqorithn which will create the bilinear surface patch describ>rl in
Eq. (6-19) is given belo.tl as a BASIC language subpro:Jram.
BlUNEAf, 100 SUB"BIUNEAR":N , ~I , P( ,),0( , ) ' BILINEAR SURFACF 110 ' N:N~IBER (F IhCRFJIDiTS Of U 120 ' t-::N\n'IBER Cf' ThCRf.MENTS (F W 'P(, )=CO~TAHIS THE COORDINATES OF n!E OORNER POitfTS 130
140 150
160 170 180 190 200 210 220
230 2110 250
260 270
280
' P HAS DIY.ENSIONS Of 11 1 3. FIRST COlJ».'N OOIITAINS X' COI~PONENTS , SECOND Y-<:Ot-'.PONENTS , THIRD Z-cQI-'.PONENTS ' OF THE CORNER POSITION VEX:TORS. • ' 0( , ):CONTAINS THE COORDINATES OF n!E UfTERPOl.AttD SURfACE ' C WlLL HAVE DIHE~SIONS Of (H+1HN+1) 1 3 ' TiiE FIRST 11+1 COORDINATE PAIRS COIIRESPOtiD TO U:O: CONSTANT ' THE SECOtiO M+l COORDINATE PAIRS CORRESPOND TO U:l/N:COtJSTANT ' THIRD M+1 COORDINATE PADS COIIRESPOND TO U:2/N:CONSTAN-T ' NOTE: 0 MUST BE DD'.ENSIONED I N DRIVER PROORAP MAT O=ZER((N+l)*(M+1) , 3) ' J\EDIMENS!Otl C & FILL WI'nl ZEitOS FOR 1:1 TO N+l ' SET UP U:CONSTANT LOOP LET U:(I- 1 )/(t~) 'CALC. INCREMENT FOR U FOR J: 1 TO M+ t ' SET UP ~ LOOP LET !<:K+ 1 ' 1~1CREMEh1' POINT CCUlnER LET W:(J-1 )/M ' SET W
290
LET Q(K , l ):P(1 , 1) 1 {1- U) 1 {1-W)+P{2,1 )*{1- U) 1 W+P(3 ,1 )*U*(1-W)
300
LET C(K, 1):C(K,l}+P(4,1) 1 U1 W ' CALC. X- ca-PONENT
310 320 330 3q0
LET
O(K , 2}:P(1 , 2} 1 (1-u) 1 ( 1-W) +P(2 , 2) * ( 1-U) 1 W.P(~,2)'U*(1-W)
LET 0(K , 2):Q(K , 2}+P(4 ,2) 1 U1 W ' CALC. Y-CCl·lPONENT
O(K , 3}: P(1 , 3) 1 (1- U) *( 1-W}+P{2,3 )1 (1-U}1 W+P(3 , 3) 1 U1 ( 1-W} LET O(K, 3):Q(K , 3}+P{4 , 3)' U1 W ' CALC. Z-~PONE~T
LET
NEXT J 360 NEXT 1 370 SUBEND 350
C-29 fw AL.GORrllfl FoR LrNEAR An alqor:l.ttm \llhi.ch will
~s
SmFACE
generate the linear COOns surface descrjbed by
Eq . (6-24) is given below as a BASIC language ~rogrmn .
The algorithn
that the botndary curve c:-.an be described by equally spaced (in paranet:er values) p:>ints on the OOundal:.y curves. '1he P(u,O) and P(u,l) curves are assured to be describ>rl by the sarre Ill.l1ber of points as are the P{ O,w) and P(~w) curves.
a5Sl.l1'eS
COONSLIN 100 SUB"UfiEAR COONSLHI" : N, M,A( , ) , B(,) ,C(,) , D( , }, 0( , ) ' LINEAR COONS SURFACE 110 · ~=NUMBER OF POINTS Qt, THE P(U , O) AND P(U,l) POUNDARY CURVES 120 ' M.:NUMBER a: POUlTS Qt, THE P(O ,~I) AND P(l ,~ ) EOUNDARY CURVES 130 ' A( , ):ARRAY COiflAININC THE POINTS Of\ n!E P(U,O) CUflVE 1!10 ' HAS DD'ENSIONS OF til 3 150 ' B( ,):ARRAY
• HAs DIMENSrOt,s OF tJI3
Dt\TA B.Z\SE N11NIPI.lL1\TIW ALC:XlRI'I1f>1S 229
170
' C( , )=ARRAY CONTAINING THE POINTS ON THr PC O,W) CURVE
180 190 200
' D( , ):ARRAY CONTAINING THE
' HAS D~ENSIONS OF M1 3
1 HAS
POl~TS
ON TPE
P(1,~)
CURVE
DUIENSIONS Of M' 3
'Q( ,):ARRAY OONTAINI~G THE CURVES WHICH DESCRlfE THE SUnFACE ' HAS DWEIIoSIOOS Of (N1 M) 1 3. THE FO!lMAT IS FOR A FIXED
210 220 230
'VALUE Of U. THE NFXT M 110\ot'S Cf" ~E 0 t'A'I'IflX CONTAIIJ ' THE VALUES FOR VARIABLE W. ' OOTE: 0 MUST 1£ DU:£NSI~ED W DRIVER PR(.(jFAN MAT Q:ZER(N1 M, 3) ' R£Dli-1EM0tl 0 & FilL \oilTH ZEPOS FOR K: 1 TO ~I ' CALC. SURFACE LET U=(K-1)/(11-1) FOR J= 1 TO N LfT ~~(J- 1 )/(N-1 ) LET S:S+ l FOR L= I TO 3 LET O(S ,L}:A(K ,L)• (1- W)+B(K , L) 1 W+C(J,L)*(l-U)+D(J,L)*U LET ~(S,L):O( S ,L ) -A (1,L) 1 (1 -U) ' ( 1-W)-B( 1,L ) ' ( l-U) * ~ lEf O(S , L):C(S ,L)-A(N ,L) 1 U' ( 1-W)-B(N , L) * U'~J
2~ 0
250 260 270 280 290 300 310
320 330 340 350
NEXT L
360
370
NEXT J NEXT K
380 390 SUEEND
C-30 PH
AL.Goolltt1
FoR A BICUBIC ~ACE
PATCH
An al
Eq. (6-37) is given below as a BASIC l..anguage subprogram.
'Lhe routine assures
that the boundary condition and the blending functioo rratrioes are known . eiCUBI C 100 11 0 120
130 1~0
150 160
170 180 190 200 210
220 230
240 250
260 270 28o
290 300 310
320 330
SUB"BICUBIC ": U 1, ~1 , X( , ) , Y( , ) , Z( ,), O (,) · ~-CUEIC ' U1:NO Of H~REI-lEt-.TS ALONG u-DIRECTI ON ' Wl:NO rR It:CRE~Et-.TS ALONG W- DIRECTI ON
SURFACE
' X( , ):~
X 4 ARfl AY COfJTAINil\G THE X- COHPOUEtn'S Of WE ' P OOutfDARY CONDITION 111\TRIX EO. (6- 36) ' Y(, }:4 X 4 ARRAY CONTAINI OO THE Y- COI'PONEt\TS Of TiiE ' P BOUtlDARY CONDITION 1-'ATRIX EQ. (6- 36) ' Z(, ):4 X ll ARRAY C:OttrAI NlNG THE Z-COMPOtiENTS OF THF ' P BOUtiDAHY COI4DITION l1ATRIX EO . (6- 36) ' P(,)=~ X 4 BOU~DARY CO~DITION MATRIX EO . (6- 36) ' N(,):4 X 4 BLENDING FU~CTION MATRIX EC. (6- 31) ' U( , ):1 X t1 ClTBIC BLENDING VECTOP EO. (6- 37) ' W( , ):!1 X 1 CUBIC BLENDING VECTOR EO. (6- 37) ' 0( , ):MATRIX COt\TAWUIG THE POSITIOU VECIOFS FOR 1llE ' FHCUBIC SURFACE , !-lAS Cll1EriSIOt\S Of ( U1+1)*('-1+1) Y ' FIRST COLUt-.1" IS X-COtJPOtlENT ' SECOIJD
MAT W:ZER( 1,4)
~
3~0
~:AT C=ZER((lJ1• 1)• n.1+1) , 3)
350 360 370
·~T
380
HAT t:ZEfl( 1, 1)
390
LfT U( 1, 1):2 L:."J li( 1, 2):N(2 ,3>=- 2 LET t1(2 , 1)=-3 LET N(2 , 2):3
·~T
A:ZER(ij,l)
~AT E=ZER(~,l)
~00
1110 1120 1!30
LET LET
~~
C:ZER(4 , I)
1170
LET K=K•l
480
FOR I=O TO U1
LET U2:l/U1 LET U(1 ,1): U2~3 LET U(1 , 2) =U2•U2 LET U(1 , 3)=U2
510
520 530 5110 550 560
' CALC. TRANSPOSf Of' N-t-'.ATRIX ' CI.LC. X-CXWPOHEliTS CF ' SUFFACE EUYEltfS
'SET UP U-NATfiiX •
L£T U(1 , 4):1
FOR J=O TO \:1 LET L:L+1 LI::T •/2:J/W1 LET W(1,1):~2~3 LET w( 2. 1>=W2*~.2
570 580
590 600 610
' SET UP \o\-1-'.ATRIX
LET W(3,1):W2 LET W(4 , 1):1
11./iT A::JII U
NAT E=P*A NAT C:JJ*B
620
63C
' CALC . SURFACE ElEl1BlT
f'1AT D:U* C LET C(L,K):D(l, 1) liEXT J
6110
650 660
670
t;EXT I IF K:2 THEN 720 IF K>2 WElJ 740
f.8o 690 700
710
,...AT P:Y GO TO 1170
120 730
P:Z ro 470
I'..~T
ro
740 SUSEtiD
C-31
•
IJ(2 , ~)=-1
!-I.AT M=TFt. (t:} t1.AT P:X
500
t.~..ATRIX
N(1 , 3)=N(1 ,4):U(3,3):tl(4,1):1
~50 1160
490
' SET UP
' CALC. Y-rot-'PO~Ef'ITS OF ' SURFACE fl.E}lEWI'S ' CALC. Z-OJI-IPONENTS Of I SURFACE ELE:l-'ENTS
BEziER SuRFACE I£NERATIOO Al..GOOJTHH 1\n algorithm which will generate a cartesian prcxhx:t Bczier surfaoe based
on F..q. (6-48) is given below as a BASIC language subprogxam.
Bezier curves are
usoo as the blending functials. FI:ZSUflF
100 SUo"£.EZSIJPF":X( , ) ,Y( , ),Z( ,),t.,I",P,S( , ) 110 ' X( , ):1\Rf
' I:EZIE~ SURFACF
OF 'nlE' POLYGOt. VERTICES OF nw POLYCO~ VFRTICFS OF WE' POLYCO~ VERTICFS
140 150 160 170
180 190 200 210 220 230 240
250
260 270 280 290 300
' FORf'lAT IS nlAT Si"O\\IJ IN FIGS. 6-1 2 ' N=ORDEfi OF niE DEFINING BEZIER POLYGONS ' I'I=ORDER CF niE DEfiNI NG BEZIER POLYGONS 1 P:NUI'BER CF GRID LINES FOR TilE DEf'IlUNG
'S( , ):POSITION VECTORS FOR SURFACE
ooro
GOTO 370 LET Ft:F=Y FNEND FOR C: l 70 3 LET \ 1:0
' GRID POSITION VECTORS
fi.A i B:X
' X- CCI1PONENT
HAT E=Y
'Y-W1 PO::ENT
GOTO 460
GOTO 460 tAT E.:Z ' 2-cot'POUEt.T FOP U: 0 TO 1 STEP 1/( P- 1) ' FOR FIXED U CALC. VARIOUS \t' S FOR H: 0 TO 1 STEP 11( P-1 ) LET Wl:\\1+ 1 f'OR 1:0 TO N LET Jl:(FNF(N)/(FNF(I) *ftlF(N-1))) 1 0-I*( 1-U)-(tJ- I) FOR J: 0 T0 M LET K1: (FlJF(~1)/(PNF(J ) 1 FNF(V-J)) )•w-J 1 ( 1- W) -(H-J) LET S(C , \1 1 ):S(C ,W1)+B(l~l,J+1 ) * J 1* J\1 NEXT J
5 10 520 530 5~0
580
' FOR EACH C011PONENT CALC. SURFIICE
ON C GOTO 410 ,430 , 450
500
570
300
LET fliF: 1
ll90
560
GRID
LET X=X-1 IF X:O THEN 360
460
550
rEF~lNG
LET Y=Y*X
340
390 400 4 10 420 430 1140 ll50 460 470
FOR U:CI AllD '=1 FtiR W:Q ANO \.-1 SURFACE
LET Y:l
320 330
380
& 6-1 3
' S( 1, ):X~ENTS ' S(2 , )=Y-c:ot-'PONF~'TS ' S(3 , )=Z-00-\PONEM'S ' F'ORt-IAT IS niAT FOR A FIXED VALUE Cf' U TilE tJEX7 ' fl El..El!E"TS 00t4TAIN 1liE \IALUES FOP TiiF CURVF C( U(SUP [), W) ' S HAS OlfiEt:SIO~S OF 3 X P-2 11AT B: ZER( N, H) ' REDIMOOIOf, & ZEPO INTERf'..AL Nf\TRIX ~F FNf( X) ' DEFINE f'UNCfiON TO CALC. FAC10HlAL Cf EC. ( 5- M) IF X:O THEN 340
310
350 360 370
~
' t,OTE: X, Y, Z HAVE Dl!'EIJSIOt-:5 OF N X
NEXT I NEXT W NEXT U NEXT C
590 SUBEfiO
INDEX
•
•
A/0 ccnverter, 8
Axis:
arbitrary, 43, 54, 56, 207
APr, 1 Absolute coordinates, 4, 191 Active ax>rclinates, 4, 191
nde~:2'1c::e ,
93
Axoncrnet.ric projection, 60, 62, 64, 208
Affine transfonnation, 59
Algorithms, 200
BASIC, 200
Alphan\ft!ric keytx>ard, 8
Basis, Bernstein, 140, 145
Alpham:rreric node, 190
Basis function, 140, 145
.Analog tablet, 8, 10, U, 30, 194
Be;un,
.AnalytiC ge<:llCtry 1 94
Begin frame 1 19 3
Analytic space curve, 118
Bernstein basis, 140, 145
Analytic surface, 158
Bernstein polynani.al, 140
Angular perspective, 69
BEZIER, 225
~tions,
Bezier:
55
4
Area, 34
curve, 139, 225 p:>lyycn, 176 surface, 176, 179, 230 BEZSURF I 230
Array, 4
BICUBIC, 229
Associatl.ve, 198
Bicubic lofted surface, 182
hlticycl.ic spline, 126, 129
Arbitrary axis, 43, 54, 56 Arc. paratoh.c, 108
INr.Ev. 233 Bl.cubic surface patch, 170, 173, 183, 229
Circular helix, US c~
end, 12s, 129
BIL.INEAR, 228
Clipping, 6
BiUnear Coals surface, 182
Closed curve, 93
Bilinear surface, 165, 167, 228 Bilinear transfonmtion, 47 Bistable stora
nx3es, 188 COnnutative, 198
Bits, 4
carprt:er aided &!si911, l, 157
Bivariate vector, 164
CCIIpJter aided
Blended exterior surface, 185
catprt:er graphics, 1, 3
Blend,ed interior surface, 185
Concatenation, 6, 44 COnic:
Blending curve, 134 , 143 Blending functioo, 140, 169,
1 71 , 174 Blending, parabolic, 133, 137, 223
Color CRI', 190
lMI"IUfacturing1
sections, 94, 111 ncnparametric, 102
parmretric, 103
8:>undary cooditicn matrix, 173
COnstructive surface, 165
B::>u!ldary coodltions , 91, 120, 124
Continuity, 123 , 142
174
1
second-order, ll9, 121
B::>ur.dary curve, 176
OOOtrol
BSPLINE, 226
co::NSLIN, 228
B-spline, 226
Ooans surface, 168, 170, 181, 228
djals,
8, 10
curves, 144, 148, 151
bilinear, 182
surface, 180
generalized, 181
linear, 168, 170 cabinet projection , 65
COOrdillate trode, 190
ca 11 i graphic, 12
Coordinates:
Canonical space, 194 , 195
absolute 1 4 1 191
cartesian product surface , 179, 180,
display, 6
Ca.trode ray tube , 4, 11, 16, 18, 23
txxrogeneous, 4, 38, 42, 46, 200 incra'rental , 4
Cavalier projection, 65
relative, 4, 191
center of
transfonood, 25 user, 6
184
projection , 40 , 60 , 67
Otaracte.rs, 7 CIJCIE, 216
Cross derivatives, 160
Circle, 95, 96
Cross prcdu:t, 57, 161
nonpararretric, 214
CRl' display, 73
parametric, 103, 215
CUbic, 92
Circular arc, 93 , 98, 112, 115, 218
Circular arc inte:q;olatian, ll3
Bez.ie- curve, 141 blend:in9, 172 pararretric, 170
234 INDEX
oottcns, 20
Direction oosine , 55 Directnx, 95 Discriminate, 96 Display buffer, 12 Display oontro11er, 12
Display coordinates, 6
spline, 1191 1301 220 spline, parmretric, 119 spline segrrent, 121, U3
CUrsor, 4, 10 , 20, 189
Display file, 5
CUrve:
B-spl1ne, 144, 148, 151
Distortion, 32
Bezier, 139, 225
Distributative, 198
bl~,
~t
134, 143
boundary, 176
cl osed, 93
tratrix plotter, 11, 16, 23
D>t product , 57, 161 •
l):)uble buffering, 13
cubic, Bezier, 141
Draw absolute, 192
fitting, 91, 139
Drawing f~mctions t 1881 191
9Merator, 3
DJ:un plotter t 13
mathematical, 89
raster scan, 13 0\mp blffer, 193
nonparamrt.ric, 90 pararretric, 92
I:XIal
Dyr-.amic notion ,
2 , 11, 13
plane, 89 second deqree, 91
space, 116 CUrved line Jrode t 190 CUrved surfaces, 164
Eccentricity, 95, 97 £le!l'elts 1 196
EIL.IPSE1, 216
Ellipse, 95, 97
Cyclic spline, U6, 129 , 132
parametric, 104, 216 Elliptic integral, 104, 115
Dashed l.ine rrode, 190
Encaste.red spline, 125
Data base, 4, 188
~
Olta stream, 189
End tangent, 159 , 176
Derivative, partial, 163
Enla.rgerrent 1 32 t 41
Derivatives, 162
Erase, 11
Determinant, 198
Brase node, 190
Device independent, 189
Euclidean gearetry, 59
Device space, 194, 195 Device state, 189
F.\1lers equation, 119
Explicit representation, 90
Diagonal elenents I 196
Explicit space curve, 117
Dials, control, B, 10
Exterwr blended surface, 185
conditioos , 128
Digitizer, 20 DIMETRIC, 209
File, 4
Dllretnc, 60, 63
Fitting,
projcct.l.on , 209
cutVC,
91, 139
Flat bed plotter , 13
~·I •LHJOI
fl.&Ckt•J" , 7 , II , l l
Uflll it•J t
l·'oc:u:.• , 9'>
llfl'l i c 1t
p.:-r
1'-p.ll d a, 17C,
rn •r t:'fr• •u
1
J'unctlm:
h tc•tr.:tll
pi
rl
:111 ~ , 117
~.,.....
, <1
nl:h 1 uuo:Jp , I 3
IJ luwliJ~
Tnriuiu
tlr.twmrl , Ulll , l 9 I
Ju f] IY'l:lflfl JI(HIII
f.J11ll •r ,l)
lnitl,
l ~('(j r.r..ti.J r, 1HI
11)
1
lr'l", ?i. , 11'1
11:.-~ .
~],
1
l'IJ
••1•'""' ,
1 ~4
hyp• riA" )lu•, 11 0
lrtf HJl , fll uph i t•
l ntPr. wt i\11 11t -v an • CYIIII rnl, I
lnlr•JroJI, •• II 1 t 11 , 1()1\, lJ 'j
poi nL, I &2
lnl••rc ity trr~rtulo~t ir11a, ~!
L'Wi.t..cJM•s , 8, 10 , lJ
fll ti'Y•II''li\11• I~·VlC
t r•nnir~tll cnnl m l, I 111
\1\_'Ct.o r
Vtlll~ ...) ,
tUI I 1'! 11 1111'1! I l OtiO ,
Jlil l nt~'l•IC'I
H•4
h."• ()r otiA liC.:S, I, 7
Tnl• ·r fd• •, 1118
w. •ll)ht inq 1 lit 'J
rntr rl()l blourlod ("; •Ot •r c'l l
I :71Vl C.':r~u
t:C•n.•ralizt'fl •rtJlOr ,
<'I 'Jl
Hurf.'l(Y• , JRI
{unclior., 101
l>CJ J ,Jt
~ J rt.'\ll • ll
'l lVI ' ,
t
I
'}!)
i iM ,J J
r.rc~flluc
111\11 rtP ),YJ•
t VI
,
HX~:I
J89
r.r••ph l o> :
10 1 4:.l
1
rioJa
(."Uoll ' ' •
110
1 :110
ISu•r tr ll
tk:vit"'' • IRii
II\ II
J(,')
I
lnt• rf'.tort IIIIJ I i 111·
c;r••rJhu: pnmll
I I uI
.rc, 1 11
f.rc.JJ,hir (•l(tlt ·n t , L9l
inpul, laCJ
lll'J
uri '""' , lltfi
liiU:I"(.IfJ l• L 1011 , ')0 I
1
1111 Vf' ,
( l '()rl'(•lry, ('f' ·r.,( •
lnl rJJrlloal 'tl
hll l lol•"'',
,
(,I) ,
prQ I' •I
I t •II ,
(,
I
2]()
llall•roK1 i Vf•, I, :l
llrx ~ ·, 190
P·
[,IJrY, H, 10, lO
.Jt'fol
14, HJ
U lVI • ,
IJ,, rclcrlf 'Y, I l , I I lf.lrtt.l.ar• d~ 1r1Clt•r '1'1'1" 1>1 t.lJr, 7 IIHJh
1'"' 'f d
lJ :l .t
pr an ti r, Jl , 2]
llrFno:J•I~o..t liJS C"Y "I lrlll r lol at ~ ·
. , 4,
4(, , 200 IM'I:JQH, ]17
,,,, i l 1711•, J<;ll
IR1 4l ,
JUdi .
')}
1 ~. n, '10 r.lsll! pr:intt•r , H, L l~l· t
t.i 01 Jll tit l iH•J I JO , I}
IM1!1Ch..? , /Ill
llypt
I
111
f
11'1 ,
rt/) ) ,, , 1,.,, ,.,,
pM.urt t 1 lc, I ()A , 1.17 ltypt·rl/>I tr: h~ •c.1. ioo , I I 0
h 1etvl 1 Jr,C) int t·rJ.n l. I u .. ,, lf,r. t
r''"
J.illl .. :
I o nrut 1 .lfa'
, •,
IMI , 1/0
236 INtEX parallel, 29
l>buSe, 8, 20
surface, pararretrlc, 163
f\bve absolute , 192
transfox:mation of, 27
t-11LTIOBJF.X:T, 180 ~tiple
Local tracking, 194
vertices, 146, 149
IDeal tracking node, 190 Lofted bicubic
surface,
Lofted surface, 166,
182
18~
t-:atural end, 126
Nc::llc:CrmUtati ve, 50
Loft:sn:ms spline, 119
Nonpararretric circle, 214
Longl.tude, 158
Nonpa.rarretrlc ronics , 102 tionparmnetric curve , 90
l'lachine tool, 1
NOnsi.ngular, 199
Manipulation software, 188
Nontridiagcnal matrix, 127
~~pping,
Noxrral., 160
34, 41, 162
~~them:ltical
curves, 89
•
unit surface, 161
!'\athematical spline, 119
NormllizaHm, 39 , 93
Matrix, 24, 196
Nornalized parMeters, 123
addition, 197
Nf'C!K!F. f 214
algebra, 196
t-.'Urerical control, 157
iden~ty,
25, 41, 197
inverse, 41, 199
Cbl.ique perspective, 72
inversion, 122
Cblique projectim, 65
multiplic:atim, 5 , 25, 33 , 197
Order, 196
multiplier, 2
()rt}x)g'raphic projection, 208
nontri.d.i.agc::l'la, 127
Oscil1osrope, ll
product, 177
OUtward normal, 161
square t 421 1251 196
OVerall scaling, 40
subtracticn, 197 tranSfonnation, 28, 47
PAAABOLA, 217
tridiagonal t 124 1 126
Parabola, 95
zero, 197
parametric, 106, 217
l1idpoi.nt transfoDM.tion , 28
arc, 108
Minicxxrputer, l
blending, 133, 137, 223
r-bde control , 189
Parallel lines 1 29
f>bde:
Parallelograms, 30
curved line, 190
ParaJreter, vector valued, 164
erase, 190
Pa.rarretrlc circle, 103, 215
graphics, 190
Paranetric caries, 103
1ncrerrental, 13
Panuretric cubic, 170
loc:Jsl tracking, 190
Pararcetric cubic spline, ll9
standby, 190
Parmretric curve , 92
Im.EX 237 Parametri c ell i pse, 104, 216
Points, t.race, 73
Paratrctric hyperbola , l OB , 217
Points , transfo rmation of , 25 , 28
Parametri c parabola, 106, 217
Polygon, 139, 146
Pararretric space curve , 117
Pararretric surface, 159, 162, 164 P.ARBIDID, 223
Bt..~ier,
176
Pol}"C]alal surfAce , 177
Polyncrnial, 91
Partial derivatives, 163
Bel:nstllli'\, 140
Passi ve graphics, 14, 16
spline, 147
Patch surface , 159 , 176
Position vector, 92, 120, 159 , 166, 190
Pen , 4
Potentiareters , 8 PROJ , 208 Project ion , 40
Pen and ink pl o tter, 11, 16, 23
Perspective , 4 7 angular1 69
axonamctric, 60, 62 , 64, 208
gearet.r:y , 59
cabinet, 65
oblique , 72
cava l ier, 65
projection, 60 , 66 , 67
transfo rtl\l1tion , 76, 77 , 210
center of, 40, 60 , 67 d.i.netr ic , 209 i.scr!etric, 210 oblique , 65
two-point , 71 , 75
orthographic, 208
s ingle point, 69 three point, 72
Phospror, 12
perspective, 60, 66, 67
Picture, 3, 4
st:.ereographic , 84
Picture control functions , 191. 193
Pi ecewise cubic, 120, U2 , 147
Raster scan , 11, l3 , 23
Piecewise surface, 164
Rea)
Planar bilinear surface, 166
Rec:::OI'lSt:ruction, 78 , 2ll
Plane curves , 89
laluc:tion , 41
Plane surface , 162, 165
Reflect1on , 26 , 31, 47, 51 , 53 , 202 , 206
Pl otter , dot matrix, 11, 16, 23
refresh rate, 12
Plotter, flat bed, 13
~fresh
Plotter, pen and ink , 11, 16, 23
Pel.ative ooordinates , 4, 191
Pl otting head , 4
Relaxed end , 126, 129 , 132
Point absolute, 192 , 193
lesol ution 1 23
Point at infinity, 68 , 77
Rotat ion , 31, 33, 36, 40, 43 , 47, 50 , 52, 56 , 202
Point functi.oo , 162
t.iJre, 2
tube , 11 , 19 , 23
Point, inflection , 92
ab:lut arbitraty ilX.i.!J, 207
Point-plo tting, 20, 23
al::out
Point , vanishing, 68 , 73 , 76
about y , 204 about z , 205
Points, 41 24 Points a t infinity , 4l
x, 204
Ruled surface, 166, 184
238 INDEX
Scale transfo:rnation, 48 Scaling, 26, 32, 34, 40, 47, 201, 203 Scan
conversion, 13
Scissoring, 6
curve, 91 degtee surfaces, ll7
Steroo viewer, 84 Stereographic projection, 84 Storage
tube, 11, 16, 18
Storage buffer, 193 Straight~.
27, 28 , 92
Secmd degree
Stretching I 26
Second
SUperscript notation, 171
.
Second order CXX'Itinuity, 119, Ul
Support carp.rt:e.r, 188
Shear, 26, 34, 40
Surface:
Snoothness, 123, 147
Bezier, 176, 179, 230 Coons, 168, 170, 181, 228 • analytical, 158 bicubic lofted, 182 bilinear, 165, 167 bilinear, Coons, 182
Software systan, 188
b-spline, 180
Space, canonical, 194
cartesian product, 179, 180, 184
Space curve, ll6
ccnst.r\rtive, 165
Shearing, 47, 49 Single point perspective , 69 Sketching, 136 Slcpe, 29, 163
Slope, infinite, 92, 99
analytic, US
curved, 164
explicit, ll7
exterior blended, 185
.inp1 icit, ll7
generalized Cool IS 1 181
pararretric, ll7
interior blended, 185
Space device, 194, 195
intc.qx>lated 1 166
Space, three-dirrensia'lAl, 37
linear O:x:ns, 168, 170
Space, user, 194, 195 Sphere, 158
lofted, 1661 184
no.tml.ls, 160
S[:herical surface, 159
parametric, 159, 162, 164
SPLINE, 220
pa.ra.nctric lines, 163 patch, 159, 176
Spline:
anticyc1ic, 126, 129
patch, bicubic, 170, 173, 183, 229
cubic , 119, 130, 220
patch, bilinear, 228
cubic segrrent, 121, 123
planar, bD inear, 166
cyclic, 126, 129, 132
plane, 162, 165
1oftsmans, ll9
polygmal, 177
mathsnatical., 119
representaticn, 157, 162
po1ynanial , 147 Square matrix, 42 , US, 196
secmd degree, ll7
s~
sphe.rical 1 159
rrode, 190
nll.ed, 166, 184
STERI:X> I 213
tensor product, 184
St.eroo pair, 213
vector valued 1 164
nmr:< 239 Tangent, end, 159, 176 Tangent vector , 93, 120, 122, 159, 173
Teletype, 16, 22, 23
TWist vector , 160, 173 200EnECT, 202
2DR:Jr, 203 2D6CALE, 201
Televisicn, 12
Tensor product surface, 184 Te.Irni.nal ccn trol functioos, 191, 193
'tWo-point pe.rspecuve, 71, 7".:J I.NIAPI', 1
Text absolute, 192
Lnisurf system, 179
Textual material 1 31 191
Chit sphere, 159
300fNJVl' 1 201
3DRBc:x:Nl, 211
!.hit square, 34 Unit surface normal, 161
3DREFLT, 206
L"nit vector, 62
JOOCALE, 203
Unit vectors, 159
3t:1l'RANS , 207
User coordinates, 6 User space, 194, 195
30>rn0l', 204
3DYrol', 205 JDZJOr, 205
Vanishing {X>int, 68, 73, 76
Three diJrensional data, 23
Vanishing {X>int.s, 74
Ihree cllirensimal Sp&ee , 37
Vector, 24
JPCIRI\K: I 218
cross product, 161
'lbree {X>int perspective, 72
generator, 20
'lh\Eb wheels, 20
magnibxJe, 160
Trace, 196
position, 92, 120, 159, 166, 190
Traoe {X>ints, 73, 74
tangent, 93 , 120, 122, 159, 173 twist, 160, 173 unit, 62
Track ball, 8 , 20
Transfonnaticn: affine, 59
valued functioo, 164
bilinear , 47
val ued pa.ra.neter, 164
linear, 5
unit, 159
lines , 27
Vertex definitioo, 108
rratrix, 28, 47
Viewport, 6
midp:>.int , 28
Visioo windc:M, 7
perspective 1 76 t 77 1 210 {X>ints, 25, 28
Weighting functioo, 145
scale, 48
Wi.nd:lW, visicn, 7
Transformed cx:ordi.nates , 25
~li..rxbwing,
Translation, 39, 44, 47, 54, 200, 207
Wbrk length, 4
6
Tridi.agonal rratrix, 124, 126
Trilretric, 60
zero
~ratrix,
197
•
•
Cover photos: DAVID F. ROGERS The cover photographs show parametric solut1ons of the compressible boundary layer equations. Photos 3, 4, 5 show vanous v1ews of the shear stress as a tunct1on of the pressure gradi ent, parameterized with surface temperature. Photos 1 and 2 show the surface heat transfer rates as a function of the pressure gradient, parametenzed with surface temperature (Ref. Phys. of Flu ids, Vol. 12, No. 3, 1969)
I
I I
I
0-07-053527-2
