Lecture 10

GE 161 – Geometric Geodesy The Reference Ellipsoid and the Computation of the Geodetic Posi Posi tion: tion: Curves on the Surface of the Ellipsoid

The Elliptic Arc, Azimuth, and Chord of a Normal Section Lecture No. 11 Department of Geodetic Engineering University of the Philippines a.s. caparas/06

The Elliptic Arc of the Normal Sections • Recall Recall that that the the equat equations ions giv giving ing the the linear linear and azimuth separations of the normal sections, we have used the relationship s=Nσ in order for us to express σ in terms of s and N. • We assum assumed ed that that the nor normal mal sect section ion is an arc length of a circle. • How Howeve ever, r, the the norma normall sectio section n is in in the the surface surface of the reference ellipsoid, therefore the normal section must be an elliptic arc. • Hav Having ing this, this, we shoul should d find find if the the relat relations ionship hip that we used would be valid to use in evaluating the separations of the normal sections. Lecture 10

GE 161 161 – – Geometric Geodesy

The Reference Ellipsoid and the Computation of the Geodetic Position: Curves on the the Surface Surface of of the the Ellipsoid Ellipsoid

The Elliptic Arc of the Normal Sections • In or order to to fin find d whether the our assumption is correct, we need to find the true relationship between s, N and σ. • Cons Consid ider er th the e pla plane ne containing the normal section from A to B.

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B o f i aa n d i e r M

B

s A12

S2

A N1


B o f n i a d i e r M

σ


The Elliptic Arc of the Normal Sections • Afte Afterr some some manipu manipulat lation ion,, we can can find: find:

S2

= 1 − σ η cos

N

1

1

2

2

2 1

2

1

3 2 2 A t + σ η c o s 12 1 1 1A 2 + .... 2

where: η12 = ( e ') 2 cos 2 ϕ 1

t 1 = tan ϕ 1

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The Elliptic Arc of the Normal Sections • To deriv derive e the the relat relations ionship hip betw between een s, N and σ, we must consider the differential curve ds ds:: ( ds) 2 = (S2 dσ ) 2 + ( dS 2 ) 2 σ: • Since dS 2 is negligible compared to S 2 d σ

( ds ) 2 = ( S2 dσ )2

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The Elliptic Arc of the Normal Sections • Substi Substitut tuting ing the the derive derived d equati equation on for for S S 2 and integrating ds ds::

 1 2 2 s = N1σ  1 − σ η1 cos2  6

 3 2 2 + σ η c o s t A 12 1 1 12  8  1

• This This formu formula la can can be inve inverte rted d to find find function of s: 2 s  1 s  2 1 +   η1 cos 2 σ = N1  6 N1 



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σ

as a

   η 1 t1 cos A12  12 + 8 N1  


3

1 s 

2

2


The Elliptic Arc of the Normal Sections • If we will eval evaluate uate the valu value e of the term terms s inside the bracket of the equation for any normal section, we will see that the value will be approximately equal to 1. • With With this this we can can conclu conclude de that that the the use of the relationship s=N σ for the computations with normal sections separations is justified to some degree of accuracy.

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Curves or Arcs on the Ellipsoid Ellipsoid • A curve or an arc on the surface of the ellipsoid generally connects two points on the ellipsoids surface. • We can can cl clas assi sify fy th this is basically into two categories:

Meridian Arc

1. “S “Special pecial”” Curve Curves/Arc s/Arcs s - Arc along the Meridian - Arc along the Parallel 2. “General “General”” Cur Curves ves - Normal Section/Curve Lecture 10


Parallel Arc

Normal Section


Length of Meridian Arcs • In order to find the length between two points with latitudes φ1 and φ2, the differential arc length ds=M ds= Md φ must be integrated: ϕ2

s=

∫

ϕ 2

(1 − e) Mϕ d = a 2

ϕ1

∫ (1 − e

ϕ 1

1 2

sin 2 ϕ )3 / 2

ϕ 2

ds=Md φ

ϕ 1

ϕ d

• But this represents an elliptical integral, which cannot be integrated using elementary integral functions. Lecture 10



Length of Meridian Arcs • Instead of f elementary elementary integral functions, functions , the use of series of series expansion is necessary to evaluate the length of meridian arc. • The McL McLaur aurin in Ser Series ies is used to expand the term inside the integral. • The leng length th of the the merid meridian ian arc arc after after expanding the term is given by:  

Sϕ = a (1 − e 2 )  Aϕ − Lecture 10

B C D E F  sin 2ϕ + sin 4ϕ − sin 6ϕ + sin 8ϕ − sin 10ϕ + ....  2 4 6 8 10  GE 161 161 – – Geometric Geodesy


Length of Meridian Arcs • In which: A = 1+ B= C= D= E= F=

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3 4 3

e2 +

45 64 15

e4 +

175 256 525

e6 +

11025

e8 +

43659

e10 + ....

16384 65556 2205 8 72765 10 e2 + e 4 + e6 + e + e + .... 4 16 512 2048 65536 15 4 105 6 2205 8 10395 10 e + e + e + e +.... 64 256 4096 16384 35 6 315 8 31185 10 e + e + e + .... 512 2048 131072 315 8 3465 10 e + e + .... 16384 65536 693 10 e + .... 131072 The Reference Ellipsoid and the Computation of the Geodetic Position: Curves on the the Surface Surface of of the the Ellipsoid Ellipsoid


Length of Meridian Arcs • The len length gth of the me merid ridian ian arc fro from m equator to the pole from the derived equations is: Sϕ =90 = a (1 − e 2 ) A

π 2

• For lines lines up to 400 km, km, the equation may be modified as  1  s= Mm ∆ϕ 1 + e2 (∆ϕ ) 2 cos 2ϕ m   8  • For even even shorter lines that reaches only 45 km, we may dropped the term in the bracket: s = M m ∆ϕ

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Lecture 7


The Reference Ellipsoid and the Computation of the Geodetic Position: GE 161- Geometric Geodesy Curves on the the Surface Surface of of the the Ellipsoid Ellipsoid

Length of Parallel Arc • Par Paral alle lell ar arcs cs ar are e arcs of circle so the length of this arc can be computed using the arclength formula for circular arcs. • Th The e len lengt gth h of of arc arc or th the e distance between two points on the same parallel having longitudes λ 1 and λ 2 is given by:

p

λ 2

λ 1 φ

L=p∆λ =Ncos =Ncosφ∆λ

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Lecture 7



The Azimuth, Chord and Length of the Normal Section • •

•

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The norm The normal al sec sectio tion n azim azimut uth h can can be com compu pute ted d if we we know the latitudes and longitudes of two points on the surface of the ellipsoid. The Th e whole whole pr proc oces ess s of comp comput utin ing g of the the nor norma mall sect sectio ion n azimuth, chord or distance involves two steps (1) Conve Converting rting the geodetic geodetic coordinates coordinates to cartesian cartes ian coordi coordinates. nates. (2) Substitutio Substitution n of the cartesian coord coordinates inates to the equations giving the normal section azimuth, chord and length. Note No te tha thatt this this pro proce cedu dure re is is just just one one of of the the seve severa rall possible procedures for computing normal section azimuth, chord and length.



The Normal Section Azimuth • Recall Recall the the coordi coordina nate te conver conversio sion n from from geodetic to cartesian… • If we ha have ve two two point points s with with lati latitud tudes es φ1 and φ2 and with longitudes λ 1 and λ 2, the cartesian carte sian coor coordina dinates tes (assuming (assuming points are on the surface of the ellipsoid) are: x1 = N1 cosϕ1 cos λ 1

x2 = N2 cosϕ2 cos λ 2

y1 = N 1 cosϕ1 sin λ 1

y2 = N 2 cosϕ2 sin λ 2

z1 = N1(1− e2 )s )siinϕ 1

z2 = N 2 (1− e2 ) si sin ϕ 2

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The Normal Section Azimuth • However However,, in comp computi uting ng the the normal normal sect section, ion, we may assume that the first point is on the meridian of origin such that: x1 = N1 cos ϕ 1

y’

y1 = 0 z1 = N 1 (1 − e 2 ) si sin ϕ 1

y

x 2 = N 2 cos ϕ 2 cos ∆λ y 2 = N 2 sin ϕ2 cos ∆λ

∆λ

z 2 = N 2 (1 − e ) si sin ϕ 2 2

x’; λ=0 x

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The Normal Section Azimuth The azimuth of the normal section can be derived following this figure: [(x2-x1)sinφ1+(z2-z1)cosφ1] y2 B

A 12

(z2-z1)

A 12

Normal Line at A

Chord AB

A φ1

y2 Lecture 10

(x2-x1)


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The Normal Section Azimuth • It follow follows s that that the the normal normal secti section on azimu azimuth th is: tan A12 =

y2

 x2 − x1 sin ϕ1 + z2 − z1 c os ϕ 1 

• The cho chord rd betw betwee een n the the two poi points nts is is simply computed as: s=

Lecture 10

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y22 + ( x2 − x1 )2 + ( z2 − z1 )2



The Normal Section Distance •

We are are not not actual actually ly intere interested sted in the the chord chord dista distance, nce, but in the the actua actuall normal section distance s. However this can be computed using the chord distance:  1  s 3  s   5  s  µ1  s  3µ 2  s  +   + + .... s = s 1 +   +   +     5  2r    6  2r  40  2r  112  2r  2  2r  2

where:

4

r = µ 1 =

6

3

4

x12 + y12 + z12 e '2 sin2 sin2ϕ 1 cos A12 2 2 1+η 1 cos A12

 sin 2ϕ1 − cos2 ϕ 1 cos2 A12   1+η 12 cos2 A12  

µ 2 = e '  2

•

For lines lines up to 100 km, this this equation equation reduc reduces es (with (with an accura accuracy cy of 1 cm) to:

 1  s 2  s = s 1 +   + ....  6  2r  

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Reference: • Rapp Rapp,, Ric Richa hard rd R. R.,, Geometric Geodesy , Ohio State University, Ohio State USA.

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Lecture 10

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