Lecture 9

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

Normal Normal Sections, Sections, Unique Unique Normal Normal Sections, Sections, and and Reciprocal Reciprocal Normal Normal Sections Sections

Lecture No. 9 Department of Geodetic Engineering University of the Philippines a.s. caparas/06

Normal Sections • Rec ecal alll tha thatt we we hav have e defined a normal section as a curve formed by the intersection of the plane that contains the normal at a given point to the surface of the ellipsoid • Ph Phys ysic ical ally ly,, the the no norm rmal al section can be viewed when an optical instrument such as a theodeolite theod eolite or total total station station is set-up above a point • A no norm rmal al pl plan ane e is th the e plane swept out by the the moving the telescope in the vertical direction Lecture Lecture 9

Normal Plane

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

Normal Sections • By si sighting on on a distant point, we define a plane that contains the normal at the observation site, and passes through the observed site • Th The e in inte ters rsec ecti tion on of this plane with the ellipsoid forms the normal section from the observation to the observed point Lecture Lecture 9

A

B



Normal Sections • Cons Consid ider er th the e norm normal al lin line e to point B • Th This is no norm rmal al li line ne wi will ll intersect the minor axis at some point • No Now w con consi side derr the the no norm rmal al line at point B • Th The e nor norma mall line line at po poin intt B will intersect the minor axis at a point different from the point of intersection of the normal line at point A and the minor axis Lecture Lecture 9


A

B


Normal Sections • Cons Consid ider er the the tw two o nor norm mal plane to the two points • We can can se see e tha thatt tw two normal planes will nor coincide in any way

B

• Thus Thus,, the the tw two o nor norm mal planes will create two different normal sections

A

• And And if we ha hav ve tw two o normal planes, we have two normal sections

Lecture Lecture 9



Reciprocal Normal Sections • In ge gene nera ral, l, ifif we we hav have e two two points on the ellipsoid whose latitudes and longitudes are different, there exist two different normal section that contain both points • Th The e nor norma mall sec secti tion on fr from om point A to point B and the normal section from point B to point A • The hese se tw two o no norrmal sections is known as the reciprocal normal sections Lecture Lecture 9


B

A


Complication of having Reciprocal Normal Sections • The pr prese sen nce of of reciprocal normal sections creates a problem in when observation are used in the computations • We ca can n see see th that at wi with th th the e observed interior angles of the triangle, we cannot have a closed figure • Th Ther eref efor ore, e, th theo eore reti tica cally lly,, no matter how good our observations are, we still cannot have a closed observed polygon on the surface of the ellipsoid Lecture Lecture 9



Unique Normal Section • Howeve However, r, there there are certa certain in cases in which the normal section between two points is unique • There There are are two two case cases s in which there exist a unique normal section between two points: 1.When the two points are on the same meridian 2.When the two points are on the same parallel Lecture Lecture 9



Separation Between RNS •

We can express the differences between the RNS in terms of the quantities that separates them • There ar are tw two pr principal separations between the RNS. • Howe Ho weve ver, r, a th thir ird d qu quan anti tity ty is needed to consider to at the two principal separation • The se separations be betw twe een RNS are: 1. Ang ngle le in be betw twee een n 2. Li Line near ar Sep Sepa arat atio ion n 3. Az Azim imut uth h Se Sepa para rati tion on

Lecture Lecture 9

Linear Separation

B

Angle between the normal section planes

A


Azimuth Separation


Angle between the RNS • The angl angle e betwe between en the inte intersec rsecting ting norm normal al section planes denoted by f is given by: = e2σ cos A12 cos 2 ϕ m sin A12 f=

f=

Lecture Lecture 9

1 2

1 2

e2

2

eσ cos2 ϕ m sin 2 12A

s N1

cos 2ϕ m sin2A12



Linear Separation between RNS • The line linear ar separa separatio tion n betwee between n the recip reciproc rocal al normal normal section denoted by d is generally given by: d=

e2

s(σ -θ ) 2cos 2ϕ msin2A12

4 • The ma maxim ximum um line linear ar s sepa eparat ration ion occ occur ur when when θ=σ/2, the equation becomes:

d=

d=

e2 16

e2

s

sσ 2co cos 2ϕ msin2A12

s 2

16 N 12

Lecture Lecture 9

cos 2ϕ msin2A12



Linear Separation between RNS As a numerical example: example: • Fo Forr a li line ne wh who ose φm=45°N and A12=45°:

Lecture Lecture 9

s

200 km

100 km

dmax m

0.050 m 0.006 m


50 km 0.0008


Azimuth Separation between RNS • The azim azimuth uth sepa separat ration ion bet betwee ween n the reciprocal normal section denoted by ∆ is given by: ∆=

e 2σ 2cos 2ϕ m sin2A12 4

Lecture Lecture 9

=

e 2  s 



2

2 c o s ϕ m sin2A12 

4  N 1 



Azimuth Separation between RNS As a numerical example: example: • Fo Forr a li line ne wh who ose φm=45°N and A12=45°: s ∆”

Lecture Lecture 9

200 km 0.36” 0.023”

100 km

50 km

0.09”



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

Lecture Lecture 9



Lecture 9

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