Computations on the Spheroid

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Lecture 6

• Computations on the Spheroid

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Lines on the spheroid

• Between two points on the spheroid, there can be
  an infinite number of lines.
• chord - straight line between points A and B
• normal section - line of sight of a theodolite
  perfectly levelled at A and pointing at B
• geodesic - shortest distance between two points
  on the spheroid

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The Geodesic
insert picture from Cross page 24

• on a sphere great circle
• curve of double curvature
• subdivides the angle between the two normal
  sections in the ratio 21
• a-A is negligible, necessary for long line
  
• difference between normal section length and
  geodesic length often negligible

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Formulae for computing on the ellipsoid

• gives the relationship between two points on the
  surface of the ellipsoid
• seven quantities involved, f1, f2, l1, l2, a12,
  a21, s
• direct problem - f2, l2, a21
• inverse problem - s, a12, a21

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Angles between curves on the spheroid
insert pic page 27 Cross

• azimuth of geodesic from 1 to 2 differs from the
  normal section azimuth by a small amount -
  negligible
• the reverse azimuth a21 differs from the direct
  azimuth a12 by Da, the spherical or meridian
  convergence
• not 180o unless both points are on the same
  meridian or on the equator it is not negligible
• meridians parallel at the equator and the angle
  between them increases until Dl at the poles

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Formulae for computing on the ellipsoid

• long line formulae
• short line formulae, medium line formulae and
  long line formulae
• formulae categorised depending on their mode of
  derivation
• normal section formulae
• geodesic formulae
• formulae selected depends on
• latitude of the survey
• precision required
• lengths of lines involved
• computational aids available

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Normal section formulae

• derived by taking a sphere which touches the
  ellipsoid at some convenient point, such as the
  standpoint of the midpoint of the line or at the
  equator
• these formulae include
• Clarke Robbins formulae (best known)
• Puissants formulae
• Gauss mid-latitude formulae (simplest)

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Computation by the mid latitude formulae The
Direct Problem
Given latitude and longitude of a point (f 1,
l1) and the geodetic azimuth (a 12) and geodetic
distance to a second point (s), Calculate the
latitude and longitude of the second point (f 2,
l2) and the reverse azimuth (a 21).
1st iteration
2nd iteration
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Computation by the mid latitude formulae The
Inverse Problem
1ppm for distances up to 30km
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Computation by the Sodanos formulae The
Direct Problem
Tanb1 (Tanf1)(1-f) If f1 gt45 then use
Cotb1
(Cotf1)/(1-f) cosb0 Cosb1 Sina12 g Cosb1
Cosa12 m1 (1( e2/2)sin2b1) (1-Cos2b0) Fs
(radians) s/b a1 (1( e2/2)sin2b1) (sin2b1
CosFs g Sinb1 SinFs) F0 Fs Term1 Term2
Term3 Term4 Term5 Where, with F in
radians Term1 a1(-(e2/2 )SinFs) Term2
m1(-(e2/4) Fs (e2/4)SinFs CosFs) Term3
a12((5e4/8)SinFs CosFs) Term4 m12((11e4/64)
Fs - (13e4/64)SinFs CosFs (e4/8) Fs Cos2Fs
(5e4/32)SinFs Cos3Fs) Term5 a1m1((3e4/8)SinF
s (e4/4)Fs CosF s (5e4/8)SinFs Cos2Fs) Cot
a21 (g CosF 0 - Sinb1 SinF 0)/Cosb0 CotL
(Cosb 1 CosF0 - Sinb 1 SinF0 Cosa 12)/ (SinF0
Sina 12) w (Term1 Term2 Term3) Cosb 0
L Where Term1 (-fFs) Term2 a1((3f2/2)
SinFs) Term3 m1((3f2/4) Fs (3f2/4) SinF s
CosFs) l2 l1 w Sinb2 Sinb1 CosF 0 g
SinF0 Cosb 2 ((Cosb0)2 (g CosF 0 - Sinb1 SinF
0)2)½ Tanb2 Sinb 2 / Cosb2 Tanf2 Tanb2 / (1
f)
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Computation by the Sodanos formulae The
Inverse Problem
Tanb1 (Tanf1)(1-f) Tanb2 (Tanf2)(1-f) If f
gt45 then use Cotb1 (Cotf1)/(1-f) Cotb2
(Cotf2)/(1-f) w l2 - l1 SinF
(term12term22)½ Where Term1 Sinw Cosb2 Term2
Sinb2 Cosb 1 Sinb1 Cosb 2 Cosw c (Cosb1
Cosb 2 Sinw)/SinF m 1 c2 s b(Term1 Term2
Term3 Term4 Term5 Term6) where, with F in
radians, Term1 F(1ff2) Term2 Sinb1 Sinb 2
(ff2)SinF - (f2/2) F 2/SinF Term3 m-(F
(ff2)/2) ((ff2)/2)SinF CosF (f2/2) F
2/TanF Term4 (Sinb1 Sinb 2 )2-(f2/2)SinF
CosF Term5 m2(f2/16) F (f2/16)SinF CosF -
(f2/2) F 2 /TanF - (f2/8)SinF Cos3F Term6
Sinb1 Sinb 2 m(f2/2) F2/SinF (f2/2)SinF
Cos2F L c(Term1 Term2 Term3) w Where,
with F in radians, Term1 (ff2) F Term2 Sinb1
Sinb 2 -(f2/2)SinF - (f2) F 2/SinF Term3
m-(5f2/4) F (f2/4)SinF CosF f2 F
2/TanF Cota12 (Sinb 2 Cosb1 - CosL Sinb1 Cosb
2)/(SinL Cosb2) Cota21 (Sinb 2 Cosb1 CosL -
Sinb1 Cosb 2)/(SinL Cosb1)
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Vector solution of direct problem

• Dot Product

• Cross Product

Other Comps