Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 207

9
=

S
M cos A

ʔ B ﾼ

S
N sin A sec B

ʔ L ﾼ

: ð 5 : 52 Þ

;

S
N sin A tan B

ʔ A ﾼ

Clairaut's Equation for the Geodesic

Integrating the differential equations of geodesics gives Clairaut's equation. It
provides the basis for solving the geodetic problems over long distance.
From ( 5.51 ) we can get:

sin A sin B
N cos B

dA ﾼ dS,

M

with ( 5.49 ), dS ﾼ

c os A dB, we obtain:

sin A
cos A

M sin BdB
N cos B

dA ﾼ ,

where, as shown in Fig. 5.27 , P is a point on the surface of the ellipsoid, PP 0 is the
arc element of the meridian, and PP 0 ﾼ MdB. The difference in the radius of the

parallel at point P and P 0 is dr. When P 0 moves to P, the latitude increases and the
radius of parallel decreases. Let PP 0 P 00 be a small right-angled plane triangle; we
get:

M sin BdB ﾼ dr,

with r ﾼ N cos B, substituting into the above equation gives:

sin A
cos A

dr
r :

dA ﾼ

We obtain:

r cos AdA þ sin Adr ﾼ 0,

dr
r :

cot AdA ﾼ

After integration, it can be rewritten as: