Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 255

sin ʻ 0

sin m ﾼ cos u 1 cos u 2

˃ 1 :

ð 5 : 107 Þ

sin

In addition, applying the cotangent theorem to the triangle P 0 1 P 0 2 N 0 , replacing ʻ

with ʻ 0 gives:

9
=
; :

cot A 1 ﾼ tan u 2 cos u 1 csc ʻ 0 sin u 1 cot ʻ 0

ʻ 0
cos u 1 tan u 2 sin u 1 cos ʻ 0

sin

ð 5 : 108 Þ

tan A 1 ﾼ

In this way the estimated value A 1 of A 1 is obtained.
In the right triangle P 0 1 P 0 2 Q 1 , we have:

9
=

sin m cot A 1
sin u 1

cot M ﾼ

: ð 5 : 109 Þ

sin u 1

;

sin m tan A 1

tanM ﾼ

Hence, the estimated value of M is obtained.
We calculate m according to ( 5.107 ) and compute ʱ 0 and ʲ 0 according to
( 5.100 ) and ( 5.92 ); the required accuracy is the same as for the direct solution.
Finally, we calculate the longitude difference on the spheroid using the
expression:

h i

0 0 sin

ʻ ﾼ l þ sin m ʱ ˃ þ ʲ ˃ cos 2M ð þ ˃ Þ : ð 5 : 110 Þ

Solution of Spherical Triangles

1. Find ˃

As shown in Fig. 5.43 , applying the cosine law to the spherical triangle
P 0 1 P 0 2 N 0 gives:

cos ˃ ﾼ sin u 1 sin u 2 þ cos u 1 cos u 2 cos ʻ: ð 5 : 111 Þ

2. Find A 1 and A 2
As shown in Fig. 5.43 , applying the cotangent law to the spherical triangle
P 0 1 P 0 2 N 0 gives: