Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 247

=
; :

sin m ﾼ cos u 1 sin A 1

tan u 1
cos A 1

ð 5 : 94 Þ

tanM ﾼ

3. Convert S into ˃ :

˃ ﾼ ʱ S þ ʲ sin ˃ cos 2M ð þ ˃ Þ þ ʳ sin 2 ˃ cos 4M ð þ 2 ˃ Þ: ð 5 : 95 Þ

The right-hand side of the above equation has the quantity ˃ that needs to be

computed, hence iteration is needed.

The first-time approximation takes:

˃ 0 ﾼ ʱ S :

The approximation of i times is:

˃ i ﾼ ʱ S þ ʲ sin ˃ i1 cos 2M ð þ ˃ i1 Þ þ ʳ sin 2 ˃ i1 cos 4M ð þ 2 ˃ i1 Þ ,

until the required accuracy is satisfied. For instance, if we want ʔ S < 0.3 m, then

10 6 ); if we want 0.01 00 (i.e., 2.8

we need | ˃ i ˃ i 1 | < ʔ S < 0.03 m, then we

10 7 ).

0.001 00 (i.e., 2.8

need | ˃ i ˃ i 1 | <

For precise solutions, ʱ , ʲ , and ʳ are calculated according to the expressions

below:

=

0 1

p

1 e 0 2

k 2
4 þ

7k 4
64

15k 6
256

ʱ ﾼ ˁ

þ

@ A

1

a

0 1

k 2
4

k 4
8 þ

37k 6
512

@ A

ʲ ﾼ ˁ

:

;

0
@

1
A

k 4
128

k 6
128

ʳ ﾼ ˁ

e 0 2 cos 2 m k 2

ﾼ

For approximate solutions in meters, the above equations are accurate to the
term of k 4 , while for approximate solutions in hectometers, the ʳ term can also be

neglected.