Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 176

tan B

e 2 z ﾼ x 1 :

Substituting the above equation into the elliptic equation, one obtains the
parametric equations in terms of parameter B, as follows:

9
=

a
1

a
W cos B

x ﾼ p cos B ﾼ

e 2 sin 2 B

:

e 2 e 2

ð Þ
1

a 1 a 1 ð Þ

;

z ﾼ p sin B ﾼ sin B

W

e 2 sin 2 B

p

1

In the above equations, W ﾼ e 2 sin 2 B is introduced.

Let PK P ﾼ N; from Fig. 5.7 , we have:

x ﾼ PK P cos B ﾼ N cos B

:

z ﾼ PQ sin B

Comparing the above two equations yields:

9
=
;

a
W

PK P ﾼ N ﾼ

: ð 5 : 18 Þ

PQ ﾼ N 1 ð e 2 Þ

QK P ﾼ PK P PQ ﾼ Ne 2

5.3.3 Transformation Between Geodetic and Cartesian
Coordinates

As shown in Fig. 5.8 , in the right triangle PK p P 3 , K p P 3 ﾼ (N+H)cosB, and

OP 2 ﾼ K p P 3 , according to ( 5.18 ). Hence, in the right triangle OP 1 P 2 :

X ﾼ OP 2 cos L ﾼ ð N þ H Þ cos B cos L,

Y ﾼ OP 2 sin L ﾼ ð N þ H Þ cos B sin L :

In the right triangle PQP 2 :

þ

e 2 PQ ﾼ N 1 H :

Hence:

þ sin B

e 2 Z ﾼ PP 2 ﾼ PQ sin B ﾼ N 1 H :

The formulae to compute geodetic Cartesian coordinates using geodetic coordi-
nates are given by: