Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 175

Fig. 5.7 The relationship
between x, z, and B

x-axis is 90 + B. The slope of the tangent at P is equal to the first-order derivative
of the curve at this point, and we have:

dz
dx ﾼ

tan 90 o

ð þ B Þ ﾼ cot B : ð 5 : 17 Þ

So, the connection between x, z, and B can be found. Taking the derivative of the
equation of meridian and substituting the above equation, the expressions of the two
coordinate systems can be obtained.
The elliptic equation is:

x 2
a 2

z 2
b 2 ﾼ

þ 1 :

Taking the derivative with respect to x gives:

2x
a 2 þ

2z
b 2

dz
dx ﾼ

0,

or

b 2
a 2

x
dz

y ﾼ

dx :

a

p

e 2

With ( 5.17 ) and b ﾼ 1 , one obtains: