Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 184

h i

dx 2

3
2

dz

1 þ

P

R A ﾼ :

d 2 z
dx 2

P

Since P is the coordinate origin, the z-axis and the x-axis are the normal and
tangent to the normal section, respectively; obviously:

9
=

x P ﾼ z P ﾼ 0

0 1

dz
dx

: ð5:29Þ

@ A P ﾼ

0

;

Hence, the radius can be expressed as:

1
d 2 z

R A ﾼ

dx 2 P :

It is obvious that solving R A is made easier due to the establishment of the new

d 2 z
dx 2

coordinate system P-xyz. Here, is the curvature of the normal section in an

P

arbitrary direction at point P. Taking the derivative with respect to x in ( 5.28 )
repeatedly, and with ( 5.29 ), we obtain:

dx 2

d 2 z

2e 0 2 cos 2 B cos 2 A

2 2N

P þ

ﾼ 0 :

Namely:

P ﾼ

e 0 2 cos 2 B cos 2 A
N

d 2 z
dx 2

1 þ

:

Hence, the formula for the radius of curvature of the normal section in an
arbitrary direction is expressed as:

N

R A ﾼ

e 0 2 cos 2 B cos 2 A :

ð5:30Þ

1 þ

This formula indicates that R A is not only dependent on the latitude B of a given
point but also on the azimuth A of the normal section. However, it is independent of
the longitude L of the point. The point P is assumed on the initial meridian while
deriving the formula. It is applicable everywhere in the world. Given the location of
the point, its latitude B will be known. At this point, both N and cosB are the specified
constants. In this case, R A varies only with the azimuth A of the normal section.