Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 180

Fig. 5.10 Normal section
in an arbitrary direction and
the coordinate system

Overview

The intersection of a plane containing the normal to the ellipsoid with the ellipsoid
surface will form the normal section. To solve simultaneous equations where one is
the equation of an ellipsoid and the other is the equation of normal section plane
will give the equation of the normal section, which is a plane curve. The radius of
curvature of the normal section can be obtained according to the formula for radius
of curvature of the plane curve.
In Fig. 5.10 we establish the Cartesian coordinate system O-XYZ with the origin
at the center O of the ellipsoid. In this coordinate system, the equation of the
ellipsoid is given by:

X 2
a 2 þ

Y 2
a 2 þ

Z 2
b 2 ﾼ

1 : ð 5 : 23 Þ

Let P be a point on the ellipsoid. The curvature of the normal section at any point
of the parallel passing through P on the rotational ellipsoid is the same in the same
direction. In order to simplify derivations of formulae, let P be on the initial
meridian plane. PK is the normal passing through point P and P 1 PP 2 is the normal
section passing through P in an arbitrary direction. Given the geodetic azimuth A,
the equation of P 1 PP 2 has to be found.
Because the normal section plane P 1 PP 2 is intersecting the coordinate plane
O-XYZ, one can imagine the complexity of its equations, making it inconvenient to
solve the equation of the normal section. In the meantime, the normal section is
represented by the space curve, so we cannot mechanically apply the formula for
radius of curvature of a plane curve. To simplify the equation of the normal section
plane, one needs to establish a new coordinate system. The curvature of a curve is a
measure of how “curved” a curve is and it is independent of the choice of coordinate
systems. In the newly established coordinate system, let a certain coordinate plane
coincide with this normal section plane. Meanwhile, in order to compute the radius
of curvature, assume that the origin of the newly established coordinate system
P-xyz coincides with P, the z-axis coincides with the normal at point P, and the