Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 204

Fig. 5.24 The shape of
geodesics

taken into consideration in computation (called correction from normal section to
geodesic). The difference in length between the normal section and the geodesic is
slight. If the geodetic latitude at a given point B 0 , the geodetic azimuth of the ﾼ

45 , and the length of the side S side A 100 km, then the difference in the

length between the normal section and the geodesic

ﾼ ﾼ

ʔ S ﾼ 0.000001 mm, which can

be shown to be negligible in practical cases.
On the meridian and equator, the geodesic coincides with the reciprocal normal
sections and both the geodesic and the reciprocal normal sections coincide with the
meridian and the equator. On the parallel circle, although the normal section and the
reverse normal section coincide, the geodesic, the normal sections, and the parallel
do not coincide.

Differential Equations of the Geodesic

Differential equations of geodesics are the differential expressions of the relation-
ships between the geodesic distance, geodetic longitude, latitude, and azimuth.
As shown in Fig. 5.26 , let P be an arbitrary point on the geodesic. Its longitude is
L, latitude is B, and geodetic azimuth is A. Let PP 1 be the arc element of the
geodesic dS. From point P to point P 1 , its longitude changes into L+dL, latitude is
B+dB, and azimuth is A+dA.
From Fig. 5.26 , the arc element of the meridian P 0 P 1 ﾼ MdB, and the arc

element of the parallel PP 0 ﾼ N cos BdL. PP 0 P 1 is a right ellipsoidal trian-

gle. Since it is infinitesimal, the right ellipsoidal triangle can be considered a right-
angled plane triangle. Hence, one obtains:

rdL ﾼ