Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 240

Fig. 5.42 Relationship between ellipsoidal (a) and spherical (b) projections

cos u sin A 1 ﾼ C :

Then we see that:

cos u 2 sin A 0 2 : cos u 1 sin A 1 ﾼ ð 5 : 79 Þ

Comparing ( 5.78 ) and ( 5.79 ), we have:

0
2 ﾼ

A 0 2 :

ʱ ð 5 : 80 Þ

The above equation shows that, in Bessel's solution of geodetic problems, the
azimuth of this geodesic remains the same after projection.
Four elements (u 1 , u 2 , A 1 , and A 2 ) of the six corresponding elements on the
ellipsoid and the spherical surface have so far been determined, and the rest are not
yet known, including the relationship between

and S. Hence,
the expressions for the meridian arc elements and the parallel arc elements on the
ellipsoid and the auxiliary sphere according to the differential equations of geo-
desics can be written on the ellipsoid as:

ʻ and l, and between ˃

,

dS cos A ﾼ MdB

ð 5 : 81 Þ

dS sin A ﾼ N cos Bdl

and on the auxiliary sphere as:

,

d ˃ cos A ﾼ du

ð 5 : 82 Þ

d ˃ sin A ﾼ cos ud ʻ

where d is measured by angle unit.

It follows from the above two sets of formulae that:

˃