Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 206

The above two differential equations are obtained under the condition that the
triangle is infinitesimal. It applies to any curves on the surface of the ellipsoid, of
course including geodesics. However, it is not exclusively applicable to geodesics.
In order to derive the differential equations exclusive to geodesics, the definition of
geodesics should be taken into consideration.
In Fig. 5.26 , PP 1 is the arc element of point P 1 on the geodesic. Let P 1 P 2 be
another arc element adjacent to P 1 . In accordance with the definition of geodesics,
PP 1 and P 1 P 2 are on the same normal section plane at point P 1 . Hence, PP 1 P 2 is the
arc element of the normal section at point P 1 . Its orthogonal projection onto the
tangent plane at point P 1 is a straight-line element.
We draw a tangent plane to a surface at point P 1 , and a line tangent to the
meridian through P 1 and P. Because P and P 1 are infinitely near points, the two
tangent lines can be considered to meet at point T on the extension of the minor axis.
The plane defined by P 1 T and PT can also be considered the tangent plane to the
surface at point P 1 . Hence, the arc element of the geodesic PP 1 P 2 can be considered
a straight line on the tangent plane. ∠ TP 1 P 2 ﾼ A +dA. It is an exterior angle of

the plane triangle TPP 1 . One can obtain:

A þ dA ﾼ A þ ∠ P 1 TP with dA ﾼ ∠ P 1 TP :

,

Since dS is the arc element, P 0 is infinitely close to P 1 ; then, P 0 can be considered
on the plane tangent to the surface at point P 1 . Hence, with the minor sector TPP 0 ,
one obtains:

rdL
PT ﾼ

N cos BdL
PT

dA ﾼ :

It follows from the right triangle K P PT that PT ﾼ N cot B, and substitution into

the above equation yields:

dA ﾼ sin BdL :

Inserting ( 5.50 ) into the above equation gives:

sin A
N

dA ﾼ tan BdS : ð 5 : 51 Þ

Equations ( 5.49 ), ( 5.50 ), and ( 5.51 ) are generally referred to as the three
differential equations of geodesics. They are the precondition for computing geo-
detic coordinates on the surface of the ellipsoid. If an increment is employed to
replace the differential, one can obtain the formulae for approximately computing
the differences in geodetic latitude, longitude, and azimuth as follows: