Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 233

calculation of geodetic coordinates along the first-order triangulation chain, the
geodetic longitude and latitude should be accurate to 0.0001 00 .
In the first-order triangulation, the final result of the azimuth is accurate to 0.01 00 .
Hence, in solutions of geodetic problems the geodetic azimuth is accurate to 0.001 00 .
The above discussions of computational accuracy are mainly concerned with the
adjustment of astro-geodetic networks and the calculation of coordinates of the
first-order geodetic points. When applied in other situations, the accuracy of
computation should be determined by the applications. For instance, in navigation
application, the geodetic longitude, latitude, and azimuth can be accurate to 0.1 00 ,
and the distance needs to be accurate only to 10 m.
To understand the rationale for solving geodetic problems, we must first provide
the method for point-by-point

integration for the direct solution of geodetic

problems.
We divide the length of P 1 P 2 (the interval between P 1 and P 2 ) in Fig. 5.40 into
n sections. The differences in longitude, latitude, and azimuth between the two
endpoints of each small section are dL, dB, and dA, respectively, satisfying the
conditions of differential equations of geodesics. Hence, the direct solution is given
by:

9
=
;

ð L 2 ð S

X

n

sin A
N

sin A i
N i

L 2 L 1 ﾼ dL ﾼ sec BdS sec B i ʔ S i

L 1 0

iﾼ1

ð B 2 ð S

X

n

cos A
M

cos A i
M i ʔ

B 2 B 1 ﾼ dB ﾼ dS S i

:

B 1 0

iﾼ1

ð A 2 180 ð S

X

n

sin A
N

sin A i
N i

A 2 A 1 180 ﾼ dA ﾼ tan BdS tan B i ʔ S i

A 1 0

i ﾼ 1

ð 5 : 69 Þ

Repeated computations of the above equations will solve the short-distance
direct geodetic problems and also enable a high degree of accuracy. The above
equations have shown that as n increases, so does the accuracy of the computations .

5.6.2 Series Expansions of the Solution of the Geodetic
Problem

In Fig. 5.40 , at the given point P 1 (L 1 ,B 1 ), when the geodesic azimuth A 1 is
determined, the geodetic longitude, latitude, and azimuth of an arbitrary point on
the geodesic are the functions of the geodesic distance S, namely: