da tan y2ani d9 (3.1.2)
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For the solution of a system of differential equations we need
the value of all variables in the starting point Po and the value of
the independent variable in the terminal point. In the case of a
geodesic we need the values 90, Xo, A0 and so o. When transferring
geographical coordinates, these values are given, moreover the
value of the independent variable s in the point Pi is given.
In the reverse problem however, A0 and s are not given but 91
and Xi. But it is possible to compute on a sphere an approximation
of At, with the formulas of Gauss-Delambre (3.1.1)
A \{A\ 4- d1o) arctan(tan ^(Xi Xo).
cos |(?i 9o)
sin i(<pi 90) I
AA ^(A\ A\) arctan(tan |(Xi Xo).
sin 1(91 90) I (3-1-1)
cos ^(91 90)
A\ A &A
In this formulas we denote by
A V A\ the azimuth in point Po respectively Pi computed
on the sphere. These are the first approximations
of the azimuths Ao and Ai.
As independent variable we can choose 9 or Xthe corresponding
systems of differential equations are (3.1.2) and (3.1.3).
dx d9
V2 cos 9
ds V3 cos A dtp
V2 cos 9
d(D jdX
r tan A
dA sin 9 dX
ds dX.
V sin A
(3.1.3)
In these formulas we have used the following notation:
V c/N f(i e'2 cos29)
c aAjb
a, b major and minor semi-axis of the ellipsoid.
The choice between (3.1.2) and (3.1.3) depends on the value of
A and 9.
