143
The solution of our geodetic problems is now rather simple; start
in a given point and compute with the method of Runge-Kutta a
point on a distance A on the line, if this point is not the terminal
point of our line, repeat the proces by taking another step.
A difficulty is that the values of all variables in a starting point
must be known. In some cases, e.g. the computation of a geodesic
between two given points, one of these values is unknown; instead
of this initial value a boundary value is given. This problem can
be solved in an iterative way, see 3.1, 4.1 and 4.2.
The method is quite general and can be used for different types
of curves (geodesies, normal sections etc.). The choice of the type
of curves to be used is reflected in the choice of the differential
equations. In this paper we shall only discuss geodesies.
A similar method has been developed by A. Schödlbauer
([4] and [5]). For the computation of a step he uses classical methods,
instead of the method of Runge-Kutta.
1.3 Numerical solution of differential equations.
Differential equations of the first order can be integrated in a
numerical way. The principle of this method of solving dy
i{x,y)dx is, to start in a given point x0,yoand to compute a point
x,ywithout knowing y y{x).
The solution is: compute a point xo h,yi) by some method,
and compute again with this point a next point (x0 2,h,yi). The
computing of a next point can be executed by the method of
Euler, Heun, Runge-Kutta and others (see [1], page 142, [2]
page 233).
For the computation of geodesies we have a system of simul
taneous differential equations. This system can also be solved with
the above mentioned methods. The methods of Euler and Heun
however, are in general insufficiently accurate for our purposes.
Using these methods means taking a short step size which causes
big accumulative truncation and rounding-off errors.
For this reason we have chosen the method of Runge-Kutta
of the fourth order. This method allows us to take steps of about
50 kilometres, giving an accuracy of about a millimetre.
The accuracy can be influenced by the step size. Within certain
limits it can be said that computing geodesies with half the step
size improves the accuracy by a factor 10 or more.
A disadvantage of the method of Runge-Kutta is that no simple
expressions are known for the precise truncation errors ([2]).
2. The transfer of geographical coordinates
2.1 Description of the method.
The transfer of geographical coordinates is a direct application
of 1.2. A system of differential equations is given for the geodesic.
