P1P3 and the geodesic P2P3. Only the boundary value is unknown.
This boundary value can he 93, >.3, S23 or S13; we can compute this
boundary value with an iterative process. The easiest way is to
compute the boundary value S23 or S13. An approximation s'23 of
S23 can be computed on a sphere with a formula of the spherical
trigonometry. With the method of 2.1 we compute the geodesic
PzP's and with the method of 3.1 we compute the geodesic P'sPi.
We have now the computed azimuth PiP'3 and the given azimuth
P1P3. With the difference cL4i we have to compute ds.
Just as in 3.1 we replace the ellipsoidal triangle PiPsP's by a
spherical one and compute with the formulas of spherical trigonom
etry an approximation of ds. With the new approximation of
S23 we compute again the geodesies P2P'3 and P3'Pi. Only if the
difference dAi is greater than 8, the process is repeated. This
method is valid for large triangles and is also a general one.
4.2 Intersection by lengths.
This problem is analogous to that of 4.1. Instead of the angles
on and a2 are given here the lengths P1P3 and P2P1 or the ratios of
the lengths P1P2 to P1P3 and P2Pi to P2P3.
In this case we do not know the value of A 23 for computing the
geodesic P2P3. This value can also be computed in an iterative
way. With the formulas of spherical trigonometry an approximation
of A23 can be computed; with this value we compute according
to the method of 2.1 the geodesic P2P'3 and with the method of
3.1 the geodesic P'3P 1. We have now computed the distance PiP'3
given is P1P3. With the difference ds we can compute dA. As in
4.1 we compute an approximation of dA on the sphere and we
repeat the process until ds is less than s.
This method is also a general one. For the computation of these
problems the procedures described in 2.1 and 3.1 can be used.
Integrated procedures for this type of computation have not yet
been written. Similar methods have been developed by A. Schödl-
bauer in [5].
i5i
p',
ds.
P
Figure 7.
