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Calculation of Geodesics

My aim is to find a nice planar visualization of the geodesics in an orbifold. I will restrict my attention to 2-orbifolds which have only a finite number of cone points, including the cone point of order infinity, i.e. to orbifolds of the form . I will visualize the orbifold as the usual plane with some cones sticking up.

Here is an algorithm showing how to visualize geodesics on an orbifold given a point and an initial direction .

Step#1:

Find Euler number of the orbifold to determine the type of the universal covering.

Step#2:

Find one of the fundamental domains of the covering.

Step#3:

Find a ``consistent'' conformal mapping from the orbifold to the chosen fundamental domain and the inverse of this mapping . By consistent I mean that the map should `glue' together only equivalent points on the boundary of the fundamental domain.

Step#4:

Draw the geodesic in the covering starting at in the direction (we know how to do it).

Step#5:

Whenever you hit the boundary of the fundamental domain, use the corresponding symmetry (in case of the cone point this symmetry is rotation) to reflect or rotate geodesic back in the fundamental domain.

Step#6:

Map this geodesic back in the orbifold.

The most difficult step is step#3. In each case you have to do something special. For all Euclidean and spherical orbifolds (there are only finitely many of them) you can find a nice mapping. You prove this by looking at all the possible cases. For example, for all Euclidean orbifolds the Schwartz-Christoffel formula and Schwartz reflection principle are the only things you need. In particular, in the case the mapping is the usual sine function, in the case the mapping is . The most difficult and almost unsolvable case is presented by the infinitely many hyperbolic orbifolds where generally you have to map something onto the region bounded by some number of circular arcs.

I implemented the hyperbolic case . (This involved choosing a sufficiently symmetrical configuration and using, besides the tools mentioned above, the modular function and Lagrange interpolation.) Here the fundamental domain is a zero-angle triangle with cone points of order in the middle of each side and you have to map the outside of three rays onto this triangle.