The cut loci (from a point) on various toroidal surfaces of revolution have been calculated and visualized.
We have defined a possible candidate for a polygonal approximation to cut loci in general. This approximation can be found in a constructive way and we also have a general method to refine it. There are then two elements in this project which need further elaboration. One is to prove rigorously that the polygonal approximation actually does converge to the cut locus under repeated refinement, and the other is to implement the methods in a computer program, and thus allow visualization of the cut-locus from points on any given 2-manifold.
From this concrete outset it is very interesting to speculate and conjecture what kinds of global geometric information is actually encoded in the (metric) structure of the cut locus from a point in a given Riemannian manifold. One feature, which has been clearly observed, is that on the ordinary "circular" torus, the cut locus from any point contains at most two conjugate points.
Click here for a JAVA applet.
This data comes from a prototype C++ program which makes considerable use of high order automatic differentiation to both generate the geodesic equations and compute derivatives of solutions of these equations with respect to changes in initial angle and distance from the starting point. The program is capable of handling surfaces topologically equivalent to a torus which may not be surfaces of revolution. It does this by first computing a piecewise polynomial approximation to the distance map from the starting point, and then numerically inverting this map. The inverse is in general multivalued.
Points on the cut locus are found by bisection in arc length along a fixed geodesic passing through the starting point. One first shoots as far as the computed distance map allows in a given direction, and inverts the distance map. If the chosen distance and direction do not correspond to the solution of shortest length from the starting point, the process is repeated with a shorter distance (otherwise with a greater distance). This process continues until a distance has been found at which the numerical inversion can no longer tell which path is shortest. The point at this distance is taken to be an approximation to a point on the cut locus.