The picture below shows the caustic of the point (0,0) on the surface z=x^{4} + y^{4}. The gray background picture shows the curvature of the metric, which is k(u,v) = 144 u^{2}v^{2}/(1+16 u^{6} + 16v^{6})^{2}. We have computed with Mathematica 20'000 geodesic paths starting at the origin and run each over the time interval [0,300]. Along each geodesic the Jacobi differential equation f'' + K(c(t)) f = 0 with initial conditions f(0)=0,f'(0)=1 has been integrated and points marked, where f is zero. These yellow points are on the caustic of the point (0,0). Only the boundary of the caustic free region in the center is the cut value set. The caustic looks remarkably regular. |