HCRP project 2009
Caustic on graph of quartic parabola
Caustic computation
Office: SciCtr 434

The picture below shows the caustic of the point (0,0) on the surface z=x4 + y4. The gray background picture shows the curvature of the metric, which is k(u,v) = 144 u2v2/(1+16 u6 + 16v6)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.

Click to see larger

Click to see larger
The left picture shows the caustic of (0,0) on the surface z=x2 + y2 in polar coordinates. The image of this picture is seen to the right. The map is the exponential map. The caustic consists of points, where expp is not invertible.
Questions and comments to knill@math.harvard.edu
Oliver Knill | Department of Mathematics | Harvard University