The picture shows some geodesics on a surface S
{ z = f(x,y) = x^{4}+y^{4} }.
It illustrates the map exp_{x} from a disk
D={x^{2}+y^{2} ≤ 10 } in the tangent space T_{x}M at
the point x=(0.1,0.2,f(x,y)) in S.
The program has computed the caustic (yellow)
by integrating up the Jacobi field equations J'(t) = - K(r(t)) J(t) along
geodesics r(t). The function K(r(t)) is the curvature. It is positive everywhere on S
in this case even so the manifold is non-compact (its soul is a point).
There were 24'000 geodesic paths drawn and 240 are shown.
The curve r(t) hits the caustic at the point r(t) if J(t)=0.