Geodesics on a torus of revolution follow paths determined by Clairaut’s relation, which constrains the angle a geodesic makes with the parallels based on the distance from the axis of symmetry. This calculator lets you specify initial conditions and computes the resulting geodesic by numerical integration, displaying it on the torus surface. Different initial directions produce qualitatively different behaviors: some geodesics wind densely around the torus while others close up.
Controls: Click and drag to orbit, scroll to zoom.
