Riemannian Geodesics on Surfaces

Geodesics — shortest paths on curved surfaces — computed via ODE integration of the geodesic equations.

Surface

Parameters

Geodesics on a sphere are great circles. Geodesics from the same pole reconverge at the antipole — this is a conjugate point (focal set).

On a torus, geodesics can be periodic or dense (Kronecker), depending on the direction.

On a paraboloid, geodesics spiral outward from the apex.