Accurate High-Order Derivatives of Geodesic Paths on Smooth Surfaces

We propose a new approach for the accurate numerical computation of high-order derivatives along geodesic curves on surfaces. The method is based on the observation that for geodesics the Darboux frame and the Frenet–Serret frame are locally equal up to a constant rotation around the tangent. It computes derivatives of arbitrary order from the result of the numerical method employed for computing the geodesic. Since it does not rely on finite difference approximations, no additional discretization errors are introduced. Applications of the method include motion planning of autonomous vehicles and geometric modeling with developable surfaces.