A new geometric algorithm to generate smooth interpolating curves on Riemannian manifolds

Abstract

This paper presents a new geometric algorithm to construct a C k- smooth spline curve that interpolates a given set of data (points and velocities) on a complete Riemannian manifold. Although based on a modification of the de Casteljau procedure, our algorithm is implemented in three steps only independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function which is defined as soon as the degree of smoothness is fixed.