A New Geometric Algorithm to Generate Smooth Interpolating Curves on Riemannian Manifolds

Abstract

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, this new algorithm is implemented in only three steps, 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.