The De Casteljau Algorithm on Lie Groups and Spheres

Abstract

We examine the De Casteljau algorithm in the context of Riemannian symmetric spaces. This algorithm, whose classical form is used to generate interpolating polynomials in $$\mathbb{R}^n $$, was also generalized to arbitrary Riemannian manifolds by others. However, the implementation of the generalized algorithm is difficult since detailed structure, such as boundary value expressions, has not been available. Lie groups are the most simple symmetric spaces, and for these spaces we develop expressions for the first and second order derivatives of curves of arbitrary order obtained from the algorithm. As an application of this theory we consider the problem of implementing the generalized De Casteljau algorithm on an m-dimensional sphere. We are able to fully develop the algorithm for cubic splines with Hermite boundary conditions and more general boundary conditions for arbitrary m.