Geometric mean, splines and de Boor algorithm in geodesic spaces

TL;DR

This paper extends classical spline algorithms and geometric means to geodesic spaces, enabling advanced geometric modeling on manifolds with applications demonstrating their properties and equivalences in Riemannian contexts.

Contribution

It introduces generalized de Boor and de Casteljau algorithms for geodesic spaces and explores weighted geometric means as an alternative approach to splines.

Findings

Bezier curves in Riemannian manifolds have coinciding endpoint tangents.

The generalized algorithms are applicable to various geodesic spaces.

Weighted geometric means offer a new perspective for spline construction.

Abstract

We extend the concepts of de Casteljau and de Boor algorithms as well as splines to geodesic spaces and present some applications in geometric modeling. The concept of weighted geometric mean provides another approach to splines. We compare the corresponding Bezier curves and show that for Riemannian manifolds, their endpoint tangents coincide.