Differentiable Piecewise-Bézier Surfaces on Riemannian Manifolds

Abstract

We generalize the notion of Bézier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve ${\mathcal{C}}^0$- and ${\mathcal{C}}^1$-continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, ${\mathcal{C}}^1$-continuity cannot be ensured by simple conditions on the Bézier control points: it requires an adaptation of the Bézier spline evaluation scheme. Finally, we propose an algorithm to optimize the Bézier control points given a set of points to be interpolated by a Bézier surface spline. We show computational examples on the sphere, the special orthogonal group, and two Riemannian shape spaces.