Differentiable piecewise-Bézier surfaces on Riemannian manifolds

We generalize the notion of Bézier surfaces and surface splines to Riemannian man-ifolds. To this end we put forward and compare three possible alternative definitions of Bézier surfaces. We furthermore investigate how to achieve C0- and C1-continuity of Bézier surface splines. Unlike in Euclidean space and for one-dimensional Bézier splines on manifolds, C1-continuity cannot be ensured by simple conditions on the Bézier con-trol 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