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