Paper ID sheet UCL-INMA-2015.10


Differentiable piecewise-Bézier surfaces on Riemannian manifolds

P.-A. Absil, Pierre-Yves Gousenbourger, Paul Striewski, Benedikt Wirth
We generalize the notion of B\'ezier surfaces and surface splines to Riemannian manifolds. To this end we put forward and compare three possible alternative definitions of B\'ezier surfaces. We furthermore investigate how to achieve $\C^0$- and $\C^1$-continuity of B\'ezier surface splines. Unlike in Euclidean space and for one-dimensional B\'ezier splines on manifolds, $\C^1$-continuity cannot be ensured by simple conditions on the B\'ezier control points: it requires an adaptation of the B\'ezier spline evaluation scheme. Finally, we propose an algorithm to optimize the B\'ezier control points given a set of points to be interpolated by a B\'ezier surface spline. We show computational examples on the sphere, the special orthogonal group and two Riemannian shape spaces.
Key words
Composite Bézier surface; Riemannian manifold; differentiability conditions; bending energy
Accepted for publication in SIAM Journal on Imaging Sciences, 2016