Differentiable piecewise-Bézier interpolation on Riemannian manifolds

We propose a generalization of classical Euclidean piecewiseBézier surfaces to manifolds, and we use this generalization to compute a C1-surface interpolating a given set of manifold-valued data points associated to a regular 2D grid. We then propose an efficient algorithm to compute the control points defining the surface based on the Euclidean concept of natural C2-splines and show examples on different manifolds. Fig. 1: C1-Bézier spline surface on the Riemannian space of shells interpolating the red shapes. The Bézier surface (gray shapes) is driven by the control points (green).