B/Surf: Interactive Bézier Splines on Surfaces

B\'ezier curves provide the basic building blocks of graphic design in 2D. In this paper, we port B\'ezier curves to manifolds. We support the interactive drawing and editing of B\'ezier splines on manifold meshes with millions of triangles, by relying on just repeated manifold averages. We show that direct extensions of the De Casteljau and Bernstein evaluation algorithms to the manifold setting are fragile, and prone to discontinuities when control polygons become large. Conversely, approaches based on subdivision are robust and can be implemented efficiently. We define B\'ezier curves on manifolds, by extending both the recursive De Casteljau bisection and a new open-uniform Lane-Riesenfeld subdivision scheme, which provide curves with different degrees of smoothness. For both schemes, we present algorithms for curve tracing, point evaluation, and point insertion. We test our algorithms for robustness and performance on all watertight, manifold, models from the Thingi10k repository, without any pre-processing and with random control points. For interactive editing, we port all the basic user interface interactions found in 2D tools directly to the mesh. We also support mapping complex SVG drawings to the mesh and their interactive editing.