b/Surf: Interactive Bezier Splines on Surface Meshes

We present a practical framework to port Bezier curves to surfaces. We support the interactive drawing and editing of Bezier 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 implement manifold extensions of the recursive de Casteljau bisection, and an open-uniform Lane-Riesenfeld subdivision scheme. For both schemes, we present algorithms for curve tracing, point evaluation, and approximated point insertion. We run bulk experiments to test our algorithms for robustness and performance, and we compare them with other methods at the state of the art, always achieving correct results and superior performance. 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.

Automated summary

We propose a practical framework for converting Bezier curves to surfaces. Our framework supports interactive drawing and editing of Bezier splines on manifold meshes with millions of triangles, using repeated manifold averages. We demonstrate that direct extensions of evaluation algorithms to the manifold setting are fragile, while subdivision-based approaches are robust and efficient. Our framework includes algorithms for curve tracing, point evaluation, and approximated point insertion, achieving correct results and superior performance compared to state-of-the-art methods.