  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.
