Subdivision schemes are used to generate smooth curves by iteratively refining an initial control polygon. The simplest such schemes are corner cutting schemes, which specify two distinct points on each edge of the current polygon and connect them to get the refined polygon, thus cutting off the corners of the current polygon. While de Boor (1987) shows that this process always converges to a Lipschitz continuous limit curve, no matter how the points on each edge are chosen, Gregory and Qu (1996) discover that the limit curve is continuously differentiable under certain constraints. We extend these results and show that the limit curve can even be curvature continuous for specific sequences of cut ratios. • Proof that non-uniform corner cutting schemes can generate curvature continuous limit curves. • Corner cutting rules for generating cubic B-splines as limit curves. • Corner cutting rules for generating cubic non-uniform γ -B-splines as limit curves. • Corner cutting rules for generating cubic non-uniform rational γ -B-splines as limit curves.

Splines in the manifold setting have been defined as extensions from the standard Euclidean setting, but they are far more complicated. Alternative approaches, which are equivalent in the Euclidean case, lead to different results in the manifold case; the existence conditions are often quite restrictive; and the necessary computations are rather involved. All difficulties stem from the peculiar nature of the geodesic distance: in general, shortest geodesics may be not unique and the dependence on their endpoints may not be smooth; and distances cannot be computed in closed form. The former issue may impose strong limitations on the placement of control points. While the latter may greatly complicate the computations. Nevertheless, some recent results suggest that splines on surfaces may have practical impact on CAGD applications. We review the literature on this topic, accounting for both theoretical results and practical implementations.

The Riemannian center of mass (a.k.a. Karcher mean or Fréchet mean) provides the equivalent to the Euclidean affine average on manifolds. In spite of its many potential applications in computer graphics and geometric modeling, there exist surprisingly few algorithms to compute it. We present a direct method for computing the Riemannian center of mass on a triangle mesh. Our method works in the polyhedral metric and uses a piecewise-linear interpolation of gradients of the distance fields from a set of control points. We present applications for tracing splines on a surface, comparing to other methods at the state of the art, and showing that we produce quality results while supporting user interaction.

We present a practical framework to port Bézier curves to surfaces. We support the interactive drawing and editing of Bézier 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.

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C1 and C2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one.

Abstract We present a novel method to compute the cut locus of a distance function encoded on a polygonal mesh. Our method exploits theoretical findings about the cut locus and – with a combination of analytic, geometric and topological tools – it is able to compute a topologically correct and geometrically accurate approximation of it. Our result can be either restricted to the mesh edges, or aligned with the real cut locus. Both outputs may be useful for practical applications. We also provide a convenient tool to optionally prune the weak branches of the cut locus, simplifying its structure. Our approach supersedes prior art, in that it is easier to use and also orders of magnitude faster. In fact, it depends on just one parameter, and it flawlessly operates on meshes with high genus and very high element count at interactive rates. We experiment with different datasets and methods for geodesic distance estimation. We also present applications to local and global surface parameterization.