Polyhedral meshes consisting of triangles, quads, and pentagons and polar configurations cover all major sampling and modeling scenarios. We give an algorithm for efficient local, parallel conversion of such meshes to an everywhere smooth surface consisting of low-degree polynomial pieces. Quadrilateral facets with 4-valent vertices are ‘regular ’ and are mapped to bi-cubic patches so that adjacent bi-cubics join C 2 as for cubic tensor-product splines. The algorithm can be implemented in the vertex and geometry shaders of the GPU pipeline and does not use the fragment shader. Its implementation in DirectX 10 achieves conversion plus rendering at 659 frames per second with 42.5 million triangles per second on input of a model of 1300 facets of which 60 % are not regular.

We describe and analyze a subdivision scheme that generalizes bicubic spline subdivision to control nets with polar structure. Such control nets appear naturally for surfaces with the combinatorial structure of objects of revolution and at points of high valence in subdivision meshes. The resulting surfaces are C 2 except at a finite number of isolated points where the surface is C 1 and the curvature is bounded.

Efficient substitutes for subdivision surfaces

This paper derives strong relations that boundary curves of a smooth complex of patches have to obey when the patches are computed by local averaging. These relations restrict the choice of reparameterizations for geometric continuity. In particular, when one bicubic tensor-product B-spline patch is associated with each facet of a quadrilateral mesh with n-valent vertices and we do not want segments of the boundary curves forced to be linear, then the relations dictate the minimal number and multiplicity of knots: For general data, the tensor-product spline patches must have at least two internal double knots per edge to be able to model a G 1-conneced complex of C 1 splines. This lower bound on the complexity of any construction is proven to be sharp by suitably interpreting an existing surface construction. That is, we have a tight bound on the complexity of smoothing quad meshes with bicubic tensor-product B-spline patches. 1

Abstract. We present an algorithm for completing a C 2 surface of up to degree bi-6 by capping an n-sided hole with polar layout. The cap consists of n tensor-product patches, each of degree 6 in the periodic and degree 5 in the radial direction. To match the polar layout, one edge of these patches is collapsed. We explore and compare with alternative constructions, based on more pieces or using total-degree, triangular patches. 1