Differentiable parameterization of Catmull-Clark subdivision surfaces

Abstract

Subdivision-based representations are recognized as important tools for the generation of high-quality surfaces for Computer Graphics. In this paper we describe two parameterizations of Catmull-Clark subdivision surfaces that allow a variety of algorithms designed for other types of parametric surfaces (i.e., B-splines) to be directly applied to subdivision surfaces. In contrast with the natural parameterization of subdivision surfaces characterized by diverging first order derivatives around extraordinary vertices of valence higher than four, the derivatives associated with our proposed methods are defined everywhere on the surface. This is especially important for Computer-Aided Design (CAD) applications that seek to address the limitations of NURBS-based representations through the more flexible subdivision framework.

Parametrization of General Catmull-Clark Subdivision Surfaces and its Applications

“…The advantage of this algorithm is that it is possible to define evaluation for parametric families of rules without considering excessive number of special cases, while improving numerical stability of calculation. In addition to Stam's approach, two different parametrizations of Catmull-Clark subdivision surfaces have been proposed by Boier-Martin and Zorin [4]. The motivation of their work is to provide parametrization techniques that are differentiable everywhere.…”

“…The motivation of their work is to provide parametrization techniques that are differentiable everywhere. Although all the natural parameterizations of subdivision surfaces are not C-1around extraordinary vertices of valence higher than four [4], the resulting surfaces are still C-2 almost everywhere. Moreover, despite of the fact that the partial derivatives diverge around an extraordinary vertex, in this paper, we will show that an standardized normal vector can be calculated explicitly everywhere.…”

“…It is known the first partial derivatives of S(u, v) at (0,0) diverge in a natural parametrization [4]. However, knowing the directions of them is sufficient in many applications.…”

“…Parametric forms are good for design and representation, discrete forms are good for machining and tessellation. Hence, we have a representation scheme that is good for all graphics and CAD/CAM applications.Research work for subdivision surfaces has been done in several important areas, such as surface interpolation [8,[19][20][21][22], surface evaluation [4,[13][14][15][16], surface trimming [9], boolean operations [3], and mesh editing [17]. However, powerful evaluation and analysis techniques for subdivision surfaces have not been fully developed yet.…”

Isogeometric shape optimisation of shell structures using multiresolution subdivision surfaces

“…The algorithm proposed by Stam [29,30] provides a spline based parameterisation of subdivision surfaces so that the properties of arbitrary surface points can be evaluated, cf (7). There are also alternative parameterisations available, see [32,33].…”