On Moments of Sets Bounded by Subdivision Surfaces

The volume enclosed by subdivision surfaces, such as Doo-Sabin, Catmull-Clark, and Loop has recently been derived. Moments of higher degree d are more challenging because of the growing number of coefficients in the d + 3-linear forms. We derive the intrinsic symmetries of the tensors, and thereby reduce the complexity of the problem. Our framework allows to compute the 4-linear forms that determine the centroid defined by Doo-Sabin, and Loop surfaces, including Loop with sharp creases. For Doo-Sabin surfaces, we also establish the tensors of rank 5 that determine the inertia for valences 3, and 4. When the subdivision weights are rational, the centroid, and inertia are obtained in exact, symbolic form. In practice, the formulas are restricted to meshes with a certain maximum valence of a vertex. The first author dedicates this work to the memory of Andrew Ladd, Nik Sperling, and Leif Dickmann. The article and additional resources are available at www.hakenberg.de. The first author was partially supported by personal savings accumulated during his visit to the Nanyang Technological University as a visiting research scientist in 2012-2013. He’d like to thank everyone who worked to make this opportunity available to him. Introduction A subdivision scheme S is a mesh refinement procedure. Starting with an initial mesh M, the repeated application of the subdivision scheme results in an increasingly dense mesh SnM. The algorithm is designed so that the sequence of meshes converges to a piecewise smooth surface S¶M. Due to these properties, subdivision is a popular technique to design and represent surfaces in computer graphics. [Catmull/Clark 1978] and [Doo/Sabin 1978] introduced the first subdivision schemes intended for the refinement of quad meshes. In the limit, large parts of the surface have piecewise polynomial parameterization. Later, [Loop 1987] designed a subdivision scheme for triangular meshes. The smoothness characteristics of the limit surface produced by the schemes are well-understood, see [Reif 1995].