Volume Enclosed by Subdivision Surfaces

We present a framework to derive the coefficients of trilinear forms that compute the exact volume enclosed by subdivision surfaces. The coefficients depend only on the local mesh topology, such as the valence of a vertex, and the subdivision rules. The input to the trilinear form are the initial control points of the mesh. Our framework allows us to explicitly state volume formulas for surfaces generated by the popular subdivision algorithms Doo-Sabin, Catmull-Clark, and Loop. The trilinear forms grow in complexity as the vertex valence increases. In practice, the explicit formulas are restricted to meshes with a certain maximum valence of a vertex. The approach extends to higher order momentums such as the center of gravity, and the inertia of the volume enclosed by subdivision surfaces. 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.