Mass Properties of Triangulated Solids and Their Derivatives

This supplemental material describes the computation of mass properties of triangulated solids and their derivatives w.r.t. surface vertices. We start by briefly reviewing the volume integrals for mass, center of mass, and moment of inertia. Thereafter, we reduce the volume to surface integrals using the Divergence theorem, resulting in analytical expressions for a volume bounded by a triangulated surface. We then discuss derivatives of these analytical surface integrals w.r.t. vertices. We provide pseudo code for both mass properties and their derivatives for the reader’s convenience. The resulting routines serve as fundamental building blocks for optimizing moment of inertia for spinnable objects.