Truncation Error for Simplex Propagation

Cube propagation has been suggested as an alternative to Jacobian integration for estimating the volume change under a deformation in three dimensions [Pai et al., 2013]. Cube propagation estimates the change in volume by approximating a three dimensional volume as a mesh of tetrahedra, which covers the interior of the volume and approximates the boundary as piece-wise triangular surface patches, and then estimate the change in volume under deformation of the volume as the sum of the change of volumes of the tetrahedra. This is an instance of the more general simplex counting, and in this technical report we derive the truncation error for simplex counting in 2 and 3 dimensions. In the appendix, we give a short review of numerical quadrature in 1 and 3 dimensions.