A Penetration Metric for Deforming Tetrahedra using Object Norm In this paper, we propose a novel penetration metric, called deformable penetration depth PD d , to define a measure of interpenetration between two linearly deforming tetrahedra using the object norm 1. First of all, we show that a distance metric for a tetrahedron deforming between two configurations can be found in closed form based on object norm. Then, we show that the PD d  between an intersecting pair of static and deforming tetrahedra can be found by solving a quadratic programming (QP) problem in terms of the distance metric with nonpenetration constraints. We also show that the PD d  between two, intersected, deforming tetrahedra can be found by solving a similar QP problem under some assumption on penetrating directions, and it can be also accelerated by an order of magnitude using precalculated penetration direction. We have implemented our algorithm on a standard PC platform using an offtheshelf QP optimizer, and experimentally show that both the staticdeformable and deformabledeformable tetrahedra cases can be solvable in from a few to tens of milliseconds. Finally, we demonstrate that our penetration metric is threetimes smaller (or tighter) than the classical, rigid penetration depth metric in our experiments.

Parametrization quantization with free boundaries for trimmed quad meshing The generation of quad meshes based on surface parametrization techniques has proven to be a versatile approach. These techniques quantize an initial seamless parametrization so as to obtain an integer grid map implying a pure quad mesh. Stateoftheart methods following this approach have to assume that the surface to be meshed either has no boundary, or has a boundary which the resulting mesh is supposed to be aligned to. In a variety of applications this is not desirable and nonboundaryaligned meshes or gridparametrizations are preferred. We thus present a technique to robustly generate integer grid maps which are either boundaryaligned, nonboundaryaligned, or partially boundaryaligned, just as required by different applications. We thereby generalize previous work to this broader setting. This enables the reliable generation of trimmed quad meshes with partial elements along the boundary, preferable in various scenarios, from tiled texturing over design and modeling to fabrication and architecture, due to fewer constraints and hence higher overall mesh quality and other benefits in terms of aesthetics and flexibility.