A Penetration Metric for Deforming Tetrahedra using Object Norm

TL;DR

This paper introduces a new deformable penetration depth metric for tetrahedra using object norm, enabling efficient and tighter inter-penetration measurement for deforming objects in simulation and collision detection.

Contribution

We propose a novel deformable penetration depth metric based on object norm, with a closed-form distance measure and quadratic programming solutions for deforming tetrahedra.

Findings

The metric can be computed in milliseconds.

It is three times tighter than classical rigid penetration metrics.

The algorithm is efficient and suitable for real-time applications.

Abstract

In this paper, we propose a novel penetration metric, called deformable penetration depth PDd, to define a measure of inter-penetration between two linearly deforming tetrahedra using the object norm. 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 PDd 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 non-penetration constraints. We also show that the PDd 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 pre-calculated penetration direction. We have implemented our algorithm on a standard PC platform using an off-the-shelf QP optimizer, and experimentally show that both the static/deformable and deformable/deformable tetrahedra cases can be solvable in from a few to tens of milliseconds. Finally, we demonstrate that our penetration metric is three-times smaller (or tighter) than the classical, rigid penetration depth metric in our experiments.