Repulsive Surfaces

TL;DR

This paper introduces a numerical framework for optimizing surface geometry with collision avoidance, utilizing a discretized tangent-point energy and novel acceleration techniques for triangle meshes.

Contribution

It develops a discretization of tangent-point energy for triangle meshes and introduces a new acceleration scheme avoiding multiresolution hierarchies.

Findings

Effective collision avoidance in surface optimization.

Accelerated computation via fractional Sobolev inner product.

Applicability to visualization and geometric modeling.

Abstract

Functionals that penalize bending or stretching of a surface play a key role in geometric and scientific computing, but to date have ignored a very basic requirement: in many situations, surfaces must not pass through themselves or each other. This paper develops a numerical framework for optimization of surface geometry while avoiding (self-)collision. The starting point is the tangent-point energy, which effectively pushes apart pairs of points that are close in space but distant along the surface. We develop a discretization of this energy for triangle meshes, and introduce a novel acceleration scheme based on a fractional Sobolev inner product. In contrast to similar schemes developed for curves, we avoid the complexity of building a multiresolution mesh hierarchy by decomposing our preconditioner into two ordinary Poisson equations, plus forward application of a fractional differential operator. We further accelerate this scheme via hierarchical approximation, and describe how to incorporate a variety of constraints (on area, volume, etc.). Finally, we explore how this machinery might be applied to problems in mathematical visualization, geometric modeling, and geometry processing.