Keenan Crane

CARNEGIE MELLON UNIVERSITY

Repulsive Surfaces

ACM Transactions on Graphics 2021

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.

Video

Talk Video

repulsive-surfaces—implementation in C++

The authors thank Saul Schleimer and Henry Segerman for helpful discussions about topological examples, and Nick Stadie for perspective on molecular symmetries. This work was supported by a Packard Fellowship, NSF Award 1943123, and gifts from Autodesk, Activision Blizzard, Adobe, Disney, and Facebook. The third author was supported by DFG-Project 282535003: Geometric curvature functionals: energy landscape and discrete methods.

@article{Yu:2021:RS,
author = {Yu, Chris and Brakensiek, Caleb and Schumacher, Henrik and Crane, Keenan},
title = {Repulsive Surfaces},
journal = {ACM Trans. Graph.},
volume = {40},
number = {6},
year = {2021},
publisher = {ACM},
address = {New York, NY, USA},
}

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