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Repulsive Curves

Published: 05 May 2021 Publication History

Abstract

Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This article develops efficient algorithms for (self-)repulsion of plane and space curves that are well-suited to problems in computational design. Our starting point is the so-called tangent-point energy, which provides an infinite barrier to self-intersection. In contrast to local collision detection strategies used in, e.g., physical simulation, this energy considers interactions between all pairs of points, and is hence useful for global shape optimization: local minima tend to be aesthetically pleasing, physically valid, and nicely distributed in space. A reformulation of gradient descent based on a Sobolev-Slobodeckij inner product enables us to make rapid progress toward local minima—independent of curve resolution. We also develop a hierarchical multigrid scheme that significantly reduces the per-step cost of optimization. The energy is easily integrated with a variety of constraints and penalties (e.g., inextensibility, or obstacle avoidance), which we use for applications including curve packing, knot untangling, graph embedding, non-crossing spline interpolation, flow visualization, and robotic path planning.

Supplementary Material

yu (yu.zip)
Appendix, image and software files for Repulsive Curves
40-2-3439429-Article10 (40-2-3439429-article10.mp4)
Presentation at SIGGRAPH Asia '21

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cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 40, Issue 2
April 2021
174 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/3454118
Issue’s Table of Contents
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Publication History

Published: 05 May 2021
Accepted: 01 November 2020
Revised: 01 October 2020
Received: 01 June 2020
Published in TOG Volume 40, Issue 2

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Author Tags

  1. Computational design
  2. shape optimization
  3. curves
  4. knots

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