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Repulsive Curves
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DOI：10.1145/3439429.png)
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Abstract
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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.
Keywords:
Computational design
shape optimization
curves
knots
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