Repulsive Curves

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 paper 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.

Paper: Download here (44.1 MB)

Video: Download here (16.7 MB)

Code: Github