# Repulsive Curves

arXiv: https://arxiv.org/abs/2006.07859 · HTML: https://mlsanity.com/paper/2006.07859/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2020.

## Built on by (most cited first, 4 of 4)

- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
- [Variational autoencoders with latent high-dimensional steady geometric flows for dynamics](https://mlsanity.com/paper/2410.10137/) (2024)
- [Single-Line Drawing Generation via Semantics-Driven Optimization](https://mlsanity.com/paper/2606.01910/) (2026)

## Abstract (arXiv)

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.