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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\n            <jats:italic>tangent-point energy<\/jats:italic>\n            , 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\n            <jats:italic>Sobolev-Slobodeckij inner product<\/jats:italic>\n            enables us to make rapid progress toward local minima\u2014independent 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.\n          <\/jats:p>","DOI":"10.1145\/3439429","type":"journal-article","created":{"date-parts":[[2021,5,6]],"date-time":"2021-05-06T05:44:44Z","timestamp":1620279884000},"page":"1-21","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":40,"title":["Repulsive Curves"],"prefix":"10.1145","volume":"40","author":[{"given":"Chris","family":"Yu","sequence":"first","affiliation":[{"name":"Carnegie Mellon University, Pittsburgh, PA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Henrik","family":"Schumacher","sequence":"additional","affiliation":[{"name":"RWTH Aachen University, Aachen, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Keenan","family":"Crane","sequence":"additional","affiliation":[{"name":"Carnegie Mellon University, Pittsburgh, PA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2021,5,5]]},"reference":[{"key":"e_1_2_2_1_1","doi-asserted-by":"crossref","unstructured":"M. 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