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Our starting point is the tangent-point energy of Buck & Orloff (1995), which for each pair of points x and y on a curve considers the radius r of the smallest sphere tangent to x and passing through y. By penalizing the reciprocal of the radius 1/r(x,y), the energy provides an infinite barrier to collisions—while naturally ignoring points that are close in space merely because they are close along the curve itself.
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Untangling the highly tangled Freedman unknot (top left) to the unit circle. For the same wall clock time, standard L2 gradient descent makes almost no progress, whereas conventional Sobolev descent fails to smooth out low (H1) or high (H2) frequencies. By carefully matching the inner product to the energy, our fractional Hs descent quickly flows to the circle.
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We can pack curves into a domain by penalizing their proximity to its boundary, and progressively increasing edge length. (Here we render curves with a non-circular cross section, which is not modeled by the energy.)
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Patterns obtained by constraining a collection of repulsive curves to a surface and increasing their lengths (initial states shown above their final configurations).
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Allowing curve endpoints to slide freely over constraint surfaces (left) enables design tasks like arranging networks of muscles or muscle fibers (right).
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Just as repulsive potentials are used to equally-distribute points, we can compute collections of equally-spaced curves (here constrained to a region via a fixed curve potential).
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Topologically intricate loops can be difficult to draw by hand---the sketches at left (drawn by Nathan Dunfield) illustrate Dehn-Thurston coordinates. At right we generate an equispaced version of this curve by flowing a rough sketch, subject to an implicit surface constraint.
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Traditional 2D graph drawing algorithms based on nodal proximity may cause edges to cross
(left) or position nodes extremely close together
(center); these layouts were produced by the popular
Graphviz library. By treating edges as repulsive curves, we can obtain graph drawings that are both more compact and more legible
(right).
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Isometric embedding: by jittering 2D drawings of non-planar graphs (which necessarily have crossings), curve repulsion with length constraints yields nicely spaced embeddings in R3 with prescribed edge lengths.
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Left: a crude initial topology for a synthetic vascular network is optimized to achieve more uniform delivery of nutrients throughout a volume. Right: plotting the maximum collision-free thickness helps to visualize the improvement in uniformity.
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As with Bézier curves, we can also control curve tangents at both interior and endpoints. Here we flow a polygonal curve (left), to a smooth interpolant with fixed points (red), and fixed points and tangents (blue).
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Standard curve interpolation methods in 2D drawing programs can cause curves to self-intersect (center), even when the control polygon (left) does not. By starting from the control polygon and constraining the control points, we obtain a smooth, non-intersecting interpolant (right).
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Top-left: In this path planning scenario, an initial trajectory brings the four agents dangerously close together. Bottom-left: By treating trajectories as curves in space-time, our system provides solutions that maximally avoid collisions, making them more robust to control errors. Right: Finding 2D trajectories is equivalent to optimizing a 3D braid with fixed endpoints constrained to an extrusion of the given environment. This same construction can easily be generalized to 3D environments.
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Encouraging curve tangents to align with a given vector field improve the quality of streamline visualization. Here, a random set of curve segments (top) aligns itself with a rotational vector field; we can also optimize randomly sampled streamlines (bottom) to improve their spacing.
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Untangling a pair of earbuds via repulsion (see supplemental video).
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Local minimizers of the tangent-point energy E2α. When α = 2 the tangent-point energy is scale-invariant and can exhibit “tight spots”; for larger values of α local interactions are penalized more than distant ones.
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To accelerate evaluation of the tangent-point energy, we build a bounding volume hierarchy that partitions both positions (left) and tangent directions (right), here drawn as a curve on the unit sphere.
