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How to peel self-intersecting onions: https://arxiv.org/abs/1909.00263
arxiv.orgHomotopic curve shortening and the affine curve-shortening flowWe define and study a discrete process that generalizes the convex-layer
decomposition of a planar point set. Our process, which we call "homotopic
curve shortening" (HCS), starts with a closed curve (which might
self-intersect) in the presence of a set $P\subset \mathbb R^2$ of point
obstacles, and evolves in discrete steps, where each step consists of (1)
taking shortucts around the obstacles, and (2) reducing the curve to its
shortest homotopic equivalent.
We find experimentally that, if the initial curve is held fixed and $P$ is
chosen to be either a very fine regular grid or a uniformly random point set,
then HCS behaves at the limit like the affine curve-shortening flow (ACSF).
This connection between HCS and ACSF generalizes the link between "grid
peeling" and the ACSF observed by Eppstein et al. (2017), which applied only to
convex curves, and which was studied only for regular grids.
We prove that HCS satisfies some properties analogous to those of ACSF: HSC
is invariant under affine transformations, preserves convexity, and does not
increase the total absolute curvature. Furthermore, the number of
self-intersections of a curve, or intersections between two curves
(appropriately defined), does not increase. Finally, if the initial curve is
simple, then the number of inflection points (appropriately defined) does not
increase.