同伦曲线缩短与仿射曲线缩短流
Homotopic curve shortening and the affine curve-shortening flow
摘要：
我们定义并研究了一个离散过程，推广了平面点集的凸层分解。我们的过程，我们称之为“同伦曲线缩短”（HCS），开始于一个封闭的曲线（这可能是自相交）在一组$P\subset \mathbb R^2$的点障碍的存在下，并在离散的步骤，其中每一步包括（1）采取捷径周围的障碍，（2）减少曲线到其最短的同伦等价。我们发现，实验上，如果初始曲线是固定的和$P$被选为一个非常好的规则网格或均匀随机点集，然后HCS的行为在极限像仿射曲线缩短流（ACSF）。ACSF和HCS之间的这种联系概括了ACSF和凸层分解之间的联系（Eppstein等人，2017年;考尔德和智能，2020年），这是有限的凸曲线。我们证明了HCS满足类似于ACSF的一些性质：HCS在仿射变换下不变，保持凸性，并且不增加总绝对曲率。此外，曲线的自相交或两条曲线之间的相交（适当定义）的数量不会增加。最后，如果初始曲线是简单的，那么拐点的数量（适当定义）不会增加。
英文摘要：
We 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 shortcuts 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 ACSF and HCS generalizes the link between ACSF and convex-layer decomposition (Eppstein et al., 2017; Calder and Smart, 2020), which is restricted to convex curves. We prove that HCS satisfies some properties analogous to those of ACSF: HCS 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.
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