Homotopic curve shortening and the affine curve-shortening flow (1909.00263v5)

Abstract: 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.

• The paper introduces a novel HCS process that extends ACSF to handle non-convex and self-intersecting curves.
• The authors combine experimental and theoretical analyses to show that HCS mimics ACSF behavior on fine grids and random point sets.
• The study demonstrates that a vertex-release algorithm in HCS preserves key geometric properties like affine invariance and non-increment of curvature.

An Examination of Homotopic Curve Shortening and its Relationship to Affine Curve-Shortening Flow

This essay reviews the paper "Homotopic curve shortening and the affine curve-shortening flow" by Sergey Avvakumov and Gabriel Nivasch. The paper introduces a discrete process termed Homotopic Curve Shortening (HCS), designed to generalize the convex-layer decomposition of planar point sets. This process has significant implications for understanding the affine curve-shortening flow (ACSF), particularly concerning its application to curves that are non-convex or self-intersecting.

Summary of the Paper

The focal point of the paper is the HCS process, which initiates with a closed curve amidst a set of planar obstacles and iteratively seeks shortcuts around these obstacles while reducing the curve to its shortest homotopic equivalent. The primary motivation behind this approach is to bridge the conceptual gap between ACSF—typically applied to convex curves—and more general cases involving non-convex or self-intersecting curves.

Experimental and Theoretical Alignment with ACSF

The authors explore the relationship between HCS and ACSF both experimentally and theoretically. Through experimentation, it is observed that when the obstacle set is either a fine grid or a random point set, HCS appears to emulate ACSF at the limit. This finding is pivotal as it extends prior research linking ACSF solely to convex curves through convex-layer decomposition.

Key properties of ACSF, such as affine invariance, preservation of convexity, and non-increment of absolute curvature, are retained in HCS, highlighting the method’s robustness and utility. Moreover, the HCS process ensures that the number of self-intersections and inflection points does not increase with each iterational step.

Implications and Conjectures

The paper posits conjectures that align HCS closely with ACSF outcomes as the granularity of the grid or the density of random points increase. These conjectures, supported by experimental evidence, propose strong links between the evolution of closed curves under HCS and the expected properties under ACSF.

Computational and Algorithmic Approach

The authors have conceptualized HCS with a strong algorithmic foundation, implementing a vertex-release method for curve shortening. The HCS method is not only theoretically insightful but also computationally feasible, as demonstrated by the authors' experiments and their published code bases. This computational feasibility enhances its applicability across various domains where polygonal approximation of homotopic paths is necessary.

Conclusion

The introduction and analysis of HCS present a significant advancement in the discrete approximation of ACSF. By enabling the paper of non-convex and self-intersecting curves through a generalizable process, this research furnishes valuable insights into curve evolution dynamics. Moreover, the well-supported experimental results underscore the relevance of HCS in theoretical explorations and practical applications, particularly in areas like computer graphics and computational geometry.

This paper contributes to the theoretical enrichment of curve shortening methodologies while simultaneously paving the way for further investigations into discrete geometric processes. Its proposals and evidentiary findings invite subsequent research endeavors aiming to illuminate the intriguing facets of curve dynamics in complex environments.