From Curves to Words and Back Again: Geometric Computation of Minimum-Area Homotopy

TL;DR

This paper unifies geometric and algebraic methods to compute the minimum homotopy area of closed curves, providing a new geometric proof of existing algorithms and the first polynomial-time algorithm for minimal self-overlapping decomposition.

Contribution

It establishes a unified framework connecting geometric and algebraic word representations of curves, and introduces the first polynomial-time algorithm for minimal self-overlapping decomposition.

Findings

Unified framework for geometric and algebraic curve representations

New geometric proof of Nie's minimum homotopy area algorithm

First polynomial-time algorithm for minimal self-overlapping decomposition

Abstract

Let $\gamma$ be a generic closed curve in the plane. Samuel Blank, in his 1967 Ph.D. thesis, determined if $\gamma$ is self-overlapping by geometrically constructing a combinatorial word from $\gamma$. More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of $\gamma$ by constructing a combinatorial word algebraically. We provide a unified framework for working with both words and determine the settings under which Blank's word and Nie's word are equivalent. Using this equivalence, we give a new geometric proof for the correctness of Nie's algorithm. Unlike previous work, our proof is constructive which allows us to naturally compute the actual homotopy that realizes the minimum area. Furthermore, we contribute to the theory of self-overlapping curves by providing the first polynomial-time algorithm to compute a self-overlapping decomposition of any closed curve $\gamma$ with minimum area.