From curves to words and back again: geometric computation of minimum-area homotopy

Abstract

Let gamma be a generic closed curve in the plane. The area of a homotopy is the area swept by the homotopy. We consider the problem of computing the minimum null-homotopy area of gamma. 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 More recently, Zipei Nie, in an unpublished manuscript, computed the minimum homotopy area of gamma by constructing a combinatorial word algebraically.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. Next, we describe the first polynomial implementation of an algorithm to compute the minimum homotopy area of a piecewise linear closed curve in the plane. We discuss how minimum homotopy area can be used as a similarity measure for curves and include experiments that compare the runtime of our algorithm to an implementation of the Frechet distance. We then extend our algorithm for computing the minimum homotopy area in the plane to homotopic, non-intersecting, non-contractible curves on an orientable surface with positive genus. Finally, we consider the inverse problem of determining which combinatorial Blank words correspond to closed curves in the plane. We solve a special case of this problem and give an exponential algorithm to the general case.