On Minimum Area Homotopies of Normal Curves in the Plane

Abstract

In this paper, we study the problem of computing a homotopy from a planar curve $C$ to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau's problem. Chambers and Wang studied the special case that $C$ is the concatenation of two simple curves, and they gave a polynomial-time algorithm for computing a minimum homotopy in this setting. We study the general case of a normal curve $C$ in the plane, and provide structural properties of minimum homotopies that lead to an algorithm. In particular, we prove that for any normal curve there exists a minimum homotopy that consists entirely of contractions of self-overlapping sub-curves (i.e., consists of contracting a collection of boundaries of immersed disks).