  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).
