Constructive Whitney-Graustein Theorem: Or How to Untangle Closed Planar Curves

The classification of polygons is considered in which two polygons are regularly equivalent if one can be continuously transformed into the other such that for each intermediate polygon, no two adjacent edges overlap. A discrete analogue of the classic Whitney-Graustein theorem is proven by showing that the winding number of polygons is a complete invariant for this classification. Moreover, this proof is constructive in that for any pair of equivalent polygons, it produces some sequence of regular transformations taking one polygon to the other. Although this sequence has a quadratic number of transformations, it can be described and computed in real time. 1. Why a circle differs from a figure-of-eight. First consider closed planar curves that are smooth. Intuitively, a "kink" on such a curve is a point without a unique tangent line. It seems obvious that there is no continuous deformation of figure-of-eight to a circle in which all the intermediate curves remain kink-free (see Fig. 1). Figure 2 shows another curve that clearly has a kink-free deformation to a circle. Let us make this precise. By a (closed planar) curve we mean a continuous function 2 is the Euclidean plane. The curve C is regular if the first derivative C'(t) is defined and not equal to zero for all t[0, 1], and 2 be a homotopy between curves Co and C1, i.e., h is a continuous function and each Cs :[0, 1]-> E 2 (0-< s-< 1) is a curve, where we define Cs(t)= h(s, t) (t[0, 1]). The homotopy is regular if each C is regular. Two regular curves are regularly equivalent if there is a regular homotopy between them. A classical result known as the Whitney-Graustein theorem [5], [1] says that two curves are regularly equivalent if and only if they have the same winding number (up FIG. 1. Transforming a figure-of-eight to a circle" kink appears.