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 no two adjacent edges overlap. A discrete analogue of the classic Whiney-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.