Reducing simple polygons to triangles - A proof for an improved conjecture

Abstract

An edge of a simple closed polygon is called eliminating if it can be translated in parallel towards the interior of the polygon to eliminate itself or one of its neighbor edges without violating simplicity. [3] presents an algorithm that reduces a polygon P with n vertices to a triangle by a sequence of O(n) parallel edge translations, of which n - 3 translate an eliminating edge; the algorithm is used in [3] for computing morphs between polygons. It is conjectured in [3] that in each simple closed polygon there exists at least one eliminating edge, i.e. n - 3 edge translations are sufficient for the reduction of P. Also the computation of eliminating edges remains an open problem in [3]. In this paper we prove that in each simple closed polygon there exist at least two eliminating edges; this lower bound is tight since for all n ≥ 5 there exists a polygon with only two eliminating edges. Furthermore we present an algorithm that computes in total O(nlogn) time using O(n) space an eliminating edge for each elimination step. We thus obtain the first non-trivial algorithm that computes for P a sequence of n - 3 edge translations reducing P to a triangle.