POLYGONS HAVE EARS

Authors

G.H. Meisters, University of Nebraska-Lincoln

Document Type

Article

Date of this Version

6-1975

Citation

The American Mathematical Monthly, Vol. 82, No. 6 (Jun. - Jul., 1975), pp. 648-651 Published by: Mathematical Association of America

Comments

Copyright 1975 Mathematical Association of America

Abstract

We refer to a simple closed polygonal plane curve with a finite number of sides as a Jordan polygon. We assume the truth of the famous Jordan Curve Theorem only for Jordan polygons. (For elementary proofs see Appendix 2 of Chapter V of [4] or Appendix B1 of [5].) Three consecutive vertices V₁, V₂, V₃ of a Jordan polygon P = V₁V₂V₃V₄... V_nV₁ (n > 4) are said to form an ear (regarded as the region enclosed by the triangle V₁V₂V₃) at the vertex V₂ if the (open) chord joining V₁ and V₃ lies entirely inside the polygon P. We say that two ears are non-overlapping if their interior regions are disjoint; otherwise they are overlapping. If we remove or cut off an ear V₁V₂V₃ (by drawing the chord V₁V₃) from the Jordan polygon P, then there remains the Jordan polygon P' = V₁V₃V₄ ... V_nV₁ which has one less vertex than P.

The property of Jordan polygons expressed by the following theorem seems to provide a particularly simple and conceptual bridge from the Jordan Curve Theorem for Polygons to the Triangulation Theorem for Jordan Polygons; at least simpler perhaps than that given in Appendix B2 of [5].