Acute Triangulations of Trapezoids and Pentagons

Maddalena Cavicchioli

doi:10.1155/2013/747128

Abstract

An acute triangulation of a polygon is a triangulation whose triangles have all their angles less than . The number of triangles in a triangulation is called the size of it. In this paper, we investigate acute triangulations of trapezoids and convex pentagons and prove new results about such triangulations with minimum size. This completes and improves in some cases the results obtained in two papers of Yuan (2010).

Highlights

Introduction and PreliminariesA triangulation of a planar polygon is a finite set of nonoverlapping triangles covering the polygon in such a way that any two distinct triangles are either disjoint or intersect in a single common vertex or edge
Burago and Zalgaller [1] and, independently, Goldberg and Manheimer [2] proved that every obtuse triangle can be triangulated into seven acute triangles and this bound is the best possible
Historical notes, and problems about acute triangulations of polygons and surfaces can be found in the survey paper [9]

Summary

Introduction and Preliminaries

A triangulation of a planar polygon is a finite set of nonoverlapping triangles covering the polygon in such a way that any two distinct triangles are either disjoint or intersect in a single common vertex or edge. Cassidy and Lord [3] showed that every square can be triangulated into eight acute triangles and eight is the minimum number. This remains true for any rectangle as proved by Hangan et al in [4]. Let K denote a family of planar polygons, and for K ∈ K, let f(K) be the minimum size of an acute triangulation of K. We discuss acute triangulations of trapezoids and convex pentagons and prove new results of such triangulations with minimum size. We get the following characterization of the right trapezoids: they are the only trapezoids needing exactly six triangles and one interior vertex for an acute triangulation of minimum size.

Characterizations of Trapezoids

Acute Triangulations of Pentagons