A geometric approach to the bisection method

Abstract

The bisection method is the consecutive bisection of a triangle by the median of the longest side. This paper introduces a taxonomy of triangles that precisely captures the behavior of the bisection method. Our main result is an asymptotic upper bound for the number of similarity classes of triangles generated on a mesh obtained by iterative bisection, which previously was known only to be finite. We also prove that the number of directions on the plane given by the sides of the triangles generated is finite. Additionally, we give purely geometric and intuitive proofs of classical results for the bisection method. © Springer-Verlag 2004.