A mathematical proof of how fast the diameters of a triangle mesh tend to zero after repeated trisection

Abstract

The Longest-Edge (LE) trisection of a triangle is obtained by joining the two points which divide the longest edge in three with the opposite vertex. If LE-trisection is iteratively applied to an initial triangle, then the maximum diameter of the resulting triangles is between two sharpened decreasing functions. This paper mathematically answers the question of how fast the diameters of a triangle mesh tend to zero as repeated trisection is performed, and completes the previous empirical studies presented in the MASCOT 2010 Meeting (Perdomo et al., 2010).