Dynamics of the Longest-Edge Altitude Bisection Algorithm

TL;DR

This paper analyzes the geometric properties and dynamics of the longest-edge altitude bisection (LEAB) algorithm for triangulation refinement, revealing its contraction behavior and fixed points in shape space.

Contribution

It provides a geometric characterization of LEAB, showing how it collapses shape space onto a fixed geodesic and deriving bounds on mesh size contraction.

Findings

LEAB's normalized shape space collapses onto a fixed geodesic in one step.

Explicit formulas for triangle subdivision mappings are derived.

Bounds for mesh size contraction under LEAB are established.

Abstract

We study a longest-edge based refinement scheme for triangulations, termed the longest-edge altitude bisection (LEAB), in which each triangle is subdivided by dropping the altitude from the vertex opposite to its longest edge. Using the normalized shape space of triangles introduced by Perdomo and Plaza in: Properties of triangulations obtained by the longest-edge bisection. \emph{Cent. Eur. J. Math.}, 12(12) (2014), 1796-1810, we show that LEAB admits a simple geometric description: the normalized left and right children of a triangle in focus are obtained by intersecting the geodesic of right triangles with rays issued from the endpoints of the longest edge and explicit formulas for the mappings are derived. This characterization implies an interesting observation that the associated refinement dynamics collapse the entire shape space onto the right-triangle geodesic in a single step and that every point on this geodesic is fixed. Two-sided bounds for the contraction of the mesh size (discretization parameter) are derived. Also, applications and limitations of the method are briefly discussed.