Iterated Medial Triangle Subdivision in Surfaces of Constant Curvature

TL;DR

This paper studies the recursive subdivision of geodesic triangles on constant curvature surfaces, proving angle bounds and identifying stable behaviors in angles and lengths during the process.

Contribution

It establishes uniform angle bounds and demonstrates stabilization of angles and lengths in iterated medial triangle subdivisions on constant curvature surfaces.

Findings

Angles remain bounded within a fixed interval during subdivision

Angles and lengths stabilize as subdivision progresses

Existence of a uniform positive delta for angle bounds

Abstract

Consider a geodesic triangle on a surface of constant curvature and subdivide it recursively into 4 triangles by joining the midpoints of its edges. We show the existence of a uniform $\delta>0$ such that, at any step of the subdivision, all the triangle angles lie in the interval $(\delta, \pi -\delta)$. Additionally, we exhibit stabilising behaviours for both angles and lengths as this subdivision progresses.