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Bounds of sub-triangles

Amalia Adlerteg, Linus Carlsson

TL;DR

This work analyzes angle propagation under the longest-edge trisection (LE3) in triangulations used for mesh refinement. By modeling triangles in a normalized complex-plane framework and leveraging conformal (Möbius) normalizations alongside the Poincaré half-plane metric, the authors derive both analytic and numerical bounds on the largest angle in LE3-generated triangles. They prove an explicit analytic bound $\gamma'\le\gamma\cdot c$ with $c=\dfrac{\arccos\left(-\frac{5}{2\sqrt{7}}\right)}{\pi/3}\approx2.6816$ for $z$ in the region $\Omega_1$, and establish a numerical bound $\gamma'<2.1652\gamma$ within $\Omega_2$. These results confirm the existence of a global upper bound on the largest angle produced by LE3, aiding reliable angle control in adaptive triangulations and finite element analyses.

Abstract

We show upper and lower bounds for angles in iterations of trisections of certain triangulations.

Bounds of sub-triangles

TL;DR

This work analyzes angle propagation under the longest-edge trisection (LE3) in triangulations used for mesh refinement. By modeling triangles in a normalized complex-plane framework and leveraging conformal (Möbius) normalizations alongside the Poincaré half-plane metric, the authors derive both analytic and numerical bounds on the largest angle in LE3-generated triangles. They prove an explicit analytic bound with for in the region , and establish a numerical bound within . These results confirm the existence of a global upper bound on the largest angle produced by LE3, aiding reliable angle control in adaptive triangulations and finite element analyses.

Abstract

We show upper and lower bounds for angles in iterations of trisections of certain triangulations.
Paper Structure (11 sections, 5 theorems, 18 equations, 9 figures)

This paper contains 11 sections, 5 theorems, 18 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Description and preliminary results of the LE3-method
  3. Longest edge trisection
  4. Mathematical understanding
  5. Conformal mappings
  6. Poincaré half plane metric
  7. Simulations of orbits
  8. The method of random orbit simulation
  9. Examples
  10. Results
  11. Conclusion and discussion

Key Result

Lemma 1

Let $d(\cdot,\cdot)$ denote the hyperbolic distance in the Poincaré half-plane. For every $z_1$ and $z_2$ in the space of triangles, $\Sigma$, we have

Figures (9)

  • Figure 1: A right triangle after trisection of the original triangle defined by $z$, such that $|z-2/3|\leq1/3$.
  • Figure 2: The triangle after the translation given by Equation \ref{['eq:translation']}.
  • Figure 3: The triangle after the rotation given by Equation \ref{['eq:rotation']}.
  • Figure 4: The triangle after the inversion given by Equation \ref{['eq:inversion']}.
  • Figure 5: The triangle after the magnification given by Equation \ref{['eq:magnification']}.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Lemma 1: Non-increasing property
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Theorem 5