An Angular Transformation of Triangles

TL;DR

This paper introduces an angular transformation method for triangles aimed at improving mesh quality for computer graphics and physical simulations, emphasizing mathematical properties and gradual regularization.

Contribution

The paper proposes a novel triangle transformation technique that enhances mesh regularity while maintaining mathematical rigor and gradual adjustment.

Findings

Transformation improves triangle regularity

Mathematically proven properties of the transformation

Potential for better mesh quality in simulations

Abstract

Triangles are everywhere in the virtual world. The surface of nearly every graphical object is saved as a triangular mesh on a computer. Light effects and movements of virtual objects are computed on the basis of triangulations. Besides computer graphics, triangulated surfaces are used for the simulations of physical processes, like heating or cooling of objects or deformations. The numerical method for these simulations is often the finite element method, whose accuracy depends on the quality of the triangulation. The quality of a triangle is generally determined by computing its proximity to an equilateral triangle. Namely, the triangle's inner angles should neither be too small nor too big in order to obtain reliable numerical results. Therefore, one often improves the mesh quality before any simulation. The fact that we require triangulations for accurate simulations is the main motivation for our occupation with triangle transformations. We need a triangulation method that transforms each triangle into a more regular one. However, the transformation should not regularize a particular triangle too fast as this may inhibit that the regularity a neighboring triangles can achieve. At the same time, we would like to prove the efficacy of the transformation. a property often missed by the heuristic procedures used in practice. Besides the practical motivation, the transformation itself exhibits interesting properties which can nicely be proved by basic mathematics.