Clean up your Mesh! Part 1: Plane and simplex

TL;DR

This paper explores the use of Plane-based Geometric Algebra to represent and compute properties of meshes and simplices in a unified, coordinate-free manner, enhancing understanding and practical computation in discrete geometry.

Contribution

It introduces a unified, coordinate-free framework for representing and computing geometric properties of simplices and complexes using PGA, with practical demonstrations.

Findings

Compact formulas for k-magnitudes derived from PGA norms

Unified coordinate-free formulas for volume, centroid, and inertia

Practical applications demonstrated on real-world examples

Abstract

We revisit the geometric foundations of mesh representation through the lens of Plane-based Geometric Algebra (PGA), questioning its efficiency and expressiveness for discrete geometry. We find how $k$-simplices (vertices, edges, faces, ...) and $k$-complexes (point clouds, line complexes, meshes, ...) can be written compactly as joins of vertices and their sums, respectively. We show how a single formula for their $k$-magnitudes (amount, length, area, ...) follows naturally from PGA's Euclidean and Ideal norms. This idea is then extended to produce unified coordinate-free formulas for classical results such as volume, centre of mass, and moments of inertia for simplices and complexes of arbitrary dimensionality. Finally we demonstrate the practical use of these ideas on some real-world examples.