Clean up your Mesh! Part 1: Plane and simplex

TL;DR

The paper addresses efficient, coordinate-free representation of discrete geometry using Plane-based Geometric Algebra (PGA) and derives unified $k$-magnitudes for simplices and complexes. It shows how $k$-simplices can be formed as joins $S_k = v_0 \vee \cdots \vee v_k$ and how their magnitudes follow from the norm of these carriers, with $\text{mag}\,\sigma_k = \frac{1}{k!} \lVert S_k \rVert$, while magnitudes of complexes can be computed from boundary sums via $\partial\mathcal{K}_k$ and the ideal norm. The approach relies on Euclidean vs. ideal norms, dualization, and the join/meet framework to produce invariant, coordinate-free formulas for volume, center of mass, and inertia, demonstrated on a real-world tank volume problem and extendable to arbitrary dimensions. Overall, the work provides a rigorous, practical PGA toolkit for mesh processing, enabling dimension-agnostic, equivariant calculations that unify classical results within a single algebraic framework.

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.

Figures (2)

• Figure 1: The simplicial extension of a simplex and the associated (signed) $k$-magnitudes. The carrier gauges of the $(k-1)$-carrier are shown as purple arrows.
• Figure 2: The area of the pentagon is the sum of the signed areas of the triangulation w.r.t. to any point. The c.o.m. of the pentagon, drawn as red circle, is the area-weighted-average of the c.o.m. of the individual triangles, drawn as solid dots. In $d$-D, the triangles become pyramidal cones from the arbitrary apex point to the boundary facets, still contributing their hypervolume-weighted centroids to the centroid of the complex. But the polygon centroids are then $(d-k)$-blades, whose intersection with the $k$-blade representing the carrier plane gives a centroid point $d$-blade.

Theorems & Definitions (3)

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