Clean up your Mesh! Part 1: Plane and simplex

2025-11-11

AI Summary

Plane-based Geometric Algebra (PGA) suffers from low representational efficiency and limited expressiveness for modeling discrete geometric entities—specifically, k-simplices (e.g., vertices, edges, faces) and k-complexes (e.g., point clouds, line complexes, triangle meshes)—in computational geometry. Method: We propose a unified, compact PGA-based representation framework. Our approach introduces Euclidean and ideal norms to derive a dimension-agnostic, closed-form k-metric formula—unifying length, area, volume, and higher-dimensional measures—and enables coordinate-free computation of geometric quantities such as centroids and inertia tensors. By integrating the join operator, simplex decomposition, and linear combinations, the framework supports algebraic construction and manipulation of both k-simplices and k-complexes. Results: Experiments demonstrate significant efficiency and practicality in mesh processing tasks. The method establishes a scalable, coordinate-free geometric computing paradigm for high-dimensional discrete geometry modeling.

Technology Category

Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesSearch and Optimization: Mixed Discrete/Continuous Search

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSecurity and Privacy: Large-scale security measurements

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.

Problem

• Reevaluating mesh representation efficiency using Plane-based Geometric Algebra
• Developing unified formulas for geometric properties of k-simplices
• Extending coordinate-free calculations to volumes and inertia moments

Innovation

• Plane-based Geometric Algebra for mesh representation
• Unified formulas for k-magnitudes using Euclidean norms
• Coordinate-free formulas for volume and inertia calculations