Exploiting degeneracy in projective geometric algebra

TL;DR

This paper explores the algebraic structure of projective geometric algebra, revealing how its degenerate Clifford algebra decomposes into Euclidean and ideal parts, and demonstrating geometric principles like Playfair's axiom through algebraic properties.

Contribution

It provides a detailed analysis of the internal structure of degenerate Clifford algebras in PGA, including their decomposition and group properties, offering new insights into geometric algebra frameworks.

Findings

Decomposition of Clifford algebra into Euclidean and ideal components

Derivation of Playfair's axiom from algebraic structure

Identification of the algebra as a twisted trivial extension

Abstract

The last two decades, since the seminal work of Selig, has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair's axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra $\mathrm{Cl}(V)$ is isomorphic to the twisted trivial extension $\mathrm{Cl}(V/\mathbb Fe_0)\ltimes_\alpha\mathrm{Cl}(V/\mathbb Fe_0)$, where $e_0$ is a degenerate vector and $\alpha$ is the grade-involution.