Geometric Algebras for Euclidean Geometry

Charles Gunn, Technisches Universität Berlin, Germany

We discuss and compare existing GA models for doing euclidean geometry. We begin by clarifying a set of fundamental terms which carry conflicting meanings in the literature, including $\mathbb{R}^{n}$, euclidean, homogeneous model, and duality. Equipped with these clarified concepts, we establish that the dual projectivized Clifford algebra $\mathbf{P(\mathbb{R}^*_{n,0,1})}$ deserves the title of standard homogeneous model of euclidean geometry (we also call it projective geometric algebra). We then turn to a comparison with the other main candidate for doing euclidean geometry, the conformal model. We establish that these two algebras exhibit the same formal feature set for doing euclidean geometry. We then compare them with respect to a set of practical criteria.

Euclidean Plane Geometry using Projective Geometric Algebra

Charles Gunn, Technisches Universität Berlin, Germany

On the basis of simple geometric exercises, we show how projective geometric algebra (PGA) can be used to teach euclidean plane geometry. PGA contains the traditional vector algebra approach as a subset -- but contains many more unexpected delights that should please every lover of plane geometry, and provide a gentle introduction to those curious to get some practice using PGA.