Doing euclidean plane geometry using projective geometric algebra

TL;DR

This paper introduces a novel approach to euclidean plane geometry using projective geometric algebra, demonstrating its advantages in terms of completeness, compactness, practicality, and elegance, with seamless integration of euclidean and ideal elements.

Contribution

It presents a comprehensive, coordinate-free framework for euclidean plane geometry based on PGA, highlighting its fundamental properties and applications, and comparing favorably with existing methods.

Findings

Establishes fundamental metric and non-metric properties using PGA

Demonstrates the effectiveness of PGA in various plane geometry topics

Shows PGA's advantages in completeness, compactness, and elegance

Abstract

The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief review of PGA, the article focuses on $\mathbf{P}(\mathbb{R}^*_{2,0,1})$, the PGA for euclidean plane geometry. It first explores the geometric product involving pairs and triples of basic elements (points and lines), establishing a wealth of fundamental metric and non-metric properties. It then applies the algebra to a variety of familiar topics in plane euclidean geometry and shows that it compares favorably with other approaches in regard to completeness, compactness, practicality, and elegance. The seamless integration of euclidean and ideal (aka infinite) elements forms an essential and novel feature of the treatment. Numerous figures accompany the text. For readers with the requisite mathematical background, a self-contained coordinate-free introduction to the algebra is provided in an appendix.