Projective geometric algebra: A new framework for doing euclidean geometry

TL;DR

This paper introduces projective geometric algebra (PGA), a modern, coordinate-free framework for Euclidean geometry that unifies various algebraic systems, simplifies computations, and enhances programming productivity for practitioners.

Contribution

It presents PGA as a comprehensive, efficient, and intuitive framework for Euclidean geometry, integrating multiple algebraic systems and supporting advanced operations like automatic differentiation.

Findings

PGA provides a uniform representation of geometric entities.

It offers robust, parallel-safe join and meet operations.

The framework supports efficient implementation and improved programming productivity.

Abstract

A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet operations; compact, polymorphic syntax for euclidean formulas and constructions; a single intuitive sandwich form for isometries; native support for automatic differentiation; and tight integration of kinematics and rigid body mechanics. Inclusion of vector, quaternion, dual quaternion, and exterior algebras as sub-algebras simplifies the learning curve and transition path for experienced practitioners. On the practical side, it can be efficiently implemented, while its rich syntax enhances programming productivity. The basic ideas are introduced in the 2D context; the 3D treatment focus on selected topics. Advantages to traditional approaches are collected in a table at the end. The article aims to be a self-contained introduction for practitioners of euclidean geometry and includes numerous examples, figures, and tables.