Normalization, Square Roots, and the Exponential and Logarithmic Maps in Geometric Algebras of Less than 6D

TL;DR

This paper provides an analysis and efficient algorithms for normalization, square roots, and exponential/logarithmic maps in geometric algebras of dimension less than 6, aiding their adoption in modeling 3D and 3+1D geometry.

Contribution

It offers a signature agnostic analysis and numerical implementations for common operations in all geometric algebras of dimension less than 6, focusing on popular algebras.

Findings

Efficient algorithms for normalization, square roots, exponential and logarithmic maps.

Implementation details for algebras R4, R3,1, R3,0,1, and R4,1.

Lowered threshold for adopting geometric algebra solutions.

Abstract

Geometric algebras of dimension $n < 6$ are becoming increasingly popular for the modeling of 3D and 3+1D geometry. With this increased popularity comes the need for efficient algorithms for common operations such as normalization, square roots, and exponential and logarithmic maps. The current work presents a signature agnostic analysis of these common operations in all geometric algebras of dimension $n < 6$, and gives efficient numerical implementations in the most popular algebras $\mathbb{R}_{4}$, $\mathbb{R}_{3,1}$, $\mathbb{R}_{3,0,1}$ and $\mathbb{R}_{4,1}$, in the hopes of lowering the threshold for adoption of geometric algebra solutions by code maintainers.