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

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.