A Low-Memory Time-Efficient Implementation of Outermorphisms for Higher-Dimensional Geometric Algebras

Abstract

From the beginning of David Hestenes rediscovery of geometric algebra in the 1960s, outermorphisms have been a cornerstone in the mathematical development of GA. Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. Over the last two decades, GA-based mathematical models and software implementations have been developed in many fields of science and engineering. As such, efficient implementations of outermorphisms are of significant importance within this context. This work attempts to shed some light on the problem of optimizing software implementations of outermorphisms for practical prototyping applications using geometric algebra. The approach we propose here for implementing outermorphisms requires orders of magnitude less memory compared to other common approaches, while being comparable in time performance, especially for high-dimensional geometric algebras.