A bit better: Variants of duality in geometric algebras with degenerate metrics

TL;DR

This paper explores alternative duality concepts in geometric algebras with degenerate metrics, emphasizing coordinate-free approaches and proposing a duality-neutral software implementation for geometric primitives.

Contribution

It compares double algebra duality and Hodge duality in degenerate metric algebras, highlighting the advantages of coordinate-free duality for geometric primitives.

Findings

J map is coordinate-free and supports geometric duality

Double algebra duality and Hodge duality are computationally identical

Proposes a duality-neutral software implementation with minimal overhead

Abstract

Multiplication by the pseudoscalar $\mathbf{I}$ has been traditionally used in geometric algebra to perform non-metric operations such as calculating coordinates and the regressive product. In algebras with degenerate metrics, such as euclidean PGA $P(\mathbb{R}^*_{3,0,1})$, this approach breaks down, leading to a search for non-metric forms of duality. The article compares the dual coordinate map $J: G \rightarrow G^*$, a double algebra duality, and Hodge duality $H: G \rightarrow G $, a single algebra duality for this purpose. While the two maps are computationally identical, only $J$ is coordinate-free and provides direct support for geometric duality, whereby every geometric primitive appears twice, once as a point-based and once as a plane-based form, an essential feature not only of projective geometry but also of euclidean kinematics and dynamics. Our analysis concludes with a proposed duality-neutral software implementation, requiring a single bit field per multi-vector.