On the Clifford Algebraic Description of the Geometry of a 3D Euclidean Space

TL;DR

This paper explores how the Clifford algebra l_{3,3} can describe various geometric transformations in 3D Euclidean space, unifying reflections, rotations, translations, and perspective operations.

Contribution

It introduces a Clifford algebraic framework for 3D geometry that unifies multiple transformations and defines new operations like cotranslation using Hodge duality.

Findings

Operations of reflection and rotation preserve certain subspaces.

Perspective and pseudo-perspective projections can be expressed via cotranslation.

The full algebra l_{3,3} is necessary for most transformations.

Abstract

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of the quadratic space $\mathbb{R}^{3,3}$. We show that this algebra describes in a unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using the concept of Hodge duality, we define an operation called cotranslation, and show that the operation of perspective projection can be written in this Clifford algebra as a composition of the translation and cotranslation operations. We also show that the operation of pseudo-perspective can be implemented using the cotranslation operation. An important point is that the expression for the operations of reflection and rotation in $\mathcal{C}\ell_{3,3}$ preserve the subspaces that can be associated with the algebras $\mathcal{C}\ell_{3,0}$ and $\mathcal{C}\ell_{0,3}$, so that reflection and rotation can be expressed in terms of $\mathcal{C}\ell_{3,0}$ or $\mathcal{C}\ell_{0,3}$, as well-known. However, all other operations mix those subspaces in such a way that they need to be expressed in terms of the full Clifford algebra $\mathcal{C}\ell_{3,3}$. An essential aspect of our formulation is the representation of points in terms of objects called paravectors. Paravectors have been used previously to represents points in terms of an algebra closely related to the Clifford algebra $\mathcal{C}\ell_{3,3}$. We compare these different approaches.