On the Clifford Algebraic Description of Transformations in a 3D Euclidean Space

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, rotation (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using Hodge duality, we define an operation called cotranslation, and show that perspective projection can be written in this Clifford algebra as a composition of translation and cotranslation. We also show that pseudo-perspective can be implemented using the cotranslation operation. In addition, we discuss how a general transformation of points can be described using this formalism. An important point is that the expressions for 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 is well-known. However, all the other operations mix these subspaces in such a way that these transformations 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.