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

Abstract

We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra Cℓ 3,3 of the quadratic space 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 Cℓ 3,3 preserve the subspaces that can be associated with the algebras Cℓ 3,0 and Cℓ 0,3 , so that reflection and rotation can be expressed in terms of Cℓ 3,0 or Cℓ 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 Cℓ 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 Cℓ 3,3 . We compare these different approaches.