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We discuss how transformations in a three dimensional euclidean space can bedescribed in terms of the Clifford algebra $\mathcal{C}\ell_{3,3}$ of thequadratic space $\mathbb{R}^{3,3}$. We show that this algebra describes in aunified way the operations of reflection, rotations (circular and hyperbolic),translation, shear and non-uniform scale. Moreover, using the concept of Hodgeduality, we define an operation called cotranslation, and show that theoperation of perspective projection can be written in this Clifford algebra asa composition of the translation and cotranslation operations. We also showthat the operation of pseudo-perspective can be implemented using thecotranslation operation. An important point is that the expression for theoperations of reflection and rotation in $\mathcal{C}\ell_{3,3}$ preserve thesubspaces 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 interms 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 needto 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 termsof objects called paravectors. Paravectors have been used previously torepresents points in terms of an algebra closely related to the Cliffordalgebra $\mathcal{C}\ell_{3,3}$. We compare these different approaches.

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