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

Author(s)

Vaz, Jr., Jayme ^[1] ; Mann, Stephen ^[2]

Affiliation

^[1] Univ Estadual Campinas, Dept Appl Math, Campinas, SP - Brazil

^[2] Univ Waterloo, Cheriton Sch Comp Sci, Waterloo, ON - Canada

• Document type: Journal article
• Source: Advances in Applied Clifford Algebras; v. 30, n. 4 AUG 2 2020.
• Web of Science Citations: 0

Abstract

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

• FAPESP's process: 16/21370-9 - Applications of Clifford Algebras in Computer Graphics
• Grantee: Jayme Vaz Junior
• Support Opportunities: Scholarships abroad - Research