On the Clifford Algebraic Description of Transformations in a 3D Euclidean Space
- PDF / 554,911 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 105 Downloads / 151 Views
Advances in Applied Clifford Algebras
On the Clifford Algebraic Description of Transformations in a 3D Euclidean Space Jayme Vaz∗
Jr. and Stephen Mann Communicated by Leo Dorst
Abstract. We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra C3,3 of the quadratic space R3,3 . 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 C3,3 preserve the subspaces that can be associated with the algebras C3,0 and C0,3 , so that reflection and rotation can be expressed in terms of C3,0 or C0,3 , 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 C3,3 . 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 C3,3 . We compare these different approaches. Mathematics Subject Classification. 15A66, 15A75, 68U05. Keywords. Clifford algebra, Paravectors, Geometric transformations.
1. Introduction There is a deep connection between geometry and algebra; exploiting this connection usually benefits these studies and advances both disciplines. Computer graphics is one of many areas where geometry and algebra play a key ∗ Corresponding
author. 0123456789().: V,-vol
53
Page 2 of 27
J. Vaz Jr, S. Mann
Adv. Appl. Clifford Algebras
role. The concepts of affine spaces and projective spaces are fundamental for the theoretical basis of computer graphics, and linear algebra is a fundamental tool in computer graphics for geometric computations. Nevertheless, the success of this formalism should not be seen as a hindrance to the study or the use of other algebraic systems. In fact, quaternions [1,2] were shown to be an efficient tool for interpolating sequences of orientations in computer graphics, and other studies [3] involving applications of other algebraic systems in computer graphics have appeared in the literature, together with studies trying to identify geometric spaces that may be more suitable as an ambient space for computer graphics [4,5]. Models based on Clifford algebras [6–8] have also been proposed as an alternative algebraic framework for computer graphics. Recently we proposed a new model for the description of a geometric space based on the exterior algebra of a vector space [9]. In this model, points are
Data Loading...
