Paravectors and the Geometry of 3D Euclidean Space

Abstract

We introduce the concept of paravectors to describe the geometry of points in a three dimensional space. After defining a suitable product of paravectors, we introduce the concepts of biparavectors and triparavectors to describe line segments and plane fragments in this space. A key point in this product of paravectors is the notion of the orientation of a point, in such a way that biparavectors representing line segments are the result of the product of points with opposite orientations. Incidence relations can also be formulated in terms of the product of paravectors. To study the transformations of points, lines, and planes, we introduce an algebra of transformations that is analogous to the algebra of creation and annihilation operators in quantum theory. The paravectors, biparavectors and triparavectors are mapped into this algebra and their transformations are studied; we show that this formalism describes in an unified way the operations of reflection, rotations (circular and hyperbolic), translation, shear and non-uniform scale transformation. Using the concept of Hodge duality, we define a new operation called cotranslation, and show that the operation of perspective projection can be written 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.Mathematics Subject Classification: 15A66, 15A75, 68U05.It can be proven [10] that this definition is equivalent tofor A k ∈ k (R 3 ). It follows then that ⋆ 1 = e 1 ∧ e 2 ∧ e 3 , ⋆ e 1 = e 2 ∧ e 3 , ⋆ e 2 = e 3 ∧ e 1 , ⋆ e 3 = e 1 ∧ e 2 , ⋆ e 1 ∧ e 2 = e 3 , ⋆ e 3 ∧ e 1 = e 2 , ⋆ e 2 ∧ e 3 = e 1 , ⋆ e 1 ∧ e 2 ∧ e 3 = 1. ParavectorsLet us fix the notation −1 (R 3 ) = ∅ and 4 (R 3 ) = ∅. We define k (V ) as k (R 3 ) = k−1 (R 3 ) ⊕ k (R 3 ), k = 0, 1, 2, 3, 4.

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

“…The 3-vector e 123 = e 1 e 2 e 3 has a special role since it belongs to the center of Cℓ 3,0 or Cℓ 0,3 . The sum of a scalar and a vector is a paravector, representing a weighted point [9]. Then the paravector representing an affine point with positive orientation is an element of the form P = 1 + p, where p = p i e i .…”

“…Recently we have proposed a new model for the description of a geometrical space based on the exterior algebra of a vector space [9]. In this model, points are described by objects called paravectors, line segments are described by biparavectors, plane fragments are described by triparavectors, and so on for higher dimensional spaces.…”

“…A k-paravector is a sum of a (k − 1)-vector and a k-vector, which are elements of the exterior algebra. In [9] we exploited the algebra of k-paravectors to describe some properties of a geometrical space relevant for computer graphics, in particular we studied geometric transformations of points, lines and planes. These transformations were studied in terms of an algebra of transformations that is analogous to the algebra of creation and annihilation operators in quantum theory.…”

“…There is a deep relationship between the algebra of creation and annihilation operators of fermions in quantum mechanics and the Clifford algebra of a quadratic space [10]. Since our exterior algebra model [9] is based on an algebra analogous to the algebra of creation and annihilation operators, this relationship leads us to consider the formulation of our model in terms of a Clifford algebra. In addition to the interest in the problem itself, in particular for the comparison of the usefulness of the different algebraic frameworks, this Clifford algebra formulation would also allow us to be able to compare in a more direct manner our model with other models in the literature based on Clifford algebras.…”

“…The key point here is how we identify the vector space V 3 inside the algebra Cℓ 3,3 , and this identification underpins our model. Then in section 4 we briefly discuss the ideas of [9] concerning the use of paravectors to represent points, and re-express these ideas in terms of Cℓ 3,3 . In section 5 we exploit this representation to study some transformations of points.…”