Paravectors and the Geometry of 3D Euclidean Space

TL;DR

This paper introduces paravectors and their algebraic operations to describe and transform points, lines, and planes in 3D Euclidean space, unifying various geometric transformations and projections.

Contribution

It develops a novel algebraic framework using paravectors, biparavectors, and triparavectors to model 3D geometry and transformations comprehensively.

Findings

Unified algebraic formalism for geometric transformations

Introduction of cotranslation for perspective projection

Representation of reflections, rotations, and scalings within the framework

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.