Oriented Projective Geometry with Clifford Algebra

Abstract

Classical projective geometry can be efficiently formulated using the Clifford-algebra based “geometric algebra” developed by Hestenes and Sobczyk. In this formulation, points, lines, planes, etc. in a projective space P ⁿ⁻¹ are represented by equivalence classes of vectors, bivectors, trivectors, etc. in the Clifford algebra Cℓ_n. Two k-blades A and B are in the same equivalence class iff AB = A · B ; that is, iff A and B differ by a scalar factor. We show that if the notion of equivalence is restricted, so that two blades of the same grade are equivalent iff they differ by a positive scalar factor, then the geometric objects represented inherit an orientation and provide the building blocks for oriented projective spaces. Projective concepts such as meet, join, duality, and collineation can be extended to such spaces, while new features, such as relative orientation, give these spaces a much richer structure with many important applications. Examples of computations using CLICAL are provided.