3D Oriented Projective Geometry Through Versors of R(3,3)

It is possible to set up a correspondence between 3D space and R(3,3), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of R(3,3). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented R(3,3) approach of Klawitter (Adv Appl Clifford Algebra, 24:713-736, 2014), and the R(4,4) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113-149, 2015) in terms of expressiveness and efficiency.