Advances in Applied Clifford Algebras

, Volume 26, Issue 4, pp 1137–1172 | Cite as

3D Oriented Projective Geometry Through Versors of \({\mathbb{R}^{3,3}}\)

  • Leo Dorst
Open Access


It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), 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 \({\mathbb{R}^{3,3}}\). 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 \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.


Projective geometry Oriented projective geometry Geometric algebra Homogeneous coordinates Plücker coordinates Oriented lines Projective collineation Versor Rotor Bivector generator Oriented reflection 

Mathematics Subject Classification

15A33 51M35 


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Copyright information

© Springer International Publishing 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Computer Vision Group, Informatics InstituteUniversity of AmsterdamAmsterdamThe Netherlands

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