Abstract
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.
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Doran C., Hestenes D., Sommen F., Van Acker N.: Lie groups as spin groups. J. Math. Phys. 34(8), 3642–3669 (1993)
Dorst, L., Fontijne, D., Mann. S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufman, Burlington (2009)
Fontijne, D.: Efficient Implementation of Geometric Algebra. PhD thesis, University of Amsterdam, Amsterdam (2007)
Goldman R., Mann S.: \({\mathbb{R}^{4,4}}\) as a computational framework for 3-dimensional computer graphics. Adv. Appl. Clifford Algebra 25(1), 113–149 (2015)
Hartley, R.I., Zisserman, A.: Multiple View Geometry in Computer Vision, 2nd edn. Cambridge University Press, Cambridge (2003) (ISBN 0521540518)
Klawitter D.: A Clifford algebraic approach to line geometry. Adv. Appl. Clifford Algebra 24, 713–736 (2014)
Li, H., Zhang, L.: Line geometry in terms of the null geometric algebra over \({\mathbb{R}^{3,3}}\), and application to the inverse singularity analysis of generalized Stewart platforms. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric in Practice, pp. 253–272. Springer, New York (2011)
Li, H., Huang, L., Shao, C., Dong, L.: Three-Dimensional Projective Geometry with Geometric Algebra (2015). arXiv:1507.06634v1
Porteous I.R.: Clifford Algebras and the Classical Groups. Cambridge university Press, Cambridge (1995)
Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, New York (2001)
Sabadini, I., Sommen, F.: Clifford analysis on the space of vectors, bivectors and \(\ell\)-vectors. In: Qian, T., Hempfling, T., McIntosh, A., Sommen, F. (eds.) Advances in Analysis and Geometry. Trends in Mathematics, pp. 161–185. Birkhäuser, Basel (2004)
Stolfi, J.: Oriented Projective Geometry. Academic Press, London (1991)
Artzy, R.: Linear Geometry, 3rd edn., p. 94. Dover Publications (2008). See also Wikipedia entry on ‘Squeeze Mapping’ (2015)
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Dorst, L. 3D Oriented Projective Geometry Through Versors of \({\mathbb{R}^{3,3}}\) . Adv. Appl. Clifford Algebras 26, 1137–1172 (2016). https://doi.org/10.1007/s00006-015-0625-y
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DOI: https://doi.org/10.1007/s00006-015-0625-y