R(4, 4) As a Computational Framework for 3-Dimensional Computer Graphics
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We investigate the efficacy of the Clifford algebra R(4, 4) as a computational framework for contemporary 3-dimensional computer graphics. We give explicit rotors in R(4, 4) for all the standard affine and projective transformations in the graphics pipeline, including translation, rotation, reflection, uniform and nonuniform scaling, classical and scissor shear, orthogonal and perspective projection, and pseudoperspective. We also explain how to represent planes by vectors and quadric surfaces by bivectors in R(4, 4), and we show how to apply rotors in R(4, 4) to these vectors and bivectors to transform planes and quadric surfaces by affine transformations.
KeywordsClifford Algebra Homogeneous Model Perspective Projection Rotor Quadric surfaces computer graphics
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- 1.L. Dorst, D. Fontijne, and S. Mann, Geometric Algebra for Computer Science. Morgan-Kaufmann, 2007.Google Scholar
- 3.D. Fontijne, Efficient Implmentation of Geometric Algebra. PhD thesis, Computer Science, University of Amsterdam, 2007.Google Scholar
- 4.Ron Goldman, Stephen Mann and Xiaohong Jia, Computing perspective projections in 3-dimensions using rotors in the homogeneous and conformal models of Clifford algebra. Advances in Applied Clifford Algebras (2014). DOI: 10.1007/s00006-014-0439-3.
- 5.Ron Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling. CRC Press, 2009.Google Scholar
- 7.D. Hildenbrand, D. Fontijne, C. Perwass and L. Dorst, Geometric algebra and its application to computer graphics. In Eurographics conference Grenoble, 2004.Google Scholar
- 8.Anthony Lasenby, Recent applications of conformal geometric algebra. In H. Li, P.J. Oliver and G. Sommer, editors, Computer Algebra and Geometric Algebra with Applications, volume 3519 of LNCS, Springer-Verlag 2005 pages 298–328.Google Scholar
- 9.Spencer T. Parkin, A model for quadric surfaces using geometric algebra. Unpublished, October 2012.Google Scholar
- 10.Alyn Rockwood and Dietmar Hildenbrand, Engineering graphics in geomeric algebra. In E. Bayro-Corrochano and G. Scheuermann, editors, Geometric Algebra Computing, Springer 2010, pages 53–69.Google Scholar
- 11.David Salomon, Computer Graphics and Geometric Modeling. Springer, 1999.Google Scholar
- 12.John A. Vince, Geometric Algebra for Computer Graphics. Springer-Verlag TELOS, Santa Clara, CA, USA, 1st edition, 2008.Google Scholar
- 13.Rich Wareham, Jonathan Cameron and Joan Lasenby, Applications of conformal geometric algebra in computer vision and graphics. In H. Li, P.J. Oliver, and G. Sommer, editors, Computer Algebra and Geometric Algebra with Applications. volume 3519 of LNCS, Springer-Verlag 2005, pages 329–349.Google Scholar