A Homogeneous Model for Three-Dimensional Computer Graphics Based on the Clifford Algebra for ℝ3

  • Ron Goldman


We construct a homogeneous model for Computer Graphics using the Clifford Algebra for ℝ3. To incorporate points as well as vectors within this model, we employ the odd-dimensional elements of this graded eight-dimensional algebra to represent mass-points by exploiting the pseudoscalars to represent mass. The even-dimensional elements of this Clifford Algebra are isomorphic to the quaternions, which operate on the odd-dimensional elements by sandwiching. Along with the standard sandwiching formulas for rotations and reflections, this paradigm allows us to use sandwiching to compute perspective projections.



I would like to thank Leo Dorst and Steve Mann for reading a preliminary draft of this manuscript and providing valuable comments, criticisms, and suggestions. I would also like to thank the anonymous referees for their constructive criticisms. This work is much improved as a result of the observations of these people. Any mistakes that still remain are, of course, entirely my own.


  1. 1.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge, UK (2003) zbMATHGoogle Scholar
  2. 2.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann, Amsterdam (2007) Google Scholar
  3. 3.
    Du Val, P.: Homographies, Quaternions and Rotations. Oxford Mathematical Monographs. Clarendon, Oxford (1964) zbMATHGoogle Scholar
  4. 4.
    Foley, J., van Dam, A., Feiner, S., Hughes, J.: Computer Graphic: Principles and Practice, 2nd edn. Addison Wesley, Reading (1990) Google Scholar
  5. 5.
    Goldman, R.: On the algebraic and geometric foundations of computer graphics. Trans. Graph. 21, 1–35 (2002) CrossRefGoogle Scholar
  6. 6.
    Goldman, R.N.: Understanding quaternions. Graph. Models 73, 21–49 (2011) CrossRefGoogle Scholar
  7. 7.
    Goldman, R.N.: Modeling perspective projections in 3-dimensions by rotations in 4-dimensions. Trans. Vis. Comput. Graph. (2010, to appear) Google Scholar
  8. 8.
    Mebius, J.E.: A matrix based proof of the quaternion representation theorem for four-dimensional rotations. http://arXiv:math/0501249v1 [math.GM] (2005)
  9. 9.
    Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Berlin (2009) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of Computer ScienceRice UniversityHoustonUSA

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