On the Homogeneous Model of Euclidean Geometry

  • Charles Gunn


We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for Euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2 and n=3 in detail, enumerating the geometric products between k-blades and m-blades. We identify sandwich operators in the algebra that provide all Euclidean isometries, both direct and indirect. We locate the spin group, a double cover of the direct Euclidean group, inside the even subalgebra of the Clifford algebra, and provide a simple algorithm for calculating the logarithm of group elements. We conclude with an elementary account of Euclidean kinematics and rigid body motion within this framework.


Ideal Point Euclidean Geometry Clifford Algebra Rigid Body Motion Inertia Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Ablamowicz, R.: Structure of spin groups associated with degenerate Clifford algebras. J. Math. Phys. 27, 1–6 (1986) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, V.I.: Mathematical Methods of Classical Physics. Springer, New York (1978), Appendix 2 Google Scholar
  3. 3.
    Ball, R.: A Treatise on the Theory of Screws. Cambridge University Press, Cambridge (1900) Google Scholar
  4. 4.
    Blaschke, W.: Ebene Kinematik. Teubner, Leipzig (1938) Google Scholar
  5. 5.
    Blaschke, W.: Nicht-euklidische Geometrie und Mechanik. Teubner, Leipzig (1942) MATHGoogle Scholar
  6. 6.
    Blaschke, W.: Analytische Geometrie. Birkhäuser, Basel (1954) MATHGoogle Scholar
  7. 7.
    Bourbaki, N.: Elements of Mathematics, Algebra I. Springer, Berlin (1989) MATHGoogle Scholar
  8. 8.
    Conradt, O.: Mathematical Physics in Space and Counterspace. Verlag am Goetheanum, Goetheanum (2008) MATHGoogle Scholar
  9. 9.
    Coxeter, H.M.S.: Projective Geometry. Springer, New York (1987) MATHGoogle Scholar
  10. 10.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann, San Francisco (2009) Google Scholar
  11. 11.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) MATHGoogle Scholar
  12. 12.
    Featherstone, R.: Rigid Body Dynamics Algorithms. Springer, Berlin (2007) Google Scholar
  13. 13.
    Gunn, C.: On the homogeneous model for Euclidean geometry: extended version. (2011)
  14. 14.
    Hestenes, D.: New tools for computational geometry and rejuvenation of screw theory. In: Bayro-Corrochano, E.J., Scheuermann, G. (eds.) Geometric Algebra Computing: In Engineering and Computer Science, pp. 3–35. Springer, Berlin (2010) CrossRefGoogle Scholar
  15. 15.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Fundamental Theories of Physics. Reidel, Dordrecht (1987) Google Scholar
  16. 16.
  17. 17.
    Jessop, C.M.: A Treatise on the Line Complex. Chelsea, New York (1969). Original 1903, Cambridge MATHGoogle Scholar
  18. 18.
    Klein, F.: Über Liniengeometrie und metrische Geometrie. Math. Ann. 2, 106–126 (1872) Google Scholar
  19. 19.
    Klein, F.: Vorlesungen Über Höhere Geometrie. Chelsea, New York (1927) Google Scholar
  20. 20.
    Klein, F.: Vorlesungen Über Nicht-euklidische Geometrie. Chelsea, New York (1949). Original 1926, Berlin Google Scholar
  21. 21.
    Li, H.: Invariant Algebras and Geometric Algebra. World Scientific, Singapore (2008) CrossRefGoogle Scholar
  22. 22.
    McCarthy, J.M.: An Introduction to Theoretical Kinematics. MIT Press, Cambridge (1990) Google Scholar
  23. 23.
    Perwass, C.: Geometric Algebra with Applications to Engineering. Springer, Berlin (2009) Google Scholar
  24. 24.
    Pottmann, H., Wallner, J.: Computational Line Geometry. Springer, Berlin (2001) MATHGoogle Scholar
  25. 25.
    Selig, J.: Clifford algebra of points, lines, and planes. Robotica 18, 545–556 (2000) CrossRefGoogle Scholar
  26. 26.
    Selig, J.: Geometric Fundamentals of Robotics. Springer, Berlin (2005) MATHGoogle Scholar
  27. 27.
    Study, E.: Von den bewegungen und umlegungen. Math. Ann. 39, 441–566 (1891) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Study, E.: Geometrie der Dynamen. Teubner, Leipzig (1903) MATHGoogle Scholar
  29. 29.
    von Mises, R.: Die Motorrechnung Eine Neue Hilfsmittel in der Mechanik. Z. Rein Angew. Math. Mech. 4(2), 155–181 (1924) CrossRefGoogle Scholar
  30. 30.
    Weiss, E.A.: Einführung in die Liniengeometrie und Kinematik. Teubner, Leipzig (1935) Google Scholar
  31. 31.
    Whitehead, A.N.: A Treatise on Universal Algebra. Cambridge University Press, Cambridge (1898) Google Scholar
  32. 32.
  33. 33.
    Ziegler, R.: Die Geschichte Der Geometrischen Mechanik im 19. Jahrhundert. Franz Steiner Verlag, Stuttgart (1985) MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.DFG-Forschungszentrum Matheon, MA 8-3Technisches Universität BerlinBerlinGermany

Personalised recommendations