Advertisement

Differential Geometry Revisited by Biquaternion Clifford Algebra

  • Patrick R. GirardEmail author
  • Patrick Clarysse
  • Romaric Pujol
  • Liang Wang
  • Philippe Delachartre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)

Abstract

In the last century, differential geometry has been expressed within various calculi: vectors, tensors, spinors, exterior differential forms and recently Clifford algebras. Clifford algebras yield an excellent representation of the rotation group and of the Lorentz group which are the cornerstones of the theory of moving frames. Though Clifford algebras are all related to quaternions via the Clifford theorem, a biquaternion formulation of differential geometry does not seem to have been formulated so far. The paper develops, in 3D Euclidean space, a biquaternion calculus, having an associative exterior product, and applies it to differential geometry. The formalism being new, the approach is intended to be pedagogical. Since the methods of Clifford algebras are similar in other dimensions, it is hoped that the paper might open new perspectives for a 4D hyperbolic differential geometry. All the calculi presented here can easily be implemented algebraically on Mathematica and numerically on Matlab. Examples, matrix representations, and a Mathematica work-sheet are provided.

Keywords

Clifford algebras Quaternions Biquaternions Differential geometry Rotation group SO(3) Hyperquaternion algebra 

Notes

Acknowledgments

This work was conducted in the framework of LabEx Celya and PRIMES (Physics Radiobiology Medical Imaging and Simulation).

References

  1. 1.
    Gürlebeck, K., Sprössig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York (1997)zbMATHGoogle Scholar
  2. 2.
    Vince, J.: Geometric Algebra for Computer Graphics. Springer, London (2008)CrossRefzbMATHGoogle Scholar
  3. 3.
    Snygg, J.: A New Approach to Differential Geometry Using Clifford’s Geometric Algebra. Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Casanova, G.: L’algèbre vectorielle. PUF, Paris (1976)Google Scholar
  5. 5.
    Girard, P.R., Pujol, R., Clarysse, P., Marion, A., Goutte, R., Delachartre, P.: Analytic video (2D+t) signals by Clifford Fourier Transforms in multiquaternion Grassmann-Hamilton-Clifford algebras. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets, pp. 197–219. Birkhäuser, Basel (2013)CrossRefGoogle Scholar
  6. 6.
    Girard, P.R.: Quaternion Grassmann-Hamilton-Clifford-algebras: new mathematical tools for classical and relativistic modeling. In: Dössel, O., Schlegel, W.C. (eds.) World Congress 2009, IFMBE Proceedings, vol. 25/IV, pp. 65–68 (2009)Google Scholar
  7. 7.
    Girard, P.R.: Quaternions, Clifford Algebras and Relativistic Physics. Birkhäuser, Basel (2007)Google Scholar
  8. 8.
    Girard, P.R.: Quaternions, Algèbre de Clifford et Physique Relativiste. PPUR, Lausanne (2004)zbMATHGoogle Scholar
  9. 9.
    Girard, P.R.: Quaternions, Clifford algebra and symmetry groups. In: Dorst, L., Doran, C., Lasenby, J. (eds.) Applications of Geometric Algebra in Computer Science and Engineering, pp. 307–315. Birkhäuser, Boston (2002)CrossRefGoogle Scholar
  10. 10.
    Girard, P.R.: Einstein’s equations and Clifford algebra. Adv. Appl. Clifford Algebras 9(2), 225–230 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Girard, P.R.: The quaternion group and modern physics. Eur. J. Phys. 5, 25–32 (1984)CrossRefGoogle Scholar
  12. 12.
    Gsponer, A., Hurni, J.P.: Quaternions in mathematical physics (1): Alphabetical bibliography, math-ph, 0510059V4, (1430 references) (2008). http://www.arxiv.org/abs/arXiv:mathph/0510059
  13. 13.
    Gsponer, A., Hurni, J.P.: Quaternions in mathematical physics (2): Analytical bibliography, math-ph, 0511092V3, (1100 references) (2008). http://www.arxiv.org/abs/arXiv:mathphy/0511092
  14. 14.
    Hamilton, W.R.: The Mathematical Papers. Conway, A.W. (ed.) for the Royal Irish Academy. Cambridge University Press, Cambridge (1931–1967)Google Scholar
  15. 15.
    Hamilton, W.R.: Elements of Quaternions, 2 vols. (1899–1901). Reprinted 1969, Chelsea, New York (1969)Google Scholar
  16. 16.
    Hankins, T.L.: Sir William Rowan Hamilton. Johns Hopkins University Press, Baltimore (1980)zbMATHGoogle Scholar
  17. 17.
    Crowe, M.J.: A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. University of Notre Dame, Notre Dame, London (1967)zbMATHGoogle Scholar
  18. 18.
    Grassmann, H.: Die lineale Ausdehungslehre: ein neuer Zweig der Mathematik, dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch die Statik, Mechanik, die Lehre von Magnetismus und der Krystallonomie erläutert. Wigand, Leipzig (1844). Second ed. (1878)Google Scholar
  19. 19.
    Grassmann, H.: Mathematische und physikalische Werke, 3 vols. in 6 pts. Engel, F. (ed.) Leipzig (1894–1911)Google Scholar
  20. 20.
    Grassmann, H.: Der Ort der Hamilton’schen Quaternionen in der Ausdehnungslehre. Math. Ann. 12, 375–386 (1877)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Petsche, H.-J.: Grassmann. Birkhäuser, Basel (2006)zbMATHGoogle Scholar
  22. 22.
    Clifford, W.K.: Applications of Grassmann’s extensive algebra. J. Math. 1, 350–358 (1878)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Clifford, W.K.: Mathematical Papers. Tucker, R. (ed.) pp. 266–276 (1882). Reprinted Chelsea, New-York, (1968)Google Scholar
  24. 24.
    Lagally, M.: Vorlesungen über Vektorrechnung, 5th edn. Akademische Verlagsgesellschaft, Leipzig (1956)zbMATHGoogle Scholar
  25. 25.
    Fehr, H.: Application de la Méthode Vectorielle de Grassmann à la Géométrie Infinitésimale. Carré et Naud, Paris (1899)zbMATHGoogle Scholar
  26. 26.
    Guggenheimer, H.W.: Differential Geometry. Dover, New York (1976)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Patrick R. Girard
    • 1
    Email author
  • Patrick Clarysse
    • 1
  • Romaric Pujol
    • 2
  • Liang Wang
    • 1
  • Philippe Delachartre
    • 1
  1. 1.Université de Lyon, CREATIS, CNRS UMR 5220, Inserm U1044, INSA-Lyon, Université LYON 1VilleurbanneFrance
  2. 2.Université de Lyon, Pôle de Mathématiques, INSA-LyonVilleurbanneFrance

Personalised recommendations