Clifford algebraic calculus for geometric reasoning

with application to computer vision
  • Dongming Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1360)


In this paper we report on our recent study of Clifford algebra for geometric reasoning and its application to problems in computer vision. A general framework is presented for construction and representation of geometric objects with selected rewrite rules for simplification. It provides a mechanism suitable for devising methods and software tools for geometric reasoning and computation. The feasibility and efficiency of the approach are demonstrated by our preliminary experiments on automated theorem proving in plane Euclidean geometry. We also explain how non-commutative Gröbner bases can be applied to geometric theorem proving. In addition to several well-known geometric theorems, two application examples from computer vision are given to illustrate the practical value of our approach.


Theorem Prove Clifford Algebra Outer Product Geometric Reasoning Grassmann Algebra 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Automated production of traditional proofs for constructive geometry theorems. In: Proc. 8th IEEE Symp. LICS (Montreal, June 19–23, 1993, pp. 48–56.Google Scholar
  2. 2.
    Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Automated geometry theorem proving by vector calculation. In: Proc. ISSAC '93 (Kiev, July 6–8, 1993, pp. 284–291.Google Scholar
  3. 3.
    Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine proofs in geometry. World Scientific, Singapore (1994).Google Scholar
  4. 4.
    Deguchi, K.: An algebraic framework for fusing geometric constraints of vision and range sensor data. In: Proc. IEEE Int. Conf. MFI '94 (Las Vegas, October 2–5, 1994), pp. 329–336.Google Scholar
  5. 5.
    Havel, T. F.: Some examples of the use of distances as coordinates for Euclidean geometry. J. Symb. Comput. 11: 579–593 (1991).Google Scholar
  6. 6.
    Hestenes, D.: New foundations for classical mechanics. D. Reidel, Dordrecht Boston Lancaster Tokyo (1987).Google Scholar
  7. 7.
    Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus. D. Reidel, Dordrecht Boston Lancaster Tokyo (1984).Google Scholar
  8. 8.
    Kandri-Rody, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comput. 9: 1–26 (1990).Google Scholar
  9. 9.
    Kapur, D., Mundy, J. L.: Wu's method and its application to perspective viewing. Artif. Intell. 37: 15–26 (1988).CrossRefGoogle Scholar
  10. 10.
    Li, H.: New explorations on mechanical theorem proving of geometries. Ph.D thesis, Beijing University, China (1994).Google Scholar
  11. 11.
    Li, H.: Clifford algebra and area method. Math. Mech. Res. Preprints 14: 37–69 (1996).Google Scholar
  12. 12.
    Li, H., Cheng, MA.: Proving theorems in elementary geometry with Clifford algebraic method. Preprint, MMRC, Academia Sinica, China (1995).Google Scholar
  13. 13.
    Richter-Gebert, J.: Mechanical theorem proving in projective geometry. Ann. Math. Artif. Intell. 13: 139–172 (1995).CrossRefGoogle Scholar
  14. 14.
    Stifter, S.: Geometry theorem proving in vector spaces by means of Gröbner bases. In: Proc. ISSAC '93 (Kiev, July 6–8, 1993, pp. 301–310.Google Scholar
  15. 15.
    van der Waerden, B. L.: Algebra, vol. II. Springer, Berlin Heidelberg (1959).Google Scholar
  16. 16.
    Wang, D.: On Wu's method for proving constructive geometric theorems. In: Proc. IJCAI '89 (Detroit, August 20–25, 1989), pp. 419–424.Google Scholar
  17. 17.
    Wang, D.: Geometry machines: From Al to SMC. In: Proc. AISMC-3 (Steyr, September 23–25, 1996), LNCS 1138, pp. 213–239.Google Scholar
  18. 18.
    Wong, R.: Construction heuristics for geometry and a vector algebra representation of geometry. Tech. Memo. 28, Project MAC, MIT, Cambridge, USA (1972).Google Scholar
  19. 19.
    Wu, W.-t.: Toward mechanization of geometry-Some comments on Hilbert's “Grundlagen der Geometrie”. Acta Math. Scientia 2: 125–138 (1982).Google Scholar
  20. 20.
    Wu, W.-t.: Mechanical theorem proving in geometries: Basic principles. Springer, Wien New York (1994).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Dongming Wang
    • 1
  1. 1.LEIBNIZ-IMAGGrenoble CedexFrance

Personalised recommendations