Advertisement

Some Applications of Clifford Algebra to Geometries

  • Hongbo Li
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1669)

Abstract

This paper focuses on a Clifford algebra model for geometric computations in 2D and 3D geometries. The model integrates symbolic representation of geometric entities, such as points, lines, planes, circles and spheres, with that of geometric constraints such as angles and distances, and is appropriate for both symbolic and numeric computations. Details on how to apply this model are provided and examples are given to illustrate the application.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. Bayro-Corrochano, J. Lasenby and G. Sommer, Geometric algebra: A framework for computing point and line correspondences and projective structure using nuncalibrated cameras, Proc. of International conference on Pattern Recognition ICPR’96, Vienna, Vol. 1, pp. 334–338, 1996. 157CrossRefGoogle Scholar
  2. 2.
    E. Bayro-Corrochano, K. Daniilidis and G. Sommer, Hand-eye calibration in terms of motions of lines using geometric algebra, Proc. of 10th Scandinavian Conference on Image Analysis, Lappeenranta, Vol. 1, pp. 397–404, 1997. 157Google Scholar
  3. 3.
    E. Bayro-Corrochano and J. Lasenby, A unified language for computer vision and robotics, Algebraic Frames for the Perception-Action Cycle, G. Sommer and J. J. Koenderink (eds.), LNCS 1315, pp. 219–234, 1997. 157CrossRefGoogle Scholar
  4. 4.
    E. Bayro-Corrochano and J. Lasenby, Geometric techniques for the computation of projective invariants using n uncalibrated cameras, Proc. of Indian Conference on Computer Vision, Graphics and Image Processing, New Delhi, pp. 95–100, 1998. 157Google Scholar
  5. 5.
    W. Blaschke, Anwendung dualer quaternionen auf die Kinematik, Annales Acad. Sci. Fennicae, 1–13, 1958. 157, 175Google Scholar
  6. 6.
    W. K. Clifford, Preliminary sketch of bi-quaternions, Proc. London Math. Soc. 4: 381–395, 1873. 157, 175CrossRefGoogle Scholar
  7. 7.
    A. Crumeyrolle, Orthogonal and Symplectic Clifford Algebras, D. Reidel, Dordrecht, Boston, 1990. 156zbMATHGoogle Scholar
  8. 8.
    R. Delanghe, F. Sommen and V. Soucek, Clifford Algebra and Spinor-Valued Functions, D. Reidel, Dordrecht, Boston, 1992. 156zbMATHGoogle Scholar
  9. 9.
    O. Faugeras, Three-dimensional Computer Vision, MIT Press, 1993. 157Google Scholar
  10. 10.
    O. Faugeras and B. Mourrain, On the geometry and algebra of the point and line correspondences between N images, Proc. of Europe-China Workshop on Geometrical Modeling and Invariants for Computer Vision, R. Mohr and C. Wu (eds.), pp. 102–109, 1995. 157Google Scholar
  11. 11.
    C. Doran, D. Hestenes, F. Sommen and N. V. Acker, Lie groups as spin groups, J. Math. Phys. 34(8): 3642–3669, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    T. Havel, Some examples of the use of distances as coordinates for Euclidean geometry, J. Symbolic Comput. 11: 579–593, 1991. 157, 159MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    T. Havel and A. Dress, Distance geometry and geometric algebra, Found. Phys. 23: 1357–1374, 1993. 157CrossRefMathSciNetGoogle Scholar
  14. 14.
    T. Havel, Geometric algebra and Möbius sphere geometry as a basis for Euclidean invariant theory, Invariant Methods in Discrete and Computational Geometry, N. L. White (ed.), pp. 245–256, D. Reidel, Dordrecht, Boston, 1995. 157Google Scholar
  15. 15.
    D. Hestenes, Space-Time Algebra, Gordon and Breach, New York, 1966.zbMATHGoogle Scholar
  16. 16.
    D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus, D. Reidel, Dordrecht, Boston, 1984. 156, 158, 162zbMATHGoogle Scholar
  17. 17.
    D. Hestenes, New Foundations for Classical Mechanics, D. Reidel, Dordrecht, Boston, 1987. 156, 158, 175Google Scholar
  18. 18.
    D. Hestenes and R. Ziegler, Projective geometry with Clifford algebra, Acta Appl. Math. 23: 25–63, 1991. 156, 157zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    D. Hestenes, The design of linear algebra and geometry, Acta Appl. Math. 23: 65–93, 1991. 156, 167zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    D. Hestenes, Invariant body kinematics I: Saccadic and compensatory eye movements, Neural Networks 7 (1): 65–77, 1994. 157CrossRefMathSciNetGoogle Scholar
  21. 21.
    D. Hestenes, Invariant body kinematics II: Reaching and neurogeometry, Neural Networks 7(1): 79–88, 1994. 157CrossRefGoogle Scholar
  22. 22.
    D. Hestenes, H. Li and A. Rockwood, New algebraic tools for classical geometry, Geometric Computing with Clifford Algebra, G. Sommer (ed.), Springer, 1999. 157, 158Google Scholar
  23. 24.
    B. Iversen, Hyperbolic Geometry, Cambridge, 1992. 157Google Scholar
  24. 25.
    J. Lasenby and E. Bayro-Corrochano, Computing 3D projective invariants from points and lines, Computer Analysis of Images and Patterns, G. Sommer, K. Daniilidis and J. Pauli (eds.), pp. 82–89, 1997. 157Google Scholar
  25. 26.
    H. Li, Hyperbolic geometry with Clifford algebra, Acta Appl. Math. 48: 317–358, 1997. 158zbMATHCrossRefMathSciNetGoogle Scholar
  26. 27.
    H. Li, D. Hestenes and A. Rockwood, Generalized homogeneous coordinates for computational geometry, Geometric Computing with Clifford Algebra, G. Sommer (ed.), Springer, 1999. 157, 163, 178Google Scholar
  27. 28.
    H. Li, D. Hestenes and A. Rockwood, A universal model for conformal geometries of Euclidean, spherical and double-hyperbolic spaces, Geometric Computing with Clifford Algebra, G. Sommer (ed.), Springer, 1999. 157Google Scholar
  28. 29.
    H. Li, D. Hestenes and A. Rockwood, Spherical conformal geometry with geometric algebra, Geometric Computing with Clifford Algebra, G. Sommer (ed.), Springer, 1999. 157Google Scholar
  29. 30.
    H. Li, Hyperbolic conformal geometry with Clifford algebra, submitted to Acta Appl. Math. in 1999. 157, 162Google Scholar
  30. 31.
    B. Mourrain and N. Stolfi, Computational symbolic geometry, Invariant Methods in Discrete and Computational Geometry, N. L. White (ed.), pp. 107–139, D. Reidel, Dordrecht, Boston, 1995. 157, 172, 173Google Scholar
  31. 32.
    B. Mourrain and N. Stolfi, Applications of Clifford algebras in robotics, Computational Kinematics, J.-P. Merlet and B. Ravani (eds.), pp. 41–50, D. Reidel, Dordrecht, Boston, 1995. 157, 173Google Scholar
  32. 33.
    B. Mourrain, Enumeration problems in geometry, robotics and vision, Algorithms in Algebraic Geometry and Applications, L. González and T. Recio (eds.), pp. 285–306, Birkhäuser, Basel, 1996. 173, 174Google Scholar
  33. 34.
    G. Peano, Geometric Calculus, 1888 (Translated by L. Kannenberg, 1997) 156Google Scholar
  34. 35.
    J. Seidel, Distance-geometric development of two-dimensional Euclidean, hyperbolic and spherical geometry I, II, Simon Stevin 29: 32–50, 65–76, 1952. 157, 159MathSciNetGoogle Scholar
  35. 36.
    G. Sommer, E. Bayro-Corrochano and T. Bülow, Geometric algebra as a framework for the perception-action cycle, Proc. of Workshop on Theoretical Foundation of Computer Vision, Dagstuhl, 1996. 157Google Scholar
  36. 37.
    E. Study, Geometrie der Dynamen, Leipzig, 1903. 157, 177Google Scholar
  37. 38.
    A. T. Yang and F. Freudenstein, Application of dual number quaternion algebra to the analysis of spatial mechanisms, J. Appl. Mech. 31: 300–308, 1964. 157, 177zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Hongbo Li
    • 1
  1. 1.Department of Physics and AstronomyArizona State University TempeUSA

Personalised recommendations