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The Quaternions and the Spaces S3, SU(2), SO(3), and ℝ ℙ3

  • Jean Gallier
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 38)

Abstract

In this chapter, we discuss the representation of rotations of ℝ3 in terms of quaternions. Such a representation is not only concise and elegant, it also yields a very efficient way of handling composition of rotations. It also tends to be numerically more stable than the representation in terms of orthogonal matrices.

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References

  1. 1.
    Michael Artin. Algebra. Prentice-Hall, first edition, 1991.Google Scholar
  2. 2.
    A.H. Barr, B. Currin, S. Gabriel, and J.F. Hughes. Smooth Interpolation of Orientations with Angular Velocity Constraints using Quaternions. In Computer Graphics Proceedings, Annual Conference Series, pages 313–320. ACM, 1992.Google Scholar
  3. 3.
    Marcel Berger. G´eom´etrie 1. Nathan, 1990. English edition: Geometry 1, Universitext,Springer-Verlag.Google Scholar
  4. 4.
    J.E. Bertin. Alg`ebre Lin´eaire et G´eom´etrie Classique. Masson, first edition, 1981.Google Scholar
  5. 5.
    Jean Dieudonn’e. Alg`ebre Lin´eaire et G´eom´etrie El´ementaire. Hermann, second edition, 1965.Google Scholar
  6. 6.
    R.L. Bryant. An introduction to Lie groups and symplectic geometry. In D.S. Freed and K.K. Uhlenbeck, editors, Geometry and Quantum Field Theory, pages 5–181. AMS, Providence,RI, 1995.Google Scholar
  7. 7.
    Thomas Horsch and Bert J‥uttler. Cartesian spline interpolation for industrial robots. Computer-Aided Design, 30(3):217–224, 1998.Google Scholar
  8. 8.
    Bert J‥uttler. Visualization of moving objects using dual quaternion curves. Computers & Graphics, 18(3):315–326, 1994.Google Scholar
  9. 9.
    Bert J‥uttler. An osculating motion with second order contact for spacial Euclidean motions. Mech. Mach. Theory, 32(7):843–853, 1997.Google Scholar
  10. 10.
    Bert J‥uttler and M.G.Wagner. Computer-aided design with spacial rational B-spline motions. Journal of Mechanical Design, 118:193–201, 1996.Google Scholar
  11. 11.
    Bert J‥uttler and M.G. Wagner. Rational motion-based surface generation. Computer-Aided Design, 31:203–213, 1999.Google Scholar
  12. 12.
    M.J. Kim, M.S. Kim, and S.Y. Shin. A general construction scheme for unit quaternion curves with simple high-order derivatives. In Computer Graphics Proceedings, Annual Conference Series, pages 369–376. ACM, 1995.Google Scholar
  13. 13.
    M.J. Kim, M.S. Kim, and S.Y. Shin. A compact differential formula for the first derivative of a unit quaternion curve. Journal of Visualization and Computer Animation, 7:43–57, 1996.CrossRefGoogle Scholar
  14. 14.
    Jack Kuipers. Quaternion and Rotation Sequences. Princeton University Press, first edition,1999.Google Scholar
  15. 15.
    Jerrold E. Marsden and T.S. Ratiu. Introduction to Mechanics and Symmetry. TAM, Vol. 17. Springer-Verlag, first edition, 1994.Google Scholar
  16. 16.
    F.C. Park and B. Ravani. B’ezier curves on Riemannian manifolds and Lie groups with kinematic applications. ASME J. Mech. Des., 117:36–40, 1995.CrossRefGoogle Scholar
  17. 17.
    F.C. Park and B. Ravani. Smooth invariant interpolation of rotations. ACM Transactions on Graphics, 16:277–295, 1997.CrossRefGoogle Scholar
  18. 18.
    Otto R‥oschel. Rational motion design: A survey. Computer-Aided Design, 30(3):169–178,1998.Google Scholar
  19. 19.
    Ken Shoemake. Animating rotation with quaternion curves. In ACM SIGGRAPH’85, volume 19, pages 245–254. ACM, 1985.Google Scholar
  20. 20.
    Ken Shoemake. Quaternion calculus for animation. In Math for SIGGRAPH, pages 1–19. ACM, 1991. Course Note No. 2.Google Scholar
  21. 21.
    Claude Tisseron. G´eom´etries Affines, Projectives, et Euclidiennes. Hermann, first edition,1994.Google Scholar
  22. 22.
    O. Veblen and J. W. Young. Projective Geometry, Vol. 2. Ginn, first edition, 1946.Google Scholar

Copyright information

© Springer Science+Businees Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA

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