Advertisement

New Tools for Computational Geometry and Rejuvenation of Screw Theory

  • David HestenesEmail author
Chapter

Abstract

Conformal Geometric Algebraic (CGA) provides ideal mathematical tools for construction, analysis, and integration of classical Euclidean, Inversive & Projective Geometries, with practical applications to computer science, engineering, and physics. This paper is a comprehensive introduction to a CGA tool kit. Synthetic statements in classical geometry translate directly to coordinate-free algebraic forms. Invariant and covariant methods are coordinated by conformal splits, which are readily related to the literature using methods of matrix algebra, biquaternions, and screw theory. Designs for a complete system of powerful tools for the mechanics of linked rigid bodies are presented.

Keywords

Rigid Body Base Point Computational Geometry Euclidean Geometry Clifford 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Hestenes, D.: Old Wine in New Bottles: A new algebraic framework for computational geometry. In: Bayro-Corrochano, E., Sobczyk, G. (eds.) Advances in Geometric Algebra with Applications in Science and Engineering, pp. 1–14. Birkhäuser, Basel (2001) Google Scholar
  2. 2.
    Hestenes, D., Fasse, E.: Homogeneous rigid body mechanics with elastic coupling. In: Dorst, L., Doran, C., Lasenby, J. (eds.) Applications of Geometric Algebra in Computer Science and Engineering, pp. 197–212. Birkhäuser, Basel (2002) Google Scholar
  3. 3.
    Hestenes, D.: A unified language for mathematics and physics. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and their Applications in Mathematica Physics, pp. 1–23. Reidel, Dordrecht (1986) Google Scholar
  4. 4.
    Hestenes, D.: Space-Time Algebra. Gordon & Breach, New York (1966) zbMATHGoogle Scholar
  5. 5.
    Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus, a unified language for mathematics and physics. Kluwer, Dordrecht (1984). Paperback (1985). Fourth printing 1999 zbMATHGoogle Scholar
  6. 6.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2002) Google Scholar
  7. 7.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science. Morgan Kaufmann Publ., Elsevier, San Mateo, Amsterdam (2007/2009) Google Scholar
  8. 8.
    Li, H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2008) zbMATHCrossRefGoogle Scholar
  9. 9.
    Hestenes, D.: Grassmann’s vision. In: Schubring, G. (ed.) Hermann Günther Grassmann (1809–1877)—Visionary Scientist and Neohumanist Scholar, pp. 191–201. Kluwer, Dordrecht (1996) Google Scholar
  10. 10.
    Clifford, W.K.: Mathematical Papers. Macmillan, London (1882). Ed. by R. Tucker. Reprinted by Chelsea, New York (1968) Google Scholar
  11. 11.
    Doran, C., Hestenes, D., Sommen, F., Van Acker, N.: Lie groups as spin groups. J. Math. Phys. 34, 3642–3669 (1993) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lasenby, A.: Recent applications of conformal geometric algebra. In: Li, H., et al. (ed.) Computer Algebra and Geometric Algebra with Applications. LNCS, vol. 3519, pp. 298–328. Springer, Berlin (2005) Google Scholar
  13. 13.
    Hestenes, D.: The design of linear algebra and geometry. Acta Appl. Math. 23, 65–93 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Onishchik, A., Sulanke, R.: Projective and Cayley–Klein Geometries. Springer, Berlin (2006) zbMATHGoogle Scholar
  15. 15.
    Hestenes, D.: New Foundations for Classical Mechanics. Kluwer, Dordrecht (1986). Paperback (1987). Second edition (1999) zbMATHGoogle Scholar
  16. 16.
    Selig, J.: Geometrical Methods in Robotics. Springer, Berlin (1996) zbMATHGoogle Scholar
  17. 17.
    Ball, R.S.: A Treatise on the Theory of Screws. Cambridge University Press, Cambridge (1900). Reprinted in paperback (1998) Google Scholar
  18. 18.
    Davidson, J., Hunt, K.: Robots and Screw Theory: Applications of Kinematics and Statics to Robotics. Oxford University Press, London (2004) zbMATHGoogle Scholar
  19. 19.
    Featherstone, R., Orin, D.: Dynamics. In: Siciliano, B., Khatib, O. (eds.) Springer Handbook of Robotics, pp. 35–65. Springer, Berlin (2008) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  1. 1.Arizona State UniversityTempeUSA

Personalised recommendations