New Tools for Computational Geometry and Rejuvenation of Screw Theory

  • David HestenesEmail author


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.


Rigid Body Base Point Computational Geometry Euclidean Geometry Clifford Algebra 
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  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

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