Rigid-Body Transforms Using Symbolic Infinitesimals

  • Glen Mullineux
  • Leon Simpson


There is a requirement to be able to represent three-dimensional objects and their transforms in many applications, including computer graphics and mechanism and machine design. A geometric algebra is constructed which can model three-dimensional geometry and rigid-body transforms. The representation is exact since the square of one of the basis vectors is treated symbolically as being infinite. The non-zero, even-grade elements of the algebra represent precisely all rigid-body transforms. By allowing the transform to vary, smooth motions are obtained. This can be achieved using Bézier and B-spline combinations of even-grade elements.


Control Point Basis Vector Geometric Algebra Curve Segment Smooth Motion 
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.



The work reported in the paper was carried within the Innovative Design and Manufacturing Research Centre at the University of Bath, and the second author is funded by a studentship provided via the Centre. The Centre is funded by the Engineering and Physical Sciences Research Council (EPSRC), and this support is gratefully acknowledged.


  1. 1.
    Belta, C., Kumar, V.: An SVD-based projection method for interpolation on SE(3). IEEE Trans. Robot. Autom. 18, 334–345 (2002) CrossRefGoogle Scholar
  2. 2.
    Etzel, K.R., McCarthy, J.M.: Interpolation of spatial displacements using the Clifford algebra of E 4. J. Mech. Des. 121, 39–44 (1999) CrossRefGoogle Scholar
  3. 3.
    Farin, G.: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Morgan Kaufmann, San Francisco (2001) Google Scholar
  4. 4.
    Gan, D., Liao, Q., Wei, S., Dai, J.S., Qiao, S.: Dual quaternion-based inverse kinematics of the general spatial 7R mechanism. Proc. Inst. Mech. Eng., Part C, J. Mech. Eng. Sci. 222, 1593–1598 (2008) CrossRefGoogle Scholar
  5. 5.
    Ge, Q.J.: On the matrix realization of the theory of biquaternions. J. Mech. Des. 120, 404–407 (1998) CrossRefGoogle Scholar
  6. 6.
    Ge, Q.J., Ravani, R.: Geometric construction of Bézier motions. J. Mech. Des. 116, 749–755 (1994) CrossRefGoogle Scholar
  7. 7.
    González Calvet, R.: Treatise of Plane Geometry Through Geometric Algebra. Cerdanyola del Vallès (2007) Google Scholar
  8. 8.
    Hofer, M., Pottmann, H., Ravani, B.: From curve design algorithms to the design of rigid body motions. Vis. Comput. 20, 279–297 (2004) CrossRefGoogle Scholar
  9. 9.
    Jin, Z., Ge, Q.J.: Computer aided synthesis of piecewise rational motions for planar 2R and 3R robot arms. J. Mech. Des. 129, 1031–1036 (2007) CrossRefGoogle Scholar
  10. 10.
    Jüttler, B., Wagner, M.G.: Computer-aided design with spatial rational B-spline motions. J. Mech. Des. 118, 193–201 (1996) Google Scholar
  11. 11.
    Leeney, M.: Fast quaternion slerp. Int. J. Comput. Math. 86, 79–84 (2009) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Mullineux, G.: Clifford algebra of three dimensional geometry. Robotica 20, 687–697 (2002) CrossRefGoogle Scholar
  13. 13.
    Mullineux, G.: Modelling spatial displacements using Clifford algebra. J. Mech. Des. 126, 420–424 (2004) CrossRefGoogle Scholar
  14. 14.
    Özgören, M.K.: Kinematics analysis of spatial mechanical systems using exponential rotation matrices. J. Mech. Des. 129, 1144–1152 (2007) CrossRefGoogle Scholar
  15. 15.
    Perez-Garcia, A., McCarthy, J.M.: Kinematic synthesis of spatial serial chains using Clifford algebra exponentials. Proc. Inst. Mech. Eng., Part C, J. Mech. Eng. Sci. 220, 953–968 (2006) CrossRefGoogle Scholar
  16. 16.
    Purwar, A., Jin, Z., Ge, Q.J.: Rational motion interpolation under kinematic constraints of spherical 6R closed chains. J. Mech. Des. 130 062301 (2008) CrossRefGoogle Scholar
  17. 17.
    Röschel, O.: Rational motion design—a survey. Comput. Aided Des. 30, 169–178 (1998) MATHCrossRefGoogle Scholar
  18. 18.
    Selig, J.M.: Clifford algebra of points, lines and planes. Robotica 20, 545–556 (2000) CrossRefGoogle Scholar
  19. 19.
    Simpson, L., Mullineux, G.: Exponentials and motions in geometric algebra. In: Vaclav, S., Hildenbrand, D. (eds.) International Workshop on Computer Graphics, Computer Vision and Mathematics (GraVisMa), pp. 9–16. Union Agency, Plzen (2009) Google Scholar
  20. 20.
    Srinivasen, L.N., Ge, Q.J.: Fine tuning of rational B-spline motions. J. Mech. Des. 120, 46–51 (1998) CrossRefGoogle Scholar
  21. 21.
    Vince, J.: Geometric Algebra for Computer Graphics. Springer, London (2008) MATHCrossRefGoogle Scholar
  22. 22.
    Wareham, R., Lasenby, J.: Mesh vertex pose and position interpolation using geometric algebra. In: Perales, F.J., Fisher, R.B. (eds.) Articulated Motion and Deformable Objects, 5th International Conference, AMDO 2008, pp. 122–131. Springer, Berlin (2008) CrossRefGoogle Scholar
  23. 23.
    Wu, W., You, Z.: Modelling rigid origami with quaternions and dual quaternions. Proc. R. Soc. A 466, 2155–2174 (2010) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Innovative Design and Manufacturing Research Centre, Department of Mechanical EngineeringUniversity of BathBathUK

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