Square Root and Logarithm of Rotors in 3D Conformal Geometric Algebra Using Polar Decomposition

  • Leo Dorst
  • Robert Valkenburg


Conformal transformations are described by rotors in the conformal model of geometric algebra (CGA). In applications there is a need for interpolation of such transformations, especially for the subclass of 3D rigid body motions. This chapter gives explicit formulas for the square root and the logarithm of rotors in 3D CGA. It also classifies the types of conformal transformations and their orbits. To derive the results, we employ a novel polar decomposition for the even subalgebra of 3D CGA and an associated norm-like expression.


Conformal Transformation Rigid Body Motion Geometric Algebra Invertible Element Polar Decomposition 
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.


  1. 1.
    Dorst, L.: Conformal geometric algebra by extended Vahlen matrices. In: Skala, V., Hildenbrandt, D. (eds.) GraVisMa 2009 Workshop Proceedings, pp. 72–79 (2009) Google Scholar
  2. 2.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object Oriented Approach to Geometry, Morgan Kaufmann, San Mateo (2007/2009). See Google Scholar
  3. 3.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984/1999) Google Scholar
  4. 4.
    Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. Cambridge University Press, Cambridge (2001) Google Scholar
  5. 5.
    Valkenburg, R., Dorst, L.: Estimating motors from a variety of geometric data in 3D conformal geometric algebra. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice. Springer, London (2011). Chap.  2 in this book Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Intelligent Systems LaboratoryUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Industrial Research LimitedAucklandNew Zealand

Personalised recommendations