Differential Geometry Revisited by Biquaternion Clifford Algebra

  • Patrick R. GirardEmail author
  • Patrick Clarysse
  • Romaric Pujol
  • Liang Wang
  • Philippe Delachartre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9213)


In the last century, differential geometry has been expressed within various calculi: vectors, tensors, spinors, exterior differential forms and recently Clifford algebras. Clifford algebras yield an excellent representation of the rotation group and of the Lorentz group which are the cornerstones of the theory of moving frames. Though Clifford algebras are all related to quaternions via the Clifford theorem, a biquaternion formulation of differential geometry does not seem to have been formulated so far. The paper develops, in 3D Euclidean space, a biquaternion calculus, having an associative exterior product, and applies it to differential geometry. The formalism being new, the approach is intended to be pedagogical. Since the methods of Clifford algebras are similar in other dimensions, it is hoped that the paper might open new perspectives for a 4D hyperbolic differential geometry. All the calculi presented here can easily be implemented algebraically on Mathematica and numerically on Matlab. Examples, matrix representations, and a Mathematica work-sheet are provided.


Clifford algebras Quaternions Biquaternions Differential geometry Rotation group SO(3) Hyperquaternion algebra 



This work was conducted in the framework of LabEx Celya and PRIMES (Physics Radiobiology Medical Imaging and Simulation).


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Patrick R. Girard
    • 1
    Email author
  • Patrick Clarysse
    • 1
  • Romaric Pujol
    • 2
  • Liang Wang
    • 1
  • Philippe Delachartre
    • 1
  1. 1.Université de Lyon, CREATIS, CNRS UMR 5220, Inserm U1044, INSA-Lyon, Université LYON 1VilleurbanneFrance
  2. 2.Université de Lyon, Pôle de Mathématiques, INSA-LyonVilleurbanneFrance

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