Skip to main content
SpringerLink
Log in

Hyperquaternions: An Efficient Mathematical Formalism for Geometry

  • Conference paper
  • First Online:
Geometric Science of Information (GSI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11712))

Included in the following conference series:

Abstract

Hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof), they constitute Clifford algebras endowed with an associative exterior product providing an efficient mathematical formalism for differential geometry. The paper presents a hyperquaternion formulation of pseudo-euclidean rotations and the Poincaré groups in n dimensions (via dual hyperquaternions). A canonical decomposition of these groups is developed as an extension of an euclidean formalism and illustrated by a 5D example. Potential applications include in particular, moving reference frames and machine learning.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Ungar, A.: Beyond Pseudo-Rotations in Pseudo-Euclidean Spaces. Academic Press, London (2018)

    MATH  Google Scholar 

  2. Ferreira, M., Sommen, F.: Complex boosts: a Hermitian Clifford algebra approach. Adv. Appl. Clifford Algebras 23, 339–362 (2013)

    Article  MathSciNet  Google Scholar 

  3. Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (2001)

    Book  Google Scholar 

  4. Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras: Spinor Structures. Kluwer Academic Publishers, Dordrecht (1990)

    Book  Google Scholar 

  5. Perez-Gracia, A., Thomas, F.: On Cayley’s factorization of 4D rotations and applications. Adv. Appl. Clifford Algebras 27, 523–538 (2017)

    Article  MathSciNet  Google Scholar 

  6. Richard, A., Fuchs, L., Andres, E., Largeteau-Skapin, G.: Decomposition of nD-rotations: classification, properties and algorithm. Graph. Models 73(6), 346–353 (2011). Elsevier

    Article  Google Scholar 

  7. Girard, P.R., Clarysse, P., Pujol, R., Goutte, R., Delachartre, P.: Hyperquaternions: a new tool for physics. Adv. Appl. Clifford Algebras 28, 68 (2018)

    Article  MathSciNet  Google Scholar 

  8. Moore, C.L.E.: Hyperquaternions. J. Math. Phys. 1, 63–77 (1922)

    Article  Google Scholar 

  9. Moore, C.L.E.: Rotations in hyperspace. In: Proceedings of the American Academy of Arts and Sciences, vol. 53, no. 8, pp. 651–694 (1918)

    Article  Google Scholar 

  10. Zhu, X., Xu, Y., Xu, H., Chen, C.: Quaternion convolutional neural networks. In: Ferrari, V., Hebert, M., Sminchisescu, C., Weiss, Y. (eds.) ECCV 2018. LNCS, vol. 11212, pp. 645–661. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-01237-3_39

    Chapter  Google Scholar 

  11. Girard, P.R., Clarysse, P., Pujol, R., Wang, L., Delachartre, P.: Differential geometry revisited by biquaternion clifford algebra. In: Boissonnat, J.-D., et al. (eds.) Curves and Surfaces 2014. LNCS, vol. 9213, pp. 216–242. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22804-4_17

    Chapter  Google Scholar 

  12. Girard, P.R.: Quaternions Clifford Algebras and Relativistic Physics. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  13. Girard, P.R.: Algèbre de Clifford et Physique relativiste. PPUR, Lausanne (2004)

    MATH  Google Scholar 

  14. Girard, P.R.: Einstein’s equations and Clifford algebra. Adv. Appl. Clifford Algebras 9(2), 225–230 (1999)

    Article  MathSciNet  Google Scholar 

  15. Girard, P.R.: The quaternion group and modern physics. Eur. J. Phys. 5, 25–32 (1984)

    Article  Google Scholar 

  16. Casanova, G.: L’algèbre vectorielle. PUF, Paris (1976)

    Google Scholar 

  17. Lipschitz, R.: Principes d’un calcul algébrique qui contient comme espèces particulières le calcul des quantités imaginaires et des quaternions. C.R. Acad. Sci. Paris, vol. 91, pp. 619–621, 660–664 (1880)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) and was performed within the framework of the LABEX CELYA (ANR-10-LABX-0060) of Université de Lyon, within the program“Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Author information

Authors and Affiliations

  1. Univ Lyon, INSA-LYON, Université Claude Bernard Lyon 1, UJM-Saint Etienne, CNRS, Inserm, CREATIS UMR 5220, U1206, 69621, Lyon, France

    Patrick R. Girard, Patrick Clarysse, Robert Goutte & Philippe Delachartre

  2. Pôle de Mathématiques, INSA-Lyon, Bât. Léonard de Vinci, 21 avenue Jean Capelle, 69621, Villeurbanne, France

    Romaric Pujol

Authors
  1. Patrick R. Girard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrick Clarysse
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Romaric Pujol
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robert Goutte
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Philippe Delachartre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Patrick R. Girard .

Editor information

Editors and Affiliations

  1. Sony Computer Science Laboratories, Inc., Tokyo, Japan

    Frank Nielsen

  2. Thales, Limours, France

    Frédéric Barbaresco

A Multivector Structure of \(\mathbb {H}\otimes \mathbb {H}\otimes \mathbb {H}\)

A Multivector Structure of \(\mathbb {H}\otimes \mathbb {H}\otimes \mathbb {H}\)

figure a

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Girard, P.R., Clarysse, P., Pujol, R., Goutte, R., Delachartre, P. (2019). Hyperquaternions: An Efficient Mathematical Formalism for Geometry. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science(), vol 11712. Springer, Cham. https://doi.org/10.1007/978-3-030-26980-7_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26980-7_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26979-1

  • Online ISBN: 978-3-030-26980-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation