Quadric Conformal Geometric Algebra of \({\mathbb {R}}^{9,6}\)

  • Stéphane Breuils
  • Vincent Nozick
  • Akihiro Sugimoto
  • Eckhard Hitzer
Part of the following topical collections:
  1. T.C. : Geometric Algebra for Computing, Graphics and Engineering with Yu Zhaoyuan, Guest E-i-C


Geometric Algebra can be understood as a set of tools to represent, construct and transform geometric objects. Some Geometric Algebras like the well-studied Conformal Geometric Algebra constructs lines, circles, planes, and spheres from control points just by using the outer product. There exist some Geometric Algebras to handle more complex objects such as quadric surfaces; however in this case, none of them is known to build quadric surfaces from control points. This paper presents a novel Geometric Algebra framework, the Geometric Algebra of \({\mathbb {R}}^{9,6}\), to deal with quadric surfaces where an arbitrary quadric surface is constructed by the mere wedge of nine points. The proposed framework enables us not only to intuitively represent quadric surfaces but also to construct objects using Conformal Geometric Algebra. Our proposed framework also provides the computation of the intersection of quadric surfaces, the normal vector, and the tangent plane of a quadric surface.


Quadrics Geometric algebra Conformal geometric algebra Clifford algebra 

Mathematics Subject Classification

Primary 99Z99 Secondary 00A00 


  1. 1.
    Breuils, S., Nozick, V., Fuchs, L., Hildenbrand, D., Benger, W., Steinmetz, C.: A hybrid approach for computing products of high-dimensional geometric algebras, Proceedings of the Computer Graphics International Conference, ENGAGE (Hiyoshi, Japan), CGI ’17, ACM, 43, 1–6 (2017)Google Scholar
  2. 2.
    Breuils, S., Nozick, V., Fuchs, L.: A geometric algebra implementation using binary tree. Adv. Appl. Clifford Algebras 27(3), 2133–2151 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Doran, C., Lasenby, A.: Geometric algebra for physicists. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dorst, L.: 3d oriented projective geometry through versors of \( {\mathbb{R}}^{3,3}\). Adv. Appl. Clifford Algebras 26(4), 1137–1172 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dorst, L., Fontijne, D., Mann, S.: Geometric algebra for computer science, an object-oriented approach to geometry. Morgan Kaufmann, Burlington (2007)Google Scholar
  6. 6.
    Easter, R.B., Hitzer, E.: Double conformal space-time algebra, AIP Conference Proceedings, vol. 1798, AIP Publishing, p. 020066 (2017)Google Scholar
  7. 7.
    Easter, R.B., Hitzer, E.: Triple conformal geometric algebra for cubic plane curves, Math. Methods Appl. Sci. mma.4597 (2017)Google Scholar
  8. 8.
    Easter, R.B., Hitzer, E.: Double conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2175–2199 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hitzer, E., Tachibana, K., Buchholz, S., Isseki, Y.: Carrier method for the general evaluation and control of pose, molecular conformation, tracking, and the like. Adv. Appl. Clifford Algebras 19(2), 339–364 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Juan, D., Goldman, R., Mann, S.: Modeling 3D geometry in the Clifford Algebra \({\mathbb{R}}^{4,4}\). Adv. Appl. Clifford Algebras 27(4), 3039–3062 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kanatani, K.: Understanding geometric algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics. A. K. Peters Ltd, Natick (2015)CrossRefzbMATHGoogle Scholar
  12. 12.
    Klawitter, D.: A Clifford algebraic approach to line geometry. Adv. Appl. Clifford Algebras 24(3), 713–736 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Luo, W., Yong, H., Zhaoyuan, Y., Yuan, L., Lü, G.: A hierarchical representation and computation scheme of arbitrary-dimensional geometrical primitives based on CGA. Adv. Appl. Clifford Algebras 27(3), 1977–1995 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Papaefthymiou, M., Papagiannakis, G.: Real-time rendering under distant illumination with conformal geometric algebra, Math. Methods Appl. Sci. (2017)Google Scholar
  15. 15.
    Parkin, S.T.: A model for quadric surfaces using geometric algebra, (October 2012)Google Scholar
  16. 16.
    Perwass, C.: Geometric algebra with applications in engineering, Geometry and Computing, vol. 4. Springer, Berlin (2009)zbMATHGoogle Scholar
  17. 17.
    Vince, J.: Geometric algebra for computer graphics. Springer Science & Business Media, Berlin (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Zamora-Esquivel, J.: G 6,3 geometric algebra; description and implementation. Adv. Appl. Clifford Algebras 24(2), 493–514 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire d’Informatique Gaspard-Monge, Equipe A3SIUMR 8049, Université Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  2. 2.CNRS JFLI, UMI 3527, National Institute of InformaticsTokyoJapan
  3. 3.National Institute of InformaticsTokyoJapan
  4. 4.International Christian UniversityTokyoJapan

Personalised recommendations