Transverse Approach to Geometric Algebra Models for Manipulating Quadratic Surfaces

  • Stéphane BreuilsEmail author
  • Vincent Nozick
  • Laurent Fuchs
  • Akihiro Sugimoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11542)


Quadratic surfaces gain more and more attention in the geometric algebra community and some frameworks to represent, transform, and intersect these quadratic surfaces have been proposed. To the best of our knowledge, however, no framework has yet proposed that supports all the operations required to completely handle these surfaces. Some existing frameworks do not allow the construction of quadratic surfaces from control points while some do not allow to transform these quadratic surfaces. Although a framework does not exist that covers all the required operations, if we consider all already proposed frameworks together, then all the operations over quadratic surfaces are covered there. This paper presents an approach that transversely uses different frameworks for covering all the operations on quadratic surfaces. We employ a framework to represent any quadratic surfaces either using control points or the coefficients of its implicit form and then map the representation into another framework so that we can transform them and compute their intersection. Our approach also allows us to easily extract some geometric properties.


Geometric algebra Quadratic surfaces Conformal geometric algebras 


  1. 1.
    Breuils, S., Nozick, V., Fuchs, L.: Garamon: Geometric algebra library generator. Advances in Applied Clifford Algebras Submitted (2019)Google Scholar
  2. 2.
    Breuils, S., Nozick, V., Sugimoto, A., Hitzer, E.: Quadric conformal geometric algebra of \(\mathbb{R}^{9,6}\). Adv. Appl. Clifford Algebras 28(2), 35 (2018). Scholar
  3. 3.
    Buchholz, S., Tachibana, K., Hitzer, E.M.S.: Optimal learning rates for clifford neurons. In: de Sá, J.M., Alexandre, L.A., Duch, W., Mandic, D. (eds.) ICANN 2007. LNCS, vol. 4668, pp. 864–873. Springer, Heidelberg (2007). Scholar
  4. 4.
    Doran, C., Hestenes, D., Sommen, F., Van Acker, N.: Lie groups as spin groups. J. Math. Phys. 34(8), 3642–3669 (1993)MathSciNetCrossRefGoogle 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.
    Dorst, L., Van Den Boomgaard, R.: An analytical theory of mathematical morphology. In: Mathematical Morphology and its Applications to Signal Processing, pp. 245–250 (1993)Google Scholar
  7. 7.
    Druoton, L., Fuchs, L., Garnier, L., Langevin, R.: The non-degenerate dupin cyclides in the space of spheres using geometric algebra. Adv. Appl. Clifford Algebras 24(2), 515–532 (2014). Scholar
  8. 8.
    Du, J., Goldman, R., Mann, S.: Modeling 3D geometry in the clifford algebra \(\mathbb{R}^{4,4}\). Adv. Appl. Clifford Algebras 27(4), 3039–3062 (2017). Scholar
  9. 9.
    Easter, R.B., Hitzer, E.: Double conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2175–2199 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Glassner, A.S.: An Introduction to Ray Tracing. Elsevier, Amsterdam (1989)zbMATHGoogle Scholar
  11. 11.
    Goldman, R., Mann, S.: R(4, 4) as a computational framework for 3-dimensional computer graphics. Adv. Appl. Clifford Algebras 25(1), 113–149 (2015). Scholar
  12. 12.
    Gregory, A.L., Lasenby, J., Agarwal, A.: The elastic theory of shells using geometric algebra. Roy. Soc. Open Sci. 4(3), 170065 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hestenes, D.: New Foundations for Classical Mechanics, vol. 15. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  14. 14.
    Hitzer, E.: Geometric operations implemented by conformal geometric algebra neural nodes. Preprint arXiv:1306.1358 (2013)
  15. 15.
    Leopardi, P.: A generalized FFT for Clifford algebras. Bull. Belg. Math. Soc. 11, 663–688 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Luo, W., Hu, Y., Yu, Z., 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). Scholar
  17. 17.
    Papaefthymiou, M., Papagiannakis, G.: Real-time rendering under distant illumination with conformal geometric algebra. Math. Methods Appl. Sci. 41, 4131–4147 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Parkin, S.T.: A model for quadric surfaces using geometric algebra. Unpublished, October 2012Google Scholar
  19. 19.
    Perwass, C.: Geometric Algebra with Applications in Engineering. Geometry and Computing, vol. 4. Springer, Heidelberg (2009). Scholar
  20. 20.
    Vince, J.: Geometric Algebra for Computer Graphics. Springer, Heidelberg (2008). Scholar
  21. 21.
    Zhu, S., Yuan, S., Li, D., Luo, W., Yuan, L., Yu, Z.: Mvtree for hierarchical network representation based on geometric algebra subspace. Adv. Appl. Clifford Algebras 28(2), 39 (2018). Scholar

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Authors and Affiliations

  1. 1.Laboratoire d’Informatique Gaspard-Monge, Equipe A3SI UMR 8049Université Paris-Est Marne-la-ValléeChamps-sur-MarneFrance
  2. 2.XLIM-ASALI, UMR 7252Université de PoitiersPoitiersFrance
  3. 3.National Institute of InformaticsTokyoJapan

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