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Cubic Curves and Cubic Surfaces from Contact Points in Conformal Geometric Algebra

  • Eckhard HitzerEmail author
  • Dietmar Hildenbrand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11542)

Abstract

This work explains how to extend standard conformal geometric algebra of the Euclidean plane in a novel way to describe cubic curves in the Euclidean plane from nine contact points or from the ten coefficients of their implicit equations. As algebraic framework serves the Clifford algebra Cl(9, 7) over the real sixteen dimensional vector space \(\mathbb {R}^{9,7}\). These cubic curves can be intersected using the outer product based meet operation of geometric algebra. An analogous approach is explained for the description and operation with cubic surfaces in three Euclidean dimensions, using as framework Cl(19, 16).

Keywords

Clifford algebra Conformal geometric algebra Cubic curves Cubic surfaces Intersections 

References

  1. 1.
    Abłamowicz, R.: Clifford algebra computations with maple. In: Baylis W.E. (eds), Clifford (Geometric) Algebras. Birkhäuser Boston (1996)CrossRefGoogle Scholar
  2. 2.
    Aragon-Camarasa, G., et al.: Clifford algebra with mathematica. In: Proceedings of the 29th International Conference on Applied Mathematics, Budapest (2015). Preprint: arXiv:0810.2412
  3. 3.
    Breuils, S., Nozick, V., Sugimoto, A., Hitzer, E.: Quadric conformal geometric algebra of \(\mathbb{R}^{9,6}\). Adv. Appl. Clifford Algebras 28(35), 1–16 (2018).  https://doi.org/10.1007/s00006-018-0851-1MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Breuils, S., Nozick, V., Fuchs, L.: GARAMON: geometric algebra library generator. In: Xambo-Descamps, S., et al. (eds.) Early Proceedings of the AGACSE 2018 Conference, 23–27 July 2018, Campinas, São Paulo, Brazil, pp. 97–106 (2018)Google Scholar
  5. 5.
    Buckley, A., Košir, T.: Determinantal representations of smooth cubic surfaces. Geom. Dedicata 125(1), 115–140 (2007).  https://doi.org/10.1007/s10711-007-9144-xMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann, Burlington (2007)Google Scholar
  7. 7.
    Easter, R.B., Hitzer, E.: Triple conformal geometric algebra for cubic plane curves. Math. Methods Appl. Sci. 41(11), 4088–4105 (2018).  https://doi.org/10.1002/mma.4597. Preprint: http://vixra.org/pdf/1807.0091v1.pdfMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hildenbrand, D.: Introduction to Geometric Algebra Computing. CRC Press, Taylor & Francis Group, Boca Raton (2018)zbMATHGoogle Scholar
  9. 9.
    Hitzer, E.: The Creative Peace License. https://gaupdate.wordpress.com/2011/12/14/the-creative-peace-license-14-dec-2011/. Accessed 5 Apr 2019
  10. 10.
    Hitzer, E., Tachibana, K., Buchholz, S., Yu, I.: 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).  https://doi.org/10.1007/s00006-009-0160-9MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hitzer, E., Sangwine, S. J.: Foundations of conic conformal geometric algebra and simplified versors for rotation, translation and scaling, to be publishedGoogle Scholar
  12. 12.
    Hitzer, E.: Three-dimensional quadrics in conformal geometric algebras and their versor transformations. Adv. Appl. Clifford Algebras 29, 46 (2019).  https://doi.org/10.1007/s00006-019-0964-1. Preprint: http://vixra.org/pdf/1902.0401v4.pdfMathSciNetCrossRefGoogle Scholar
  13. 13.
    Hrdina, J., Navrat, A., Vasik, P.: Geometric algebra for conics. Adv. Appl. Clifford Algebras 28(66), 21 (2018).  https://doi.org/10.1007/s00006-018-0879-2MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    De Keninck, S.: ganja.js - geometric algebra for Javascript. https://github.com/enkimute/ganja.js. Accessed 03 May 2019
  15. 15.
    Newstead, P.E.: Geometric invariant theory. In: Bradlow, S.B., et al. (eds.) Moduli Spaces and Vector Bundles, pp. 99–127. Cambridge University Press, Cambridge (2009).  https://doi.org/10.1017/CBO9781139107037.005. https://www.cimat.mx/Eventos/cvectorbundles/newsteadnotes.pdfCrossRefGoogle Scholar
  16. 16.
    Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-89068-3CrossRefzbMATHGoogle Scholar
  17. 17.
    Sangwine, S.J., Hitzer, E.: Clifford multivector toolbox (for MATLAB). Adv. Appl. Clifford Algebras 27(1), 539–558 (2017).  https://doi.org/10.1007/s00006-016-0666-x. Preprint: http://repository.essex.ac.uk/16434/1/authorfinal.pdfMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.International Christian UniversityMitaka-shiJapan
  2. 2.Technical University of DarmstadtDarmstadtGermany

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