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Foundations of Conic Conformal Geometric Algebra and Compact Versors for Rotation, Translation and Scaling


This paper explains in algebraic detail how two-dimensional conics can be defined by the outer products of conformal geometric algebra (CGA) points in higher dimensions. These multivector expressions code all types of conics in arbitrary scale, location and orientation. Conformal geometric algebra of two-dimensional Euclidean geometry is fully embedded as an algebraic subset. With small model preserving modifications, it is possible to consistently define in conic CGA versors for rotation, translation and scaling, similar to Hrdina et al. (Appl Clifford Algebras 28(66), 1–21,, 2018), but simpler, especially for translations.

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Fig. 1


  1. For further literature see the references in [1].

  2. We use for null vectors the notation \(\mathbf {e}_{o}\), and \(\mathbf {e}_{\infty }\) with added indexes 1, 2, 3, because this intuitive notation for CGA null vectors became widespread with [4], replacing the earlier notation \({\overline{n}}\) and n. The notation \({\overline{n}}\) and n with added indexes 1, 2, 3 was used in [13], but [1] consistently combined instead \(\mathbf {e}_{o}\), and \(\mathbf {e}_{\infty }\) with added indexes 1, 2, 3, etc. Avoiding the introduction of further new conventions, we adopt the notation for basis vectors following [1].

  3. The parameters \(\lambda _i\), \(i=1,2,3\), parameterize a continuous set of horospheres [6].

  4. Note that our definition (43) is different from that given in [10, equation (2)].

  5. We have introduced the lower index \(\text {C}\) for entities in the subalgebra, generated by \(\{\mathbf {e}_o, \mathbf {e}_1, \mathbf {e}_2, \mathbf {e}_{\infty }\}\), isomorphic to Cl(3, 1), the CGA of the two-dimensional Euclidean plane \(\mathbb {R}^2\).

  6. Note, that the left- and right contraction, respectively, are needed essentially. Albeit, because the two are related by reversion, \(A \rfloor B = {\widetilde{B}} \lfloor {\widetilde{A}}\), and reversion only changes signs of blades, it would be possible to use only one of the two forms of the contraction, and express the other by reordering and sign changes.

  7. For \(\{\mathbf {e}_{\infty 1}, \mathbf {e}_{o 1}\}\) and \(\{\mathbf {e}_{\infty 2}, \mathbf {e}_{o 2}\}\), we could also have chosen \(\lambda _1=\lambda _2=\sqrt{2}\) as in [10], without altering our form of the transformation versors given below. But for aesthetic reasons, we decided in (2) to simply set all three \(\lambda \) coefficients to 1. For ease of numerical implementation, the choice \(\lambda _1=\lambda _2=\sqrt{2}\) might be of advantage, but \(\lambda _3=1\) has to be preserved if the current versor formulation is to be adopted.


  1. 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).

    Article  MATH  Google Scholar 

  2. De Keninck, S.: The Coffee Shop. (Accessed 30 Aug 2019)

  3. Desmos: Implicite functions online graphing software. (Accessed 02 Feb 2019)

  4. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science, An Object-oriented Approach to Geometry. Morgan Kaufmann, Burlington (2007)

    Google Scholar 

  5. Easter, R.B., Hitzer, E.: Double conformal geometric algebra. Adv. Appl. Clifford Algebras 27(3), 2175–2199 (2017).

    MathSciNet  Article  MATH  Google Scholar 

  6. El Mir, G., Saint-Jean, C., Berthier, M.: Conformal geometry for viewpoint change representation. Adv. Appl. Clifford Algebras 24(2), 443–463 (2014).

    MathSciNet  Article  MATH  Google Scholar 

  7. Hildenbrand, D.: Introduction to Geometric Algebra Computing, 1st edn. Chapman and Hall/CRC, Boca Raton (2018)

    MATH  Google Scholar 

  8. Hitzer, E.: The Creative Peace License.

  9. 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).

    MathSciNet  Article  MATH  Google Scholar 

  10. Hrdina, J., Navrat, A., Vasik, P.: Geometric algebra for conics. Adv. Appl. Clifford Algebras 28(66), 1–21 (2018).

    MathSciNet  Article  MATH  Google Scholar 

  11. Li, H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2008)

    Book  Google Scholar 

  12. Naeve, A., Svensson, L.: Projective Geometric Computing, The 5th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa-Zihuatanejo, Mexico, June 27–July 4 (1999) (preprint).

  13. Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  14. Sangwine, S.J., Hitzer, E.: Clifford multivector toolbox (for MATLAB). Adv. Appl. Clifford Algebras 27(1), 539–558 (2017).

    MathSciNet  Article  MATH  Google Scholar 

  15. Zamora-Esquivel, J.: \(G_{6,3}\) geometric algebra; description and implementation. Adv. Appl. Clifford Algebras 24(2), 493–514 (2014).

    MathSciNet  Article  MATH  Google Scholar 

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Soli Deo Gloria (J. C. Maxwell had inscribed at the gate of his Cavendish laboratory: The works of the Lord are great, Studied by all who have pleasure in them. [Psalm 111:2, New King James Version]. D. Capkova writes in Colloquium Comenius and Descartes, Foundation Comenius Museum, Naarden, 1997, p. 15, that: “Comenius’ warning against onesided rationalism and against universal application of rationalism speaks to our time very urgently.”). The author EH requests the Creative Peace License [8] to be adhered to regarding the content of this work. Finally, we thank the anonymous reviewers for their helpful advice.

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This article is part of the Topical Collection on FTHD 2018, edited by Sirkka-Liisa Eriksson, Yuri M. Grigoriev, Ville Turunen, Franciscus Sommen and Helmut Malonek.

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Hitzer, E., Sangwine, S.J. Foundations of Conic Conformal Geometric Algebra and Compact Versors for Rotation, Translation and Scaling. Adv. Appl. Clifford Algebras 29, 96 (2019).

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  • Clifford algebra
  • Conformal geometric algebra
  • Conics
  • Versors

Mathematics Subject Classification

  • Primary 15A66
  • Secondary 11E88
  • 15A15
  • 15A09