Skip to main content
SpringerLink
Log in

On Specific Conic Intersections in GAC and Symbolic Calculations in GAALOPWeb

  • Published:
Advances in Applied Clifford Algebras Aims and scope Submit manuscript

A Correction to this article was published on 10 January 2022

This article has been updated

Abstract

We describe a possibility for geometric calculation of specific conics’ intersections in Geometric Algebra for Conics (GAC) using its operations that may be expressed as sums of products. The advantage is that no solver for a system of quadratic equations is needed and thus no numerical error is involved. We also describe specific conics connected to intersections of conics in a general mutual position. Then we show how symbolic operations may be calculated directly in GAALOPWeb software, that the basis coefficients may be read off in the appropriate basis and, moreover, the result may be immediately and truly visualized. We compare the functionality with Maple package Clifford.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Change history

Notes

  1. Since GAC is able to handle rotations correctly compared to other algebras this is presented as an example in this chapter.

  2. The GAALOPWeb visualization is based on the ganja tool of Steven de Keninck as described in [7].

References

  1. Abłamowicz, R., Fauser, B.: Mathematics of Clifford - a Maple package for Clifford and Graßmann algebras. Adv. Appl. Clifford Algebras 15 , 157–181 (2005). https://doi.org/10.1007/s00006-005-0009-9

  2. Derevianko, A.I., Korobov, V.I.: Controllability of the given switched linear system of special type. Visnyk of V.N. Karazin Kharkiv National University Ser. Mathematics, Applied Mathematics and Mechanics, vol. 89, pp. 93–101 (2019). https://doi.org/10.26565/2221-5646-2019-89-07

  3. Derevianko, A.I., Vašík, P.: Solver-ree optimal control for linear dynamical switched system by means of Geometric Algebra. arXiv:2103.13803 [math.OC]. (2021)

  4. Easter, R.B., Hitzer, E.: Double conformal geometric algebra. Adv. Appl. Clifford Algebras 27, 2175 (2017). https://doi.org/10.1007/s00006-017-0784-0

    Article  MathSciNet  MATH  Google Scholar 

  5. Hildenbrand, D., Steinmetz, C.: GAALOPWeb. Online tool. https://gaalopweb.fme.vutbr.cz/gaalopweb/

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

    MATH  Google Scholar 

  7. Hildenbrand, D.: The Power of Geometric Algebra Computing: For Engineering and Quantum Computing. CRC Press, Taylor & Francis Group, Boca Raton (2021)

    Book  Google Scholar 

  8. Hildenbrand, D., Steinmetz, C., Tichý, R.: GAALOPWeb for Matlab: an easy to handle solution for industrial geometric algebra implementations. Adv. Appl. Clifford Algebras 30(52) (2020). https://doi.org/10.1007/s00006-020-01081-9

  9. Hildenbrand, D., Franchini, S., Gentile, A., Vassallo, G., Vitabile, S.: GAPPCO: an easy to configure geometric algebra coprocessor based on GAPP programs. Adv. Appl. Clifford Algebras 27, 2115–2132 (2017). https://doi.org/10.1007/s00006-016-0755-x

    Article  MathSciNet  MATH  Google Scholar 

  10. Hrdina, J., Návrat, A., Vašík, P.: Geometric Algebra for Conics. Adv. Appl. Clifford Algebras 28(66) (2018). https://doi.org/10.1007/s00006-018-0879-2

  11. Hrdina, J., Návrat, A., Vašík, P.: Conic fitting in geometric algebra setting. Adv. Appl. Clifford Algebras 29(72) (2019). https://doi.org/10.1007/s00006-019-0989-5

  12. Hrdina, J., Návrat, A., Vašík, P., Dorst, L.: Projective geometric algebra as a subalgebra of conformal geometric algebra. Adv. Appl. Clifford Algebras 31(18) (2021). https://doi.org/10.1007/s00006-021-01118-7

  13. Lasenby, A.: Rigid body dynamics in a constant curvature space and the ‘1D-up’ approach to conformal geometric algebra. In: Dorst, L., Lasenby, J. (eds.) Guide to Geometric Algebra in Practice. Springer, London (2011)

  14. Lounesto, P.: Clifford Algebra and Spinors, 2nd edn. CUP, Cambridge (2006)

  15. Perwass, Ch.: Geometric Algebra with Applications in Engineering. Springer, New York (2009)

  16. Richter-Gebert, J.: Perspectives on Projective Geometry Berlin. Springer, Berlin, Heidelberg (2011)

    Book  Google Scholar 

  17. Zamora-Esquivel, J.: \(\mathbb{G}_{6,3}\) geometric algebra; description and implementation. Adv. Appl. Clifford Algebr. 24(2), 493–514 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Institute of Mathematics, Brno University of Technology, Brno, Czech Republic

    Roman Byrtus, Anna Derevianko & Petr Vašík

  2. Techische Universität Darmstadt, Darmstadt, Germany

    Dietmar Hildenbrand & Christian Steinmetz

Authors
  1. Roman Byrtus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anna Derevianko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Petr Vašík
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dietmar Hildenbrand
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Christian Steinmetz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr Vašík.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Roman Byrtus, Anna Derevianko and Petr Vašík were supported by a grant no. FSI-S-20-6187.

This article is part of the ENGAGE 2020 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Werner Benger, Dietmar Hildenbrand, Eckhard Hitzer, and George Papagiannakis.

The original version of this article was revised to update the formula on page 9 and some of the reference corrections.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Byrtus, R., Derevianko, A., Vašík, P. et al. On Specific Conic Intersections in GAC and Symbolic Calculations in GAALOPWeb. Adv. Appl. Clifford Algebras 32, 2 (2022). https://doi.org/10.1007/s00006-021-01182-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-021-01182-z

Mathematics Subject Classification

Keywords

Search

Navigation