Skip to main content
SpringerLink
Log in

The talented Mr. Inversive Triangle in the elliptic billiard

  • Research Article
  • Published:
European Journal of Mathematics Aims and scope Submit manuscript

Abstract

Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new “focus-inversive” family inscribed in Pascal’s limaçon. The following are some of its surprising invariants: (i) perimeter, (ii) sum of cosines, and (iii) sum of distances from inversion center (the focus) to vertices. We prove these for the \(N=3\) case, showing that this family (a) has a stationary Gergonne point, (b) is a 3-periodic family of a second, rigidly moving elliptic billiard, and (c) the loci of incenter, barycenter, circumcenter, orthocenter, nine-point center, and a great many other triangle centers are circles.

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
Fig. 9
Fig. 10

Similar content being viewed by others

Notes

  1. Any “new” invariants are dependent upon the two integrals of motion—linear and angular momentum—that render the elliptic billiard an integrable dynamical system [9].

References

  1. Akopyan, A.V.: Conjugation of lines with respect to a triangle. Journal Classical Geometry 1, 23–31 (2012)

    Google Scholar 

  2. Bialy, M., Tabachnikov, S.: Dan Reznik’s identities and more. Eur. J. Math. https://doi.org/10.1007/s40879-020-00428-7

  3. Ferréol, R.: Booth’s curve. Mathcurve Portal (2020). https://www.mathcurve.com/courbes2d.gb/booth/booth.shtml

  4. Ferréol, R.: Pascal’s limaçon. Mathcurve Portal (2020). https://mathcurve.com/courbes2d.gb/limacon/limacon.shtml

  5. Garcia, R.: Elliptic billiards and ellipses associated to the 3-periodic orbits. Amer. Math. Monthly 126(6), 491–504 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Garcia, R., Reznik, D.: Invariants of self-intersected and inversive \({N}\)-periodics in the elliptic billiard (2020). arXiv:2011.06640

  7. Garcia, R., Reznik, D., Koiller, J.: Loci of 3-periodics in an elliptic billiard: Why so many ellipses? (2020). arXiv:2001.08041

  8. Garcia, R., Reznik, D., Koiller, J.: New properties of triangular orbits in elliptic billiards. Amer. Math. Monthly (to appear)

  9. Kaloshin, V., Sorrentino, A.: On the integrability of Birkhoff billiards. Phil. Trans. Roy. Soc. 376(2131), Art. No. 20170419 (2018)

  10. Kimberling, C.: Triangle centers as functions. Rocky Mountain J. Math. 23(4), 1269–1286 (1993). https://doi.org/10.1216/rmjm/1181072493

    Article  MathSciNet  MATH  Google Scholar 

  11. Kimberling, C.: Encyclopedia of triangle centers (2019). https://faculty.evansville.edu/ck6/encyclopedia/ETC.html

  12. Odehnal, B.: Poristic loci of triangle centers. J. Geom. Graph. 15(1), 45–67 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Preparata, F., Shamos, M.I.: Computational Geometry. Texts and Monographs in Computer Science, 2nd edn. Springer, Berlin (1988)

    Google Scholar 

  14. Reznik, D., Garcia, R.: Circuminvariants of 3-periodics in the elliptic billiard. Intl. J. Geom. 10(1), 31–57 (2021)

    MathSciNet  MATH  Google Scholar 

  15. Reznik, D.S., Garcia, R., Koiller, J.: The ballet of triangle centers on the elliptic billiard. J. Geom. Graph. 24(1), 79–101 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Reznik, D., Garcia, R., Koiller, J.: Can the elliptic billiard still surprise us? Math. Intelligencer 42(1), 6–17 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reznik, D., Garcia, R., Koiller, J.: Fifty new invariants of \({N}\)-periodics in the elliptic billiard. Arnold Math. J. https://doi.org/10.1007/s40598-021-00174-y

  18. Roitman, P., Garcia, R., Reznik, D.: New invariants of Poncelet–Jacobi bicentric polygons (2021). arXiv:2103.11260

  19. Romaskevich, O.: On the incenters of triangular orbits on elliptic billiards. Enseign. Math. 60(3–4), 247–255 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schwartz, R., Tabachnikov, S.: Centers of mass of Poncelet polygons, 200 years after. Math. Intelligencer 38(2), 29–34 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stachel, H.: Expression for Joachimsthal’s constant. Private Communication (2020)

  22. Tabachnikov, S.: Geometry and Billiards. Student Mathematical Library, vol. 30. American Mathematical Society, Providence (2005). https://www.bit.ly/2RV04CK

  23. Weisstein, E.: Mathworld (2019). https://mathworld.wolfram.com

Download references

Acknowledgements

We would like to thank Arseniy Akopyan, Peter Moses, and Pedro Roitman for useful insights.

Author information

Authors and Affiliations

  1. Data Science Consulting, Rio de Janeiro, Brazil

    Dan Reznik

  2. Institute of Mathematics and Statistics, Federal University of of Goiás, Campus II, Goiânia, GO, 74690-900, Brazil

    Ronaldo Garcia

  3. Department of Mathematics, Rice University, P.O. Box 1892, Houston, TX, 77005-1892, USA

    Mark Helman

Authors
  1. Dan Reznik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ronaldo Garcia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mark Helman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dan Reznik.

Additional information

Publisher's Note

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

Ronaldo Garcia is a fellow of CNPq and a coordinator of Project PRONEX/ CNPq/ FAPEG 2017 10 26 7000 508.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Reznik, D., Garcia, R. & Helman, M. The talented Mr. Inversive Triangle in the elliptic billiard. European Journal of Mathematics 8, 1550–1565 (2022). https://doi.org/10.1007/s40879-021-00489-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40879-021-00489-2

Keywords

Mathematics Subject Classification

Search

Navigation