Skip to main content
SpringerLink
Log in

Configuration Spaces of Plane Polygons and a sub-Riemannian Approach to the Equitangent Problem

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

The equitangent locus of a convex plane curve consists of the points from which the two tangent segments to the curve have equal length. The equitangent problem concerns the relationship between the curve and its equitangent locus. An equitangent n-gon of a convex curve is a circumscribed n-gon whose vertices belong to the equitangent locus. We are interested in curves that admit 1-parameter families of equitangent n-gons.We use methods of sub-Riemannian geometry: We define a distribution on the space of polygons and study its bracket generating properties. The 1-parameter families of equitangent polygons correspond to the curves, tangent to this distribution. This distribution is closely related with the Birkhoff distribution on the space of plane polygons with a fixed perimeter length whose study, in the framework of the billiard ball problem, was pioneered by Yu. Baryshnikov, V. Zharnitsky, and J. Landsberg.

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

Access this article

Log in via an institution

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Notes

  1. This kind of generalized billiards were recently studied by A. Glutsyuk [12, 13].

References

  1. Arnold V. Ordinary differential equations. Berlin: Springer-Verlag; 2006.

    Google Scholar 

  2. Auerbach H. Sur un problem de M. Ulam concernant l’equilibre des corps flottants. Stud Math 1938;7: 121–142.

    Google Scholar 

  3. Banchoff T, Giblin P. On the geometry of piecewise circular curves. Amer Math Mon 1994;101: 403–416.

    Article  MathSciNet  MATH  Google Scholar 

  4. Baryshnikov Yu, Zharnitsky V. Sub-Riemannian geometry and periodic orbits in classical billiards. Math Res Lett 2006;13: 587–598.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bracho J, Montejano L, Oliveros D. A classification theorem for Zindler carrousels. J Dynam Control Syst 2001;7: 367–384.

    Article  MathSciNet  MATH  Google Scholar 

  6. Bracho J, Montejano L, Oliveros D. Carousels, Zindler curves and the floating body problem. Period Math Hungar 2004;49: 9–23.

    Article  MathSciNet  MATH  Google Scholar 

  7. Bruce J, Giblin P, Gibson C. Symmetry sets. Proc Roy Soc Edinb Sect A 1985;101: 163–186.

    Article  MathSciNet  MATH  Google Scholar 

  8. Davis Ph. Circulant matrices: John Wiley; 1979.

  9. Flatto L. Poncelet’s theorem. Providence: AMS; 2009.

    MATH  Google Scholar 

  10. Giblin P, Brassett S. Local symmetry of plane curves. Amer Math Mon 1985; 92: 689–707.

    Article  MathSciNet  MATH  Google Scholar 

  11. Genin D, Tabachnikov S. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. J Mod Dyn 2007; 1: 155–173.

    Article  MathSciNet  MATH  Google Scholar 

  12. Glutsyuk A. On odd-periodic orbits in complex planar billiards. J Dyn Control Syst 2014;20: 293–306.

    Article  MathSciNet  MATH  Google Scholar 

  13. Glutsyuk A. On 4-reflective complex analytic planar billiards. Preprint arXiv:1405.5990.

  14. Jeronimo-Castro J, Roldán-Pensado E. A characteristic property of the Euclidean disc. Period Math Hungar 2009;59: 215–224.

    Google Scholar 

  15. Jeronimo-Castro J, Ruiz-Hernandez G, Tabachnikov S. The equal tangents property. Adv Geom 2014;14: 447–453.

    Article  MathSciNet  MATH  Google Scholar 

  16. Landsberg J. 2006. Exterior differential systems and billiards. Geometry, integrability and quantization, 35–54, Softex, Sofia.

  17. Mauldin R. The Scottish book, Mathematics from the Scottish Café: Birhauser; 1981.

  18. Montgomery R. 2002. A tour of subriemannian geometries, their geodesics and applications, Amer. Math. Soc., Providence, RI.

  19. Odani K. On Ulam’s floating body problem of two dimension. Bulletin of Aichi Univ of Education 2009;58(Natural Sciences): 1–4.

    Google Scholar 

  20. Rademacher H, Toeplitz O. The enjoyment of mathematics: Princeton University Press; 1957.

  21. Radić M. One system of equations concerning bicentric polygons. J Geom 2009; 91: 119–139.

    Article  MathSciNet  Google Scholar 

  22. Tabachnikov S. Tire track geometry: ariations on a theme. Israel J Math 2006; 151: 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  23. Tabachnikov S. The (un)equal tangents problem. Amer Math Mon 2012; 110: 398–405.

    MathSciNet  Google Scholar 

  24. Tumanov A, Zharnitsky V. Periodic orbits in outer billiard. Int Math Res Not 2006: 17. Art. ID 67089.

  25. Várkonyi P. Floating body problems in two dimensions. Stud Appl Math 2009; 122: 195–218.

    Article  MathSciNet  MATH  Google Scholar 

  26. Wegner F. Floating bodies of equilibrium. Stud Appl Math 2003;111: 167–183.

    Article  MathSciNet  MATH  Google Scholar 

  27. Wegner F. Three problems, one solution. A web site., http://www.tphys.uni-heidelberg.de/~wegner/Fl2mvs/Movies.html.

Download references

Acknowledgments

We are grateful to R. Montgomery and V. Zharnitsky for stimulating discussions. The second author was supported by the NSF grant DMS-1105442.

Author information

Authors and Affiliations

  1. Facultad de Ingeniería, Universidad Autónoma de Querétaro, Cerro de las Campanas s/n, 76010, Querétaro, México

    Jesús Jerónimo-Castro

  2. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Serge Tabachnikov

  3. ICERM, Brown University, Box 1995, Providence, RI, 02912, USA

    Serge Tabachnikov

Authors
  1. Jesús Jerónimo-Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serge Tabachnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serge Tabachnikov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jerónimo-Castro, J., Tabachnikov, S. Configuration Spaces of Plane Polygons and a sub-Riemannian Approach to the Equitangent Problem. J Dyn Control Syst 22, 227–250 (2016). https://doi.org/10.1007/s10883-015-9269-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-015-9269-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation