Skip to main content
SpringerLink
Log in

On evolutoids of planar convex curves II

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

In this paper we continue the study of evolutoids of convex curves. We proved that if a curve is homothetic to one of its evolutoids then it is a circle. This result is analogous, for the case of evolutoids, to the planar case of the famous homothetic floating body problem which states that if a floating body is homothetic to the body itself then it is an ellipsoid. Among other things, we proved that a curve and any of its evolutoids have the same Steiner point. Moreover, some relations between evolutoids and constant angle caustics are also given, for instance, that if for a given angle the left and right evolutoids describe the same curve then the curve possesses a constant angle caustic.

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

Similar content being viewed by others

References

  1. Apostol T., Mnatsakanian M.A.: The method of sweeping tangents. Math. Gaz. 92, 396–417 (2008)

    Google Scholar 

  2. Apostol T.M., Mnatsakanian M.A.: Tanvolutes: generalized involutes. Am. Math. Mon. 117, 701–713 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bezdek K., Connelly R.: Minimal translation cover for sets of diameter 1. Period. Math. Hungar. 34, 23–27 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Braude L.: Über einige Verallgemeinerungen des Begriffes der Evolutoïde. Archiv der Math. und Physik 3, 44–52 (1912)

    Google Scholar 

  5. Chakerian G.D., Groemer H.: Convex bodies of constant width. In: Gruber, P.M., Wills, J.M. (eds.) Convexity and its Applications, Birkhäuser, Basel (1983)

    Google Scholar 

  6. Cieślak W., Miernowski A., Mozgawa W.: \({\varphi}\) -optics and generalized Bianchi-Auerbach equation. J. Geom. Phys. 62, 2337–2345 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cyr, V.: A number theoretic question arising in the geometry of plane curves and in billiard dynamics. Proc. Amer. Math. Soc. 140(9), 3035–3040 (2012)

  8. Groemer H.: On a characterization of circles and spheres. Beitr. Algebra Geom. 36, 211–218 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Groemer H.: Geometric Applications of Fourier Series and Spherical harmonics. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  10. Giblin, P.J., Warder, J.P.: Evolving evolutoids. Amer. Math. Monthly. 121(10), 871–889 (2014)

  11. Gutkin E.: Capillary floating and the billiard ball problem. J. Math. Fluid Mech. 14, 363–382 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamann M.: A note on ovals and their evolutoids. Beitr. Algebra Geom. 50, 433–441 (2009)

    MathSciNet  MATH  Google Scholar 

  13. Harazis̆vili A.B.: Some properties of the center of symmetry. Soobsch. Akad. Nauk. Gruzin SSR 89, 21–24 (1978)

    MathSciNet  Google Scholar 

  14. Inzinger R.: Über die Evolutoiden und Zwischenevolutoiden von Raumkurven. Monatsh. Math. Phys. 42, 57–84 (1935)

    Article  MathSciNet  Google Scholar 

  15. Jerónimo-Castro J.: On evolutoids of planar convex curves. Aequ. Math. 88, 97–103 (2014)

    Article  MATH  Google Scholar 

  16. Jerónimo-Castro J., Roldán-Pensado E.: A characteristic property of the Euclidean disc. Period. Math. Hungar. 59, 215–224 (2009)

    Article  Google Scholar 

  17. Jerónimo-Castro, J.: A characterization of the ellipse related to illumination bodies. to appear in Elem. Math.

  18. Khassa D.S.: Relation between maximal chords and symmetry for convex sets. J. Lond. Math. Soc. 15, 541–546 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mozgawa W., Skrzypiec M.: Some properties of secantoptics of ovals. Beitr. Algebra Geom. 53, 261–272 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schneider R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  21. Schütt C., Werner E.: The convex floating body. Math. Scand. 66, 275–290 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Schütt C., Werner E.: Homothetic floating bodies. Geom. Dedicata 49, 335–348 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Skrzypiec M.: A note on secantoptics. Beitr. Algebra Geom. 49, 205–215 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Stancu A.: The floating body problem. Bull. Lond. Math. Soc. 38, 839–846 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Toponogov V.A.: Differential Geometry of Curves and Surfaces, a Concise Guide. Birkhäuser, Boston-Basel-Berlin (2006)

    Google Scholar 

  26. Valentine F.A.: Convex Sets. McGraw-Hill, New York (1964)

    MATH  Google Scholar 

  27. Werner, E.: Floating bodies and illumination bodies. In: Proceedings of the Conference “Integral Geometry and Convexity” Wuhan 2004, World Scientific, Singapore (2006)

  28. Werner E., Ye D.: On the homothety conjecture. Indiana Univ. Math. J. 60, 1–20 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wunderlich W.: Über die Evolutoiden der Ellipse. Elem. Math. 10, 37–40 (1953)

    MathSciNet  Google Scholar 

  30. Wunderlich W.: Ebene Kinematik. Bibliographisches Institut, Mannheim (1970)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad Autónoma de Querétaro, Querétaro, Querétaro, Mexico

    V. A. Aguilar-Arteaga, R. Ayala-Figueroa, I. González-García & J. Jerónimo-Castro

Authors
  1. V. A. Aguilar-Arteaga
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Ayala-Figueroa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. I. González-García
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Jerónimo-Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Jerónimo-Castro.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aguilar-Arteaga, V.A., Ayala-Figueroa, R., González-García, I. et al. On evolutoids of planar convex curves II. Aequat. Math. 89, 1433–1447 (2015). https://doi.org/10.1007/s00010-015-0352-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-015-0352-4

Mathematics Subject Classification

Keywords

Search

Navigation