Skip to main content
SpringerLink
Log in

The singular evolutoids set and the extended evolutoids front

  • Published:
Aequationes mathematicae Aims and scope Submit manuscript

Abstract

In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most cusp singularities. By the Gauss-Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary (Theorem 2.20 in [4]) applied to the extended front of evolutoids of a hedgehog we obtain an integral equality for smooth periodic curves.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Aguilar-Arteaga, V.A., Ayala-Figueroa, R., González-Garcia, I., Jerónimo-Castro, J.: On evolutoids of planar convex curves II. Aequat. Math. 89, 1433–1447 (2015)

    Article  MathSciNet  Google Scholar 

  2. Cambraia Jr. A., Lemos, A.: On Affine Evolutoids, Quaestiones Mathematicae 1–10 (2018)

  3. Domitrz, W., Romero Fuster, M.C., Zwierzyński, M.: The geometry of the secant caustic of a planar curve. Differ. Geom. Its Appl. 78, 101797 (2021)

    Article  MathSciNet  Google Scholar 

  4. Domitrz, W., Zwierzyński, M.: The gauss-bonnet theorem for coherent tangent bundles over surfaces with boundary and its applications. J. Geom. Anal. 30, 3243–3274 (2020)

    Article  MathSciNet  Google Scholar 

  5. Domitrz, W., Zwierzyński, M.: The Gometry of the Wigner caustic and a decomposition of a curve into parallel arcs. Anal. Math. Phys. 12, 7 (2022)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. Springer, Berlin (1974)

    MATH  Google Scholar 

  8. Hamman, M.: A note on ovals and their evolutoids. Beitrage zur Algebra und Geometrie 50, 433–441 (2009)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Jerónimo-Castro, J., Rojas-Tapia, M.A., Velasco-Garcia, U., Yee-Romero, C.: As isoperimetric inequality for isoptic curves of convex bodies. Results Math. 75, 134 (2020)

    Article  Google Scholar 

  11. Martinez-Maure, Y.: Geometric Inequalities for Plane Hedgehogs, Demonstratio Mathematica, Vol. XXXII, No 1, (1999)

  12. Martinez-Maure, Y.: New notion of index for hedgehogs of \(\mathbb{R}^3\) and applications, Rigidity and related topics in Geometry. Eur. J. Combin. 31, 1037–1049 (2010)

    Article  Google Scholar 

  13. Martinez-Maure, Y.: Uniqueness results for the Minkowski problem extended to hedgehogs, Central European. J. Math. 10, 440–450 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Martins, L.F., Saji, K.: Geometric invariants of cuspidal edges. Canadian J. Math. 68(2), 445–462 (2016)

    Article  MathSciNet  Google Scholar 

  15. Martins, L.F., Saji, K., Umehara, M., Yamada, K.: Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts. Springer Proceedings in Mathematics & Statistics 154, 247–281 (2016)

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

    Article  MathSciNet  Google Scholar 

  17. Saji, K., Umeraha, M., Yamada, K.: Behavior of corank-one singular points on wave fronts. Kyushu J. Math. 62, 259–280 (2008)

    Article  MathSciNet  Google Scholar 

  18. Saji, K., Umeraha, M., Yamada, K.: The geometry of fronts. Ann. Math. 169, 491–529 (2009)

    Article  MathSciNet  Google Scholar 

  19. Saji, K., Umehara, M., Yamada, K.: Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts. J. Geom. Anal. 22(2), 383–409 (2012)

    Article  MathSciNet  Google Scholar 

  20. Saji, K., Umehara, M., Yamada, K.: An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications. J. Math. Soc. Jpn. 69(1), 417–457 (2017)

    Article  MathSciNet  Google Scholar 

  21. Umehara, M., Yamada, K.: Differential Geometry of Curves and Surfaces. World Scientific Publishing, Singapore (2017)

    Book  Google Scholar 

  22. Wolfram Research, Inc., Mathematica, Version 12.2, Champaign, IL (2020)

  23. Zwierzyśki, M.: The Improved Isoperimetric Inequality and the Wigner Caustic of Planar Ovals. J. Math. Anal. Appl. 442(2), 726–739 (2016)

    Article  MathSciNet  Google Scholar 

  24. Zwierzyśki, M.: The Constant Width Measure Set, The Spherical Measure Set and Isoperimetric Equalities for Planar Ovals, arXiv:1605.02930

  25. Zwierzyśki, M.: Isoperimetric equalities for rosettes. Int. J. Math. 31(05), 2050041 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662, Warsaw, Poland

    Michał Zwierzyński

Authors
  1. Michał Zwierzyński
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michał Zwierzyński.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zwierzyński, M. The singular evolutoids set and the extended evolutoids front. Aequat. Math. 96, 849–866 (2022). https://doi.org/10.1007/s00010-022-00873-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00010-022-00873-7

Keywords

Mathematics Subject Classification

Search

Navigation