Skip to main content
SpringerLink
Log in

Osculation in General: An Approach

  • Conference paper
  • First Online:
Geometry and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 72))

  • 1153 Accesses

Abstract

This is an essay about osculation, that is, the tangency of highest order of different types, of different objects (hyperplanes, spheres, cyclides, and others), and arbitrary hypersurfaces.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. A. Bartoszek, P. Walczak, Sz. Walczak, Dupin cyclides osculating surfaces, preprint, 2012.

    Google Scholar 

  2. G. Cairns, R. W. Sharpe and L. Webb. Conformal invariants for curves in three dimensional space forms, Rocky Mountain J. Math. 24 (1994), 933–959.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. E. Cecil, Q. Chi, and G. R. Jensen, Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. 166 (2007), 1–76.

    Article  MATH  MathSciNet  Google Scholar 

  4. G. Darboux, Sur le contact des courbes et des surfaces, Bull. sci. math. et astr., 4 (1880), 348–384.

    MATH  Google Scholar 

  5. D. DeTurck, H. Gluck, D. Pomerleano, D. S. Vick, The Four Vertex Theorem and Its Converse, Notices of the Amer. Math. Soc. 54 (2007), 192–207.

    MATH  MathSciNet  Google Scholar 

  6. J. Drach, Sur les lignes d’osculation quadrique des surfaces, C. R. Acad. Sci. Paris 224 (1947), 309–312.

    MATH  MathSciNet  Google Scholar 

  7. J. Drach, Détermination des lignes d’osculation quadrique (lignes de Darboux) sur les surfaces cubiques. Lignes asymptotiques de la surface de Bioche, C. R. Acad. Sci. Paris 226 (1948), 1561–1564.

    MATH  MathSciNet  Google Scholar 

  8. A. Fialkov, The conformal geometry of curves, Trans. Amer. Math. Soc. 51 (1942), 435–501.

    Article  MathSciNet  Google Scholar 

  9. A. Fialkov, Conformal differential geometry of a subspace, Trans. Amer. Math. Soc. 56 (1944), 309–433.

    Article  MathSciNet  Google Scholar 

  10. D. Fuchs, S. Tabachnikov, Mathematical Omnibus, Amer. Math. Soc., Providence 2007.

    MATH  Google Scholar 

  11. R. Miyaoka, Isoparametric hypersurfaces with (g,m)=(6,2), Ann. of Math. 177 (2013), 53–110.

    Article  MATH  MathSciNet  Google Scholar 

  12. S. Mukhopadhyaya, New methods in the geometry of a plane arc, Bull. Calcutta Math. Soc. 1 (1909), 21–27.

    Google Scholar 

  13. T. Olivier, J. École Polytechnique, 15 (1835), 61–91, 252–263.

    Google Scholar 

  14. R. Uribe-Vargas, Four-vertex theorems in higher-dimensional spaces for a larger class of curves than theconvex ones, C. R. Acad. Sci. Paris St’er. I Math. 330 (2000), 1085–1090.

    Article  MATH  MathSciNet  Google Scholar 

  15. Y.-C. Wong, On an explicit charcterization of spherical curves, Proc. Amer. Math. Soc. 34 (1972), 239–242.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Supported by the Polish NSC grant 6065/B/H03/2011/40.

Author information

Authors and Affiliations

  1. Wydział Matematyki i Informatyki Uniwersytet Łódzki, Łódź, Poland

    Paweł G. Walczak

Authors
  1. Paweł G. Walczak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paweł G. Walczak .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Haifa, Haifa, Israel

    Vladimir Rovenski

  2. Faculty of Mathematics, University of Łód´z', Łód´z', Poland

    Paweł Walczak

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Walczak, P.G. (2014). Osculation in General: An Approach. In: Rovenski, V., Walczak, P. (eds) Geometry and its Applications. Springer Proceedings in Mathematics & Statistics, vol 72. Springer, Cham. https://doi.org/10.1007/978-3-319-04675-4_7

Download citation

Publish with us

Policies and ethics

Search

Navigation