Skip to main content Accessibility help
×
Home

Sextactic points on a simple closed curve

  • Gudlaugur Thorbergsson (a1) and Masaaki Umehara (a2)

Abstract

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Sextactic points on a simple closed curve
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Sextactic points on a simple closed curve
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Sextactic points on a simple closed curve
      Available formats
      ×

Copyright

References

Hide All
[Ar1] Arnold, V. I., A ramified covering of CP2 → S4, hyperbolicity and projective topology, (Russian), Sib. Mat. Zh., 29 (1988), 3647; English translation in: Sib. Math. J., 29 (1988), 717726.
[Ar2] Arnold, V. I., Topological Invariants of Plane Curves and Caustics, University Lecture Series 5, American Mathematical Society, Providence, Rhode Island, 1994.
[Ar3] Arnold, V. I., Remarks on the extatic points of plane curves, The Gelfand Mathematical Seminars, 19931995, Birkhäuser, Boston (1996), 1122.
[Ba] Barner, M., Über die Mindestanzahl stationärer Schmiegebenen bei geschlossenen strengkonvexen Raumkurven, Abh. Math. Sem. Univ. Hamburg, 20 (1956), 196215.
[Bs] Basset, A. B., On sextactic and allied conics, Quart. J., 46 (1915), 247252.
[Bt] Battaglini, G., Sui punti sestatici di una curva qualunque, Atti R. Acc. Lincei, Rend. (Serie quarta) IV 2 (1888), 238246.
[Bl1] Blaschke, W., Über affine Geometrie VIII: Die Mindestzahl der sextaktischen Punkte einer Eilinie, Leipziger Berichte, 69 (1917), 321324; Also in: Gesammelte Werke, Band 4, 153156. Thales Verlag, Essen, 1985.
[Bl2] Blaschke, W., Vorlesungen über Differentialgeometrie II, Affine Differentialgeometrie, Springer-Verlag, Berlin, 1923.
[Bo] Bol, G., Projektive Differentialgeometrie, 1. Teil., Vandenhoeck & Ruprecht, Göttingen, 1950.
[Ca1] Cayley, A., On the conic of five-pointic contact at any point of a plane curve, Philosophical Transactions of the Royal Society of London CXLIX (1859), 371400; Also in: The Collected Mathematical Papers, Vol. IV, Cambridge University Press, 1891.
[Ca2] Cayley, A., On the sextactic points of a plane curve, Philosophical Transactions of the Royal Society of London CLV (1865), 548578; Also in: The Collected Mathematical Papers, Vol. V, Cambridge University Press, 1892.
[Fa] Fabricius-Bjerre, Fr., On a conjecture of G. Bol, Math. Scand., 40 (1977), 194196.
[GMO] Guieu, L., Mourre, E. and Ovsienko, V. Yu., Theorem on six vertices of a plane curve via Sturm theory, The Arnold-Gelfand Mathematical Seminars, Birkhäuser, Boston (1997), 257266.
[IS] Izumiya, S. and Sano, T., Private Communication, 1998.
[Kn] Kneser, H., Neuer Beweis des Vierscheitelsatzes, Christiaan Huygens, 2 (1922/23), 315318.
[Mi] Miranda, R., Algebraic curves and Riemann surfaces, Graduate Studies in Mathematics 5, American Mathematical Society, Providence, Rhode Island, 1995.
[Mö] Möbius, A. F., Über die Grundformen der Linien der dritten Ordnung, Abhandlungen der Königl. Sächs. Gesellschaft der Wissenschaften, math.-phys. Klasse I (1852), 182; Also in: Gesammelte Werke, vol. II, Verlag von S. Hirzel, Leipzig, 1886, 89176.
[Mu1] Mukhopadhyaya, S., New methods in the geometry of a plane arc, I, Bull. Calcutta Math. Soc., 1 (1909), 3137; Also in: Collected geometrical papers, vol. I. Calcutta University Press, Calcutta (1929), 1320.
[Mu2] Mukhopadhyaya, S., Sur les nouvelles méthodes de géometrie, C. R. Séance Soc. Math. France, année 1933 (1934), 4145.
[TU1] Thorbergsson, G. and Umehara, M., A unified approach to the four vertex theorems, II, Differential and symplectic topology of knots and curves (Tabachnikov, S., ed.), Amer. Math. Soc. Transl., Ser. 2, 190 (1999), 229252.
[TU2] Thorbergsson, G. and Umehara, M., On global properties of flexes of periodic functions, preprint (2001).
[Um] Umehara, M., A unified approach to the four vertex theorems, I, Differential and symplectic topology of knots and curves (Tabachnikov, S., ed.), Amer. Math. Soc. Transl., Ser. 2, 190 (1999), 185228.
[Vi] Viro, A. O., Differential geometry “in the large” of plane algebraic curves, and integral formulas for invariants of singularities, (Russian), Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI), 231 (1995), 255268.
[Wa] Wall, C. T. C., Duality of real projective plane curves: Klein’s equation, Topology, 35 (1996), 355362.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed