Advertisement

Annali di Matematica Pura ed Applicata

, Volume 85, Issue 1, pp 301–306 | Cite as

Some vertex theorems proved by means of moebius transformations

  • Erhard Heil
Article

Summary

The 4-and 2n-vertex theorem for plane curves is proved by mapping one point to. The minimum number of vertices for the plane clover-leaf knot is found to be 6.

Keywords

Plane Curf Moebius Transformation Vertex Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Barner,Über die Mindestzahl stationärer Schmiegebenen bei geschlossenen strengkonvexen Raumkurven, Abhandlungen Math. Sem. Univ. Hamburg19 (1955), pp. 196–215.MathSciNetGoogle Scholar
  2. [2]
    W. Blaschke,Kreis nnd Kugel, Leipzig 1916.Google Scholar
  3. [3]
    W. C. Graustein,Extensions of the four-vertex theorem, Trans. Am. Math. Soc.41 (1937), pp. 9–23.CrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    O. Haupt,Zur Verallgemeinerung des Zweischeitelsatzes bei ebenen Kurven, Arch. Math.11 (1960), pp. 294–297.CrossRefMATHMathSciNetGoogle Scholar
  5. [5]
    O. Haupt, G. Aumann, C. Pauc,Differential- und Integralrechnung, vol. 2, 2nd ed., Berlin 1950.Google Scholar
  6. [6]
    O. Haupt, H. Künneth,Geometrische Ordnungen, Berlin-Heidelberg-New York 1967.Google Scholar
  7. [7]
    E. Heil,Scheitelsätze in der euklidischen, affinen und Minkowskischen Geometrie, Darmstadt 1967 (mimeographed).Google Scholar
  8. [8]
    S. B. Jackson,Vertices of plane curves, Bull. Am. Math. Soc.50 (1944), pp. 564–578.MATHCrossRefGoogle Scholar
  9. [9]
    C. Juel,Om simple cykliske Kurver, D. Kgl. Danske Vidensk. Selsk. Skrifter (7), naturvidensk. og. math. Afd.8 (1911), pp. 367–383.Google Scholar
  10. [10]
    A. Kneser,Bemerkungen über die Anzahl der Extreme der Krümmung auf gesclossenen Kurven und über verwandte Fragen in einer nichteuklidischen Geometrie, FestschriftH. Weber 1912, pp. 170–180.Google Scholar
  11. [11]
    S. Mukhopadhyaya,Extended minimum-number theorems of cyclic and sextactic points on a plane convex oval, Math. Zeitschr.33 (1931), pp. 648–662.CrossRefMATHGoogle Scholar
  12. [12]
    A. Ostrowski.Über Evoluten und Evolventen ebener Kurven, Arch-Math.6 (1955), pp. 170–179.CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    -- --,Vorlesungen über Differential- und Integralrechnung, vol 2, 2nd ed., Basel 1961.Google Scholar
  14. [14]
    F. Schuh,Bewijs van de stelling der vier toppen, Christiaan Huygens2 (1922/23), pp. 374–375.MATHGoogle Scholar
  15. [15]
    L. Vietoris,Ein einfacher Beweis des Vierscheitelsatzes der ebenen Kurven, Arch. Math.3 (1952), pp. 304–306.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    W. Vogt,Über monotongekrümmte Kurven, Journal r.a. Math.144 (1914), pp. 239–248.MATHGoogle Scholar
  17. [17]
    K. Zindler,Über konvexe Gebilde I, Monatsh. Math. Phys.30 (1920), pp. 87–102.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1970

Authors and Affiliations

  • Erhard Heil
    • 1
  1. 1.Germania

Personalised recommendations