Journal of Geometry

, Volume 13, Issue 2, pp 126–132 | Cite as

A proof of a relation between the numbers of singularities of a closed polygon

  • Fr. Fabricius-Bjerre
Article

Abstract

Let (P) denote a closed plane polygon, te and ti the numbers of exterior and interior double supporting lines, respectively, d the number of double points and i the number of inflectional edges of (P). T.F. Banchoff has shown that these numbers satisfy the relation te−ti=d+1/2i.

A new and simple proof of this equation is presented.

Keywords

Simple Proof Double Point Supporting Line Double Supporting Closed Polygon 

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References

  1. [1]
    Banchoff, T.F., Global geometry of polygons I, The Theorem of Fabricius-Bjerre, Proc. Amer. math. Soc. 45 (1974), 237–241.Google Scholar
  2. [2]
    Fabricius-Bjerre, Fr., On the double tangents of plane curves, Math. Scand. 11 (1962), 113–116.Google Scholar
  3. [3]
    Fabricius-Bjerre, Fr., A theorem on closed polygons in the projective plane, Nord. Mat. Tids. 10 (1962), 143–146.Google Scholar
  4. [4]
    Fabricius-Bjerre, Fr., A relation between the numbers of singular points and singular lines of a plane closed curve, Math. Scand. 40 (1977), 20–24.Google Scholar
  5. [5]
    Halpern, B., Double normals and tangent normals for polygons, Proc. Amer. Math. Soc. 51 (1975), 434–437.Google Scholar

Copyright information

© Birkhäuser Verlag 1979

Authors and Affiliations

  • Fr. Fabricius-Bjerre
    • 1
  1. 1.Mathematical InstituteThe Technical University of DenmarkLyngbyDenmark

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