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


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.


Simple Proof Double Point Supporting Line Double Supporting Closed Polygon 
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.


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  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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