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1974 (Publication #15)

Global Geometry of Polygons I: The Theorem of Fabricius-Bjerre

Proc. A.M.S. 45 (1974), 237-241.

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Suppose X:[a,b] -> E^2 is a closed curve with C crossings and F inflection points (or inflection intervals), where C and F are finite. Furthermore, assume that X has finitely many double support lines. A support line is a line through two points of a curve such that each point has a neighborhood lying on one side of the line. Let I(t) be the number for which the two neighborhoods lie on opposite sides of the line, and II(T) be the number for which the two neighborhoods lie on the same side. Note that C, F, I(t), and II(t) are all zero if ZX is a convex curve. Various examples suggest that C + F/2 + I(t) - II(t) = 0, and this was proven by Halpern [2] for particular smooth curves.

This article provides an elementary proof of this conjecture for polygonal planar curves. This result will generalize to mappings of 1-manifolds into the plane. The techniques used will relate somewhat to the polyhedral analogue of catastrophe theory.