Mathematics Colloquia and Seminars

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I will be talking about a paper by Fabricius-Bjerre on the Relation Between the Numbers of Singular Points and Singular Lines of a Plane Closed Curve. We consider a smooth closed curve, and count the numbers of double tangents, double points and inflection points. There is a known relation between these numbers. This relation is that the difference between the number of exterior tangents and the number of interior tangents is equal to the number of double points plus half the number of inflection points. In this paper we prove that if in addition to the afore-mentioned singular points we allow cusps of the first kind and of the second kind (to be defined later) that we have a more general formula relating all of these numbers.