The Fabricius-Bjerre theorem states that for a generic curve in the plane, the number of crossings plus half the number of inflections plus the number of opposite-side double tangencies is equal to the number of same-side double tangencies. We will define each of the quantities referred to in the theorem and look at some examples before we give the original (and very beautiful) proof of Fabricius-Bjerre himself.
In the case of polygons with vertices in general position, similar definitions are given for crossings, opposite-side and same-side double tagencies, and inflection edges. We will sketch the proof of Tom Banchoff for the polygonal version of the Fabricius-Bjerre theorem, which is based on a deformation argument.
In the end, we will look at generic curves in the sphere, and give the Spherical Fabricius-Bjerre formula.