Invariants of curves in R P 2 and R

Let K be a smooth immersed curve in the plane. Fabricius-Bjerre [2] found the following relation among the double tangent lines, crossings, and inflections points for a generic K : T1 − T2 = C + (1/2)I where T1 and T2 are the number of exterior and interior double tangent lines of K , C is the number of crossings, and I is the number of inflection points. Here “generic” means roughly that the interesting attributes of the curve are invariant under small smooth perturbations. Fabricius-Bjerre remarks on an example due to Juel which shows that the theorem cannot be straightforwardly generalized to the projective plane. A series of papers followed. Halpern [5] re-proved the theorem and obtained some additional formulas using analytic techniques. Banchoff [1] proved an analogue of the theorem for piecewise linear planar curves, using deformation methods. Fabricius-Bjerre gave a variant of the theorem for curves with cusps [3]. Weiner [7] generalized the formula to closed curves lying on a 2–sphere. Finally Pignoni [6] generalized the formula to curves in real projective space, but his formula depends, both in the statement and in the proof, on the selection of a base point for the curve. Ferrand [4] relates the Fabricius-Bjerre and Weiner formulas to Arnold’s invariants for plane curves. Note that any formula for curves in RP2 is more general than one for curves in R2 , since one can specialize to curves in R2 by considering curves lying inside a small disk in RP2 .