The rotation index of a plane curve

The rotation index of a smooth closed plane curve is the number of complete rotations that a tangent vector to the curve makes as it goes around the curve.  According to a classical theorem of Heinz Hopf in 1935, the rotation index
of a piecewise smooth closed plane curve with no self-intersections is +1 or -1, depending on whether the curve is oriented counterclockwise or clockwise.
I will give a generalization of Hopf's theorem by allowing the curve to have self-intersections. The rotation index is then given by a localization formula,
as a sum of local indices at self-intersections.
I will also discuss possible higher-dimensional generalizations.