For these purposes a sensible curve is one that may be given in parametric form with parametric functions of bounded variation. Such a curve is the set of points {<x(p), y(p)>|0<=p<=1} where x and y are continuous functions of bounded variation, defined on the interval [0, 1].

∫y dx = ∫y(p) d(x(p))
in the sense of the Stieltjes integral, is defined for such curves and is the intersection’s area, at least for Jordan curves, which do not intersect themselves.

By parametric curves intersecting finitely I mean that there are only a finite number of pairs of parameter values where the two curves coincide.

Note that two curves may have a finite number of intersections, but not intersect finitely!

A function is of bounded variation if it can be expressed as the sum of a monotonically increasing function and a monotonically decreasing function.

Consider two closed sensible curves that intersect finitely. Let {<pi, qi>} be the set of parameter value pairs where they intersect in some irrevelant order. I claim that the intergral over the plane of the product of their winding numbers (area of their intersection, if they are simple), is wnq(<xp(0), yp(0)>)