In connection with Green's theorem, I encountered the notion of the 'directed area' inside a closed directed curve in the 2D plane (e.g., the complex plane). So the area inside the curve would be positive if the curve goes counterclockwise, and otherwise negative.
It seems this generalizes to self-intersecting curves, so that one would count an area positively for each counterclockwise curve it is in, and negatively for each clockwise curve it is in.
So, e.g., the 'directed area' of the straight-lined curve going through (-2,-2), (2,-2), (2,2), (-1,-1), (1,-1), (-2,2) (for a picture see this WolframAlpha link) would have an area of 13: 11 for the area enclosed once; plus twice the inner triangle of area 1, because that is 'twice enclosed' by the counterclockwise curve (-1,-1), (1,-1), (0,0).
So in that example, it seems that (0,-0.5) counts twice, (1,0) counts once, and (-3,0) counts zero times.
My questions:
First, what is this generalized notion of directed area for any closed curve, including intersecting ones, and where can I find more information on it? (Note that I've asked a related question here.)
And second, is there a more formal or simple way to define 'the number of times a point $\;z\;$ is counted for directed curve $\;C\;$'?
For the second question: Yes, this number is called the winding number of the curve around the respective point and has interesting representations in complex analysis and differential topology, see e.g. Wikipedia.
For the first point: Yes, there is a general formula to calculate that area. It uses Stokes in the derivation (I use the language of differential forms, if you are not familiar with it that would be a good point to start learning more about these things):
$$ A = \int_A dx \wedge dy = \int_A d(x \wedge dy) = \int_{\partial A} x \wedge dy = \int_C x dy = \int_{t_0}^{t_1} x(t) \frac{dy(t)}{dt} dt $$