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https://drive.google.com/open?id=1Y2hYkRG94whLroK20Zmce07jslyRL3zZ

I couldn't get the image to load. So above is a link to an image of the problem. I am trying to find a general solution to the intersection of the two general closed loops.

In the image Ai and Aj are just the area of each closed loops and Aij is the area of the intersection of the two closed loops. ri and rj are just parameterizations of the positions around the closed loop.

Their is a general equation for the area of a closed loop defined by a line integral and I was wondering if there might be a line integral for the area of intersection between two closed loops.