I have a parallelogram, for which I have the "half width" vectors (that is, this was a unit square that has had an arbitrary affine transformation applied to it). I have another square, which we can arbitrarily say is the axis aligned unit square.

I want to determine the area of overlap of the parallelogram with the square. Or more precisely, I want to determine the percentage overlap of the parallelogram by the square.

I can think of some relatively straight-forward ways to approximate the area using monte-carlo techniques, but I'm wondering if there's a more analytic solution.

I can imagine, for instance, using something like Sutherland-Hodgman to construct the convex hull of the overlap, and then finding the area of that convex hull. But that seems like overkill.

Plus, this would be for a pixel shader (hopefully SM2, though I might bump it up to SM3 if absolutely necessary), so I'm a bit concerned about branching. Sutherland-Hodgman and then area calculation seem pretty hard to construct without branching, because of all the different cases (though I haven't really tried yet, so it's probably doable).

Anyway, I'm just wondering if anyone has any other ideas for how to approach the problem.