algorithm - Area of Intersection of Two Rotated Rectangles

Question

I have two 2D rectangles, defined as an origin (x,y) a size (height, width) and an angle of rotation (0-360°). I can guarantee that both rectangles are the same size.

I need to calculate the approximate area of intersection of these two rectangles.

The calculation does not need to be exact, although it can be. I will be comparing the result with other areas of intersection to determine the largest area of intersection in a set of rectangles, so it only needs to be accurate relative to other computations of the same algorithm.

I thought about using the area of the bounding box of the intersected region, but I'm having trouble getting the vertices of the intersected region because of all of the different possible cases:

I'm writing this program in Objective-C in the Cocoa framework, for what it's worth, so if anyone knows any shortcuts using NSBezierPath or something you're welcome to suggest that too.

Answer 1

To supplement the other answers, your problem is an instance of line clipping, a topic heavily studied in computer graphics, and for which there are many algorithms available. If you rotate your coordinate system so that one rectangle has a horizontal edge, then the problem is exactly line clipping from there on.

You could start at the Wikipedia article on the topic, and investigate from there.