Estimating the area of a triangle-circle intersection
Working Towards an Exact solution

Just some quick thought before I must run!

Ok, let us turn the problem on its head. What if one does not calculate the area of the triangle cut by the circle. Instead let's calculate the inverse of this. The area of three segments cut out of the circle. The beauty of this approach is that we can make a single piece of code that we can repeat three times. Of course some of the time the answer will be 0 but that is OK.



Image 1: Let's calculate this problem instead.

Now it turns out we know how to calculate the inverse, it is the segment of a circle.  So we can just say that $A_{int}=A_{circle}-A_1-A_2-A_3$. 



Image 2: We know this solution.

The formula for the area of a segment is simply $A_{sector}-A_{triangle}$ and that has a well known solution in literature. 

So now all you need to do is find the intersections between the edges and the circle in a space centered on the circle, calculate the angle between those and then calculate each segment. And the only special case is if no intersection exists. I dont know if this is fast but at least code should be somewhat easy to implement.