I've been trying to figure out the most straightforward way of doing this for a while, and would like to get some advice on new approaches, as the one I was using didn't work out at all. So here it is: The stadium billiard is defined as two semicircles joined by two tangent lines, as shown in the figure attached. Given the angle of a point in the boundary, my goal is to determine the arc length of the boundary, from theta=0 to the collision angle. To do that, I have the parameters of the length of the tangent lines, and the radius of the semicircles (figure.) What approach would fit best here? Thank you all for your time.