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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.