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There are two different methods of computing a random ray bounce being compared in this section of the online book, the first one is: Pick a point $s$ anywhere inside the sphere, then compute a vector from a point $p$ on the surface to the point $s$ inside the sphere. The second method is: Pick a point $s$ anywhere on the surface of the sphere, then compute ...

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I can simplify the problem for you, but I'm not sure it's possible to do what you want in the general case. When it works, this solution will tell you which circle the ray intersects first; finding the exact intersection point is then simple. First, let's say that the ray starts at a point where $-1 < x \le 0$ and is traveling in the positive x and y ...

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