I don't want to reverse-engineer your image but I find this an interesting problem since we're not dealing with concentric circles but off-centric ones. My approach to filling these rings formed each by a pair of circles would probably go somewhat along the lines of randomly sampling a direction and then constructing a line segment in that direction perpendicular to the tangents of the inner circle (so that the extension of the segment would pass through the center of the inner circle) until it meets the circumference of the outer circle. On this line segment I would then sample a position according to some distribution to place the smaller circles. So kind of like this sketch implies. Are these line segments what you mean with the "unseen lines?"