sorry, I wasn't clear above about which breakpoints. I mean the left breakpoint of the left arc and the right breakpoint of the right arc involved in the circle event. Obviously, if the circle is centered between these breakpoints, i.e a vertical line from its center intercepts one of the involved, and the event will occur below where the sweepline is now, it's possible that they will converge. I'm pretty sure this simple test will eliminate non-converging cases and accept converging cases without involving a great deal of calculation.
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