I don’t understand the point you’re trying to make here:
Why do you want to use another sensor rather than making use of the built-in sensors on the 3pi?
The line-break in your course picture doesn’t seem like it will be as difficult to deal with as the acute angle or the angled transition from black to white. As long as you’re going straight and centered over the line when you hit the break, the 3pi should be able to detect the loss of the line while being pointed in the proper direction to pick the line up again across the gap; just program the 3pi to keep driving straight when the line suddenly disappears (but don’t do this when you lose the line to the right or the left). Note that you might need to drive the two motors at slightly different speeds in order to make the 3pi drive straight across the gap due to random variations in your 3pi’s motors. I suggest you conduct some tests to find left and right speeds close to the max speed of your line-following algorithm that cause your particular robot to drive in a straight line, and use these two speeds when the line abruptly disappears. You might also want to include code that shuts down the motors if you don’t find the line again after a certain period of time to prevent your 3pi from driving uncontrollably into a wall or spectators if things go unexpectedly wrong.
Note that you have to be careful not to confuse the sharp kink in the line with the gap, because it seems like you’re also likely to abruptly lose the line if you hit that acute angle at a high speed. You should be able to use the outer sensors on the 3pi to see the sharp-angle turn coming, but it might take some clever coding.
I’d love to see any videos you have of your 3pi competing!