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I'm not up on my chaos theory, but mathematically speaking, I believe that one must essentially prove that a point is (or is not) exactly ON the curve before examining which side of the curve it is on.
The only way I can think of doing this is to see if the coordinates of the point satisfy the equation of the curve (which, of course, may not be known = you're screwed).
In other words, it may be infinitely close to the curve without actually being on it, which means that no matter how many iterations you do, it will never fall on the triangle.
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