On 13 Jul '08, at 9:14 PM, Graham Cox wrote:
Since apps like Adobe Illustrator and Flash can perform these operations on bezier paths, clearly there are methods that exist for doing it, even though the mathematicians claim otherwise (what they are claiming is the lack of a general all-purpose solution to the arbitrary intersection of bezier paths, but for real-world graphics there wouldn't be too many pathological cases to worry about).
What the mathematicians balk at isn't the pathological cases; rather, getting exact results.
It's not possible to find the exact points of intersection of cubic curves, because it requires factoring a sixth-order polynomial, for which there are no known general techniques. The best you can do is to find approximations, using algorithms like Newton's Method. You can make the approximations as precise as you need by increasing the number of iterations. That's what existing programs do.
—Jens
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