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Point projection on Bézier curves can be easily accomplished using Newton Iteration to try to minimize the dot product between the vector connecting the point P and its projection on curve C and the curve tangent vector C' at the same parameter. When the dot product is zero we have a perpendicular projection.

Am I right saying that this approach cannot work for points on the curve itself? And that if the point P is on the curve I need to try to minimize the distance between the two points P and C?

Is there any other approach that works in all cases, both for points on the curve and away from the curve without involving a distance check? In my case it's impossible to estimate a tolerance for the distance a priori.

Thanks.

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I think your approach should work fine also for points on the curve. Since the zero comes not from the tangent vector but from the difference between the point and the projection, you're effectively just finding the zero of that difference, which should get you to the point. Why do you expect problems?

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    @devdept: It refers to the at-username that you're putting in front of your comments. For instance, in this case I got notified even though you misspelled my name :-)2011-12-23