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.