re detecting double points in bezier curves renner adobe com john renner writes in article almaden ibm com capelli vnet ibm com ron ca in ia b w w oeinck waterland wlink nl ferdinand oeinck writes i m looking for any information on detecting and or calculating a double point and or cusp in a bezier curve see maureen stone and tony derose a geometric characterization of parametric cubic curves acm tog vol no july pp i ve used that reference and found that i needed to go to their original tech report maureen stone and tony derose characterizing cubic bezier curves xerox edl december first thanks to all who replied to my original question i ve implemented the ideas from the article above and i m very satisfied with the results i needed it for my bezier curve approximation routine in some cases generating offset curves loops can occur i now have a fast method of detecting the generation of a curve with a loop although i did not follow the article above strictly the check if the fourth control point lies in the the loop area which is bounded by two parabolas and one ellips is too complicated instead i enlarged the loop area and surrounded it by for straight lines the check is now simple and fast and my approximation routine never ever outputs self intersecting bezier curves again ferdinand
