I am trying to find the the sharp corners of a NURBS curve. For this problem I define a limit curvature. I am trying to find the sections on the curve that has a curvature higher then this value. One option is to interpolate over the curve and calculate curvature for all values but it may take time and some sharp points are likely to be missed. Any ideas about how to find these sections in an effective way?
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Computing the derivative of the curvature analytically, I guess that you will find a (terrible) expression with a polynomial at the numerator. A good polynomial solver will allow you to find the roots, hence the extrema, to split the curve in sections with a monotonic curvature, and from there find the precise solutions of k=c by regula falsi or similar.
A much simpler approach is by flattening the curve (convert to a smooth polyline) and estimating the local curvature on all triples of consecutive points (using their circumscribed circle). High curvature sections will probably also be detectable by anomalies in the point density while flattening.
The benefit of flattening over uniform sampling is that it auto-adjusts the point density.
Another idea is to resort to a method of approximation of curves by circular arcs (this can be compared to a second order flattening operation). You will find a few papers on the topic (do not confuse with circle approximation by curves), but usually these methods are complex.
Maybe it is also possible to devise an analytic formula for a lower bound on the NURBS curvature in a given interval and use that to implement a bisection approach.
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This is a very old question, but: Sharp corners in a NURBS curve will be the result only of multiplicity in the knot vector, especially having a multiplicity that is related to the degree of the curve.
See this answer.
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