# Detect “cusp” in parametric curve

I'm using the word "cusp" informally here, I apologize if there is a formal definition for it. What I'm looking for is a point where the derivative is non-continuous, I think.

I have a a sequence of two-dimensional points on a parametric curve (equations for the curve itself are unknown) and I want to find "sharp" points on the curve. I'm sorry for not being more clear, I don't have the background to use all the correct terminology.

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What is a «$2$-dimensional curve»? A curve in the plane? – Mariano Suárez-Alvarez Mar 1 '12 at 19:06
Yes, sorry. I guess it's actually a parametric equation (x(t), y(t)). – sh1ftst0rm Mar 1 '12 at 19:12
You can edit your question (and its title) to make it more clear, then :) – Mariano Suárez-Alvarez Mar 1 '12 at 19:12

You can't do this exactly if you only have discrete points on the curve, but you can estimate the "cuspiness" of the curve using the curvature of a circle through three consecutive points; this will give you an estimate of the curvature of the curve at the point in the middle. The curvature of the circle through three points is given by

$$\kappa=4\frac{\sqrt{s(s-a)(s-b)(s-c)}}{abc}\;,$$

where $s$ is the semiperimeter, $s=(a+b+c)/2$.

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After a quick trip to wikipedia, I think I have a basic grasp of curvature, but I'm not clear what a, b, and c are in your equation. I assume they relate to the three points, but what are they? -Thanks. – sh1ftst0rm Mar 2 '12 at 17:12
@bmearns: Sorry, those are the distances between the points, i.e. the lengths of the sides of the triangle they form. – joriki Mar 2 '12 at 17:17
Ah, thanks for clarifying. Sounds like it might be a workable solution for me. Thanks. – sh1ftst0rm Mar 2 '12 at 17:24