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.
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, $s=(a+b+c)/2$.