What should be minimized in order to best approximate a curve? I have a smooth parametrized curve $r(t), t \in [0,1]$ in the plane and I want to find a sequence $r(t_1), r(t_2), ... r(t_n)$ which "best" approximates the curve. I'd like to know how to define "best". 
Intuitively, I expect that the points should be more densely distributed in sections of the curve with high curvature. For example, I expect that samples of a parabola will be denser near its intersection with the line of symmetry. In the extreme case of a straight line, I expect the points to be uniformly distributed on the curve.
I suspect that there is some function F of r and the n points that is minimized/maximined at the "optimal" distribution of points. Ideally, there should be a simple algorithm for calculating these points from such a function. 
Consider, for example, $F = \sum_i \kappa_i$ where $\kappa_i$ is the curvature at $r(t_i)$. It seems reasonable to want to choose $t_1, ..., t_n$ that maximize $F$, but I imagine that there is a less ad hoc choice of $F$ ... one that suggests a natural algorithm for selecting $t_1, ..., t_n$. 
As a concrete example, find an algorithm for sampling n points on the parabola described parametrically by: $r(t) = (t, t^2), 0\leq t \leq 1$.
Any ideas? References? 
 A: A classic method is the Douglas-Peucker algorithm. You start with a densely sampled polyline, and recursively decide whether to keep or discard vertices based on how much approximation error discarding would introduce.
A: This paper discusses the problem:


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*H. Imai and M. Iri. Polygonal approximations of a curve-formulations and algorithms. In G. T. Toussaint, editor, Computational Morphology, pages 71—86. North- Holland, 1988.


This paper seems interesting:


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*P.L. Rosin, "Techniques for assessing polygonal approximations of curves", IEEE Transactions Pattern Analysis and Machine Intelligence, vol. 19, no. 6, pp. 659—666, 1997. DOI: 10.1109/34.601253


For a recent paper, see


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*Pankaj K. Agarwal, Sariel Har-Peled, Nabil H. Mustafa, and Yusu Wang. 2002. Near-Linear Time Approximation Algorithms for Curve Simplification. In Proceedings of the 10th Annual European Symposium on Algorithms (ESA '02), Springer-Verlag, pp. 29—41.


Adaptive arc-length computation is a good proxy for curvature-adaptive approximation. In this context, see this answer. 
A: These are good comments and references, but I think that the method that most closely answers the call is http://www.cs.technion.ac.il/~tess/publications/curve_sampling.pdf.
Thanks! 
