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?