# 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?

• For a Bézier curve, simply picking equally spaced $t$ values will work. – Parcly Taxel May 6 '17 at 1:46
• You can check "Numerical Recipes" which have a long discussion about this topic. – Felix Marin May 9 '17 at 1:30

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.

This paper discusses the problem:

• 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:

For a recent paper, see

• 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.

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!