How to find the optimal approximation of a polyline dataset with a given number of nodes?

Question

Algorithms

It is necessary to construct a broken line (piecewise linear continuous function) with a predetermined number of nodes (points of breaks in the broken line) that best approximates this curve. Through the least squares method, I found a solution to a particular case when the nodes are fixed, but how to make the method itself give the best choice of these nodes? I don't even know how to get on. The implementation language is not important, only a formula or description of the iterative process is needed.

Answer 1

Too little data for a complete answer.
In general, a spline of the first order with the required number of nodes will suit you (this is a broken line)
Just a continuous curve (non-differentiable at break points)
The mathematics of this process can be found in the NURBS book or on the Internet For example, the theory of NURBS
splines is very deeply developed
nodes)
I didn’t look for a long time, but it seemed like NURBS ++ worked good
. There, the number of nodes and boundaries are usually parameterized.
Obviously, nodes should be placed closer to points with a minimum derivative module
(kinks, minima and maxima)