On the complexity of curve fitting algorithms

TL;DR

This paper analyzes the computational complexity of a common polynomial curve fitting algorithm, revealing conditions under which it can be simplified, especially for circle fitting but not for ellipses or hyperbolas.

Contribution

It identifies specific conditions that allow complexity reduction in polynomial curve fitting algorithms, particularly for circles.

Findings

Complexity reduction is possible for circle fitting.

No complexity reduction for ellipses or hyperbolas.

Provides precise conditions for when reduction is feasible.

Abstract

We study a popular algorithm for fitting polynomial curves to scattered data based on the least squares with gradient weights. We show that sometimes this algorithm admits a substantial reduction of complexity, and, furthermore, find precise conditions under which this is possible. It turns out that this is, indeed, possible when one fits circles but not ellipses or hyperbolas.