Metric for measuring linearity of finite set of points in $R^2$

Question

Suppose one has $n > 2$ points in $R^2$, and one wants to measure "how linear" they are. I want a metric such that (a) if all the points are in fact on the same line, the metric gives 1, (b) if all the points are equally spaced on a circle, the metric gives 0. What metrics are there that satisfy these criteria?

Answer 1

$\newcommand\R{\Bbb R}\newcommand\la{\lambda}$One good "metric" is as follows. Let $S$ be your finite set of $n\ge2$ distinct points in $\R^2$. Let $C(S)$ be the covariance matrix of the uniform probability distribution $U(S)$ over $S$: $$C(S)=\frac1n\sum_{s\in S}(s-\bar s)(s-\bar s)^\top,$$ where $\bar s:=\frac1n\sum_{s\in S}s$ is the mean of the distribution $U(S)$; as usual, we identify $\R^2$ with the set of all $2\times1$ real column matrices.

Then

• if all the points in $S$ lie on the same straight line, then the eigenvalues of $C(S)$ are $\la$ and $0$ for some real $\la>0$;
• if all the points in $S$ are equally spaced on a circle, then the eigenvalues of $C(S)$ are $\la$ and $\la$ for some real $\la>0$.

So, a good "metric" of linearity of $S$ is $$L(S)=1-\frac{\la_2(S)}{\la_1(S)},$$ where $\la_1(S)\ge\la_2(S)\ge0$ are the eigenvalues of the (positive-semidefinite nonzero) covariance matrix $C(S)$. Indeed, then $L(S)=1$ if (and only if) all the points in $S$ lie on the same straight line, and $L(S)=0$ if all the points in $S$ are equally spaced on a circle.