I am trying to determine a metric for measuring cluster stretch. Let $C$ be a cluster of points $P\_0, P\_1,...,P\_n$ in a two dimensional space with the same units.


I need a metric that will allow me to differentiate clusters that are long and thin from other clusters. 


I imagine it like a function that would be close to 0 when the cluster is shaped like a circle or a square and would be close to 1 when the cluster is shaped like a long and thin line or curve.


I thought of finding the length of the cluster by finding the shortest path between extreme points of the cluster, then finding the shortest distance from each point to the center line, then finding the mean distance between any point and the center line and then getting the ratio.


The ratio between the mean width of the cluster and its length would give a good estimation of the cluster stretch.


Unfortunately, this approach is quite difficult to implement in reality, so I thought I would ask for ideas. 


Maybe I should calculate cluster boundaries and then go from there? How about noise problems?


Thanks a lot for your help!



Determining the stretch of a cluster of points
There are many *roundness* measures that have been explored for different applications, which may give you ideas.
A good source for roundness in image processing is this paper, which analyzes
and compares several different measures:



> 
> Ritter, Nicola, and James Cooper. "New resolution independent measures of circularity." *Journal of Mathematical Imaging and Vision* 35.2 (2009): 117-127.
>  ([***Springer link***](http://link.springer.com/article/10.1007/s10851-009-0158-x).)
> 
> 
> 


The [***Wikipedia article on roundness***](https://en.wikipedia.org/wiki/Roundness_(object)) could also be useful to you.
Much depends on your application. I doubt there is one "best" measure.


