PrA

PrA is a simple ad-hoc parameter to define the "non-circularity" or eccentricity of a 2D object. This quantity is simply:

${\begin{alignedat}{2}\mathrm {PRA} ={\frac {Pr}{A}}\end{alignedat}}$

Where $P$ is the object's perimeter, $A$ is its surface area, and $r$ is an effective size (radius), computed based on the corresponding circle of the same area:

${\begin{alignedat}{2}r={\sqrt {\frac {A}{\pi }}}\end{alignedat}}$

This definition of PrA is convenient, since it provides a simple measure of eccentricity. In particular, for a circle one expects:

${\begin{alignedat}{2}\mathrm {PRA} &={\frac {Pr}{A}}\\&={\frac {(2\pi r)(r)}{\pi r^{2}}}\\&=2\end{alignedat}}$

Since a circle has the minimal perimeter (for a given area), this is a limiting value of PrA:

${\begin{alignedat}{2}\mathrm {PRA} \geq 2\end{alignedat}}$

And thus any non-circular object will have a larger PrA.