# 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. An infinitely eccentric object would have $\mathrm {PRA} \to \infty$ .

### Ellipse

If the object is an ellipse, with equation:

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ Then the width is $2a$ and height $2b$ (we assume $a\geq b$ ), the foci are $(\pm c,0)$ for ${\textstyle c={\sqrt {a^{2}-b^{2}}}}$ . The eccentricity is:

$e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}$ A circle has $e=0$ , while increasingly squashed ellipses have values of $e$ closer and closer to $1$ . The area of an ellipse is:

$A=\pi ab$ The perimeter is not analytic but can be approximated very roughly by:

$P\approx \pi (a+b)$ Which yields:

{\begin{alignedat}{2}\mathrm {PRA} &={\frac {Pr}{A}}\\&={\frac {P\left({\sqrt {\frac {A}{\pi }}}\right)}{A}}\\&\approx {\frac {\pi (a+b)}{\pi ab}}\left({\sqrt {\frac {\pi ab}{\pi }}}\right)\\&\approx {\frac {(a+b)}{ab}}{\sqrt {ab}}\\&\approx {\frac {(a+b)}{\sqrt {ab}}}\\\end{alignedat}} One can establish a relationship between eccentricity and PrA by setting $b=1$ and considering $\displaystyle a \in [1, \infnty$ :