PrA is a simple ad-hoc parameter to define the "non-circularity" or eccentricity of a 2D object. This quantity is simply:
Where is the object's perimeter, is its surface area, and is an effective size (radius), computed based on the corresponding circle of the same area:
This definition of PrA is convenient, since it provides a simple measure of eccentricity. In particular, for a circle one expects:
Since a circle has the minimal perimeter (for a given area), this is a limiting value of PrA:
And thus any non-circular object will have a larger PrA. An infinitely eccentric object would have .
If the object is an ellipse, with equation:
Then the width is and height (we assume ), the foci are for . The eccentricity is:
A circle has , while increasingly squashed ellipses have values of closer and closer to . The area of an ellipse is:
The perimeter is not analytic but can be approximated very roughly by:
One can establish a relationship between eccentricity and PrA by setting and considering :
From the quadratic equation:
Since as , we select the positive branch.