Revision as of 16:57, 12 May 2022 by KevinYager (talk | contribs) (Ellipse)
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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:

Which yields:

One can establish a relationship between eccentricity and PrA by setting and considering Failed to parse (unknown function "\infnty"): {\displaystyle a \in [1, \infnty} :