# Difference between revisions of "PrA"

KevinYager (talk | contribs) (Created page with "'''PrA''' is a simple ad-hoc parameter to define the "non-circularity" or [https://en.wikipedia.org/wiki/Ellipse eccentricity] of a 2D object. This quantity is simply: :<math>...") |
KevinYager (talk | contribs) |
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\end{alignat} | \end{alignat} | ||

</math> | </math> | ||

− | And thus any non-circular object will have a larger PrA. | + | And thus any non-circular object will have a larger PrA. An infinitely eccentric object would have <math>\mathrm{PRA} \to \infty</math>. |

+ | ===Ellipse=== | ||

+ | If the object is an [https://en.wikipedia.org/wiki/Ellipse ellipse], with equation: | ||

+ | : <math>\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1</math> | ||

+ | Then the width is <math>2a</math> and height <math>2b</math> (we assume <math>a \ge b</math>), the foci are <math>(\pm c, 0)</math> for <math display="inline">c = \sqrt{a^2-b^2}</math>. The eccentricity is: | ||

+ | : <math>e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}</math> | ||

+ | The area is: | ||

+ | : <math>A = \pi a b</math> | ||

+ | The perimeter is not analytic but can be approximated very roughly by: | ||

+ | : <math>P \approx \pi (a +b)</math> | ||

+ | Which yields: | ||

+ | :<math> | ||

+ | \begin{alignat}{2} | ||

+ | \mathrm{PRA} & = \frac{Pr}{A} \\ | ||

+ | & = \frac{P \left( \sqrt{\frac{A}{\pi}} \right) }{A} \\ | ||

+ | & \approx \frac{\pi(a+b) }{\pi a b} \left( \sqrt{\frac{\pi a b}{\pi}} \right) \\ | ||

+ | & \approx \frac{(a+b) }{ a b} \sqrt{a b} \\ | ||

+ | & \approx \frac{(a+b) }{ \sqrt{a b} } \\ | ||

+ | \end{alignat} | ||

+ | </math> |

## Revision as of 15:49, 12 May 2022

**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 .

### Ellipse

If the object is an ellipse, with equation:

Then the width is and height (we assume ), the foci are for . The eccentricity is:

The area is:

The perimeter is not analytic but can be approximated very roughly by:

Which yields: