# Difference between revisions of "PrA"

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

− | The area is: | + | A circle has <math>e=0</math>, while increasingly squashed ellipses have values of <math>e</math> closer and closer to <math>1</math>. The area of an ellipse is: |

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

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

Line 45: | Line 45: | ||

\end{alignat} | \end{alignat} | ||

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

+ | One can establish a relationship between eccentricity and PrA by setting <math>b=1</math> and considering <math>a \in [1, \infnty</math>: |

## Revision as of 15:57, 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:

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}**
: