Difference between revisions of "PrA"

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(Ellipse)
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\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>:
+
One can establish a relationship between eccentricity and PrA by setting <math>b=1</math> and considering <math>a \in [1, \infty]</math>:
 +
:<math>
 +
\begin{alignat}{2}
 +
\mathrm{PRA} & \approx \frac{(a+1) }{ \sqrt{a} } \\
 +
e & = \sqrt{1 - \frac{1}{a^2}}
 +
\end{alignat}
 +
</math>
 +
So:

Revision as of 17:05, 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 :

So: