Difference between revisions of "PrA"

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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>
 +
In particular:
 +
:<math>
 +
\begin{alignat}{2}
 +
\frac{(a+1) }{ \sqrt{a} } & = \mathrm{PRA}  \\
 +
(a+1)^2 & = \mathrm{PRA}^2 a  \\
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a^2+2a+1- \mathrm{PRA}^2 a & = 0  \\
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(1)a^2+(2-\mathrm{PRA}^2 )a+(1) & = 0  \\
 +
\end{alignat}
 +
</math>
 +
From the quadratic equation:
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:<math>
 +
\begin{alignat}{2}
 +
a & = \frac{-(2-\mathrm{PRA}^2)\pm \sqrt{(2-\mathrm{PRA}^2)^2 - 4(1)(1)} }{2(1)} \\
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  & = \frac{1}{2} \left( -2+\mathrm{PRA}^2\pm \sqrt{(2-\mathrm{PRA}^2)^2 - 4} \right)\\
 +
  & = \frac{1}{2} \left( \mathrm{PRA}^2-2 \pm \sqrt{(2-\mathrm{PRA}^2)^2 - 4} \right)\\
 +
  & = \frac{1}{2} \left( \mathrm{PRA}^2-2 \pm \sqrt{4 -4\mathrm{PRA}^2 + \mathrm{PRA}^4 - 4} \right)\\
 +
  & = \frac{1}{2} \left( \mathrm{PRA}^2-2 \pm \sqrt{\mathrm{PRA}^4 - 4\mathrm{PRA}^2} \right)\\
 +
  & = \frac{1}{2} \left( \mathrm{PRA}^2-2 \pm \mathrm{PRA} \sqrt{\mathrm{PRA}^2 - 4} \right)\\
 +
\end{alignat}
 +
</math>
 +
Since <math>a \to \infty</math> as <math>P \to \infty</math>, we select the positive branch.
 +
:<math>
 +
\begin{alignat}{2}
 +
a & = \frac{1}{2} \left( \mathrm{PRA}^2-2 + \mathrm{PRA} \sqrt{\mathrm{PRA}^2 - 4} \right)\\
 +
a^2 & = \frac{1}{4} \left( \mathrm{PRA}^2-2 + \mathrm{PRA} \sqrt{\mathrm{PRA}^2 - 4} \right)^2\\
 +
\end{alignat}
 +
</math>
 +
And so:
 +
:<math>
 +
\begin{alignat}{2}
 +
e & = \sqrt{1 - \frac{1}{a^2}} \\
 +
  & = \sqrt{1 - \frac{4}{\left( \mathrm{PRA}^2-2 + \mathrm{PRA} \sqrt{\mathrm{PRA}^2 - 4} \right)^2}} \\
 +
\end{alignat}
 +
</math>
 +
We can convert into a width:height ratio (<math>a/b</math>) as:
 +
:<math>
 +
\begin{alignat}{2}
 +
\frac{a}{b} & = \sqrt{1 - e^2} \\
 +
  & = \frac{1}{2} \left( \mathrm{PRA}^2-2 + \mathrm{PRA} \sqrt{\mathrm{PRA}^2 - 4} \right)\\
 +
\end{alignat}
 +
</math>

Latest revision as of 10:55, 31 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 :

In particular:

From the quadratic equation:

Since as , we select the positive branch.

And so:

We can convert into a width:height ratio () as: