PrA - Revision history

KevinYager at 14:55, 31 May 2022

2022-05-31T14:55:35Z

KevinYager: /* Ellipse */

2022-05-12T21:37:43Z

‎Ellipse

KevinYager: /* Ellipse */

2022-05-12T21:27:39Z

‎Ellipse

KevinYager at 21:05, 12 May 2022

2022-05-12T21:05:42Z

KevinYager: /* Ellipse */

2022-05-12T20:57:18Z

‎Ellipse

KevinYager at 20:49, 12 May 2022

2022-05-12T20:49:43Z

KevinYager: 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: : $..."$

2022-05-12T20:35:27Z

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>..."

New page

'''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>
\begin{alignat}{2}
\mathrm{PRA} = \frac{Pr}{A}
\end{alignat}
</math>
Where <math>P</math> is the object's perimeter, <math>A</math> is its surface area, and <math>r</math> is an effective size (radius), computed based on the corresponding circle of the same area:
:<math>
\begin{alignat}{2}
r = \sqrt{\frac{A}{\pi}}
\end{alignat}
</math>
This definition of PrA is convenient, since it provides a simple measure of eccentricity. In particular, for a circle one expects:
:<math>
\begin{alignat}{2}
\mathrm{PRA} & = \frac{Pr}{A} \\
& = \frac{(2 \pi r)(r)}{\pi r^2} \\
& = 2
\end{alignat}
</math>
Since a circle has the minimal perimeter (for a given area), this is a limiting value of PrA:
:<math>
\begin{alignat}{2}
\mathrm{PRA} \geq 2
\end{alignat}
</math>
And thus any non-circular object will have a larger PrA.

@@ Line 84: / Line 84: @@
 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>

@@ Line 65: / Line 65: @@
 \begin{alignat}{2}
 a & = \frac{-(2-\mathrm{PRA}^2)\pm \sqrt{(2-\mathrm{PRA}^2)^2 - 4(1)(1)} }{2(1)} \\
-   & = \frac{1}{2} \left( -2+\mathrm{PRA}^2\pm \sqrt{(2+\mathrm{PRA}^2)^2 - 4} \right)\\
+   & = \frac{1}{2} \left( -2+\mathrm{PRA}^2\pm \sqrt{(2-\mathrm{PRA}^2)^2 - 4} \right)\\
-   & = \frac{1}{2} \left( \mathrm{PRA}^2\pm \sqrt{(2+\mathrm{PRA}^2)^2 - 4} -2 \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\pm \sqrt{(2+\mathrm{PRA}^2)^2 - 4} -2 \right)\\
+   & = \frac{1}{2} \left( \mathrm{PRA}^2-2 \pm \sqrt{4 -4\mathrm{PRA}^2 + \mathrm{PRA}^4 - 4} \right)\\
 \end{alignat}
 </math>
 Since <math>a \to \infty</math> as <math>P \to \infty</math>, we select the positive branch.

@@ Line 52: / Line 52: @@
 \end{alignat}
 </math>
-So:
+In particular:

@@ Line 45: / Line 45: @@
 \end{alignat}
 </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>:

@@ Line 31: / Line 31: @@
 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:
+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>
 The perimeter is not analytic but can be approximated very roughly by:
@@ Line 45: / Line 45: @@
 \end{alignat}
 </math>

@@ Line 25: / Line 25: @@
 \end{alignat}
 </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>.

PrA - Revision history

KevinYager at 14:55, 31 May 2022

KevinYager: /* Ellipse */

KevinYager: /* Ellipse */

KevinYager at 21:05, 12 May 2022

KevinYager: /* Ellipse */

KevinYager at 20:49, 12 May 2022

KevinYager: 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: :..."

KevinYager: 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: : $..."$