Difference between revisions of "PrA"

From GISAXS
Jump to: navigation, search
(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>...")
 
Line 25: Line 25:
 
\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 16:49, 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathrm{PRA} = \frac{Pr}{A} \end{alignat} }

Where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} is the object's perimeter, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is its surface area, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is an effective size (radius), computed based on the corresponding circle of the same area:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} r = \sqrt{\frac{A}{\pi}} \end{alignat} }

This definition of PrA is convenient, since it provides a simple measure of eccentricity. In particular, for a circle one expects:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathrm{PRA} & = \frac{Pr}{A} \\ & = \frac{(2 \pi r)(r)}{\pi r^2} \\ & = 2 \end{alignat} }

Since a circle has the minimal perimeter (for a given area), this is a limiting value of PrA:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathrm{PRA} \geq 2 \end{alignat} }

And thus any non-circular object will have a larger PrA. An infinitely eccentric object would have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{PRA} \to \infty} .

Ellipse

If the object is an ellipse, with equation:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1}

Then the width is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2a} and height Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2b} (we assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \ge b} ), the foci are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\pm c, 0)} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle c = \sqrt{a^2-b^2}} . The eccentricity is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}}

The area is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \pi a b}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \approx \pi (a +b)}

Which yields:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }
Retrieved from "http://gisaxs.com/index.php?title=PrA&oldid=6123"