Metallic MOS: Difference between revisions

Line 1,059:

In particular, we see that the denominator of the weighted ratio finds itself on the same side of the ratio as the weight.

The full derivation follows. With lower bounding ratio <math>\frac{a_1}{a_2}</math> and upper bounding ratio <math>\frac{b_1}{b_2}</math>, we have a mediant of <math>\frac{\~~phia_1~~ + b_1}{\~~phia_2~~ + b_2}</math>. So then the segment from the lower bounding ratio to the mediant has length

The full derivation follows. With lower bounding ratio <math>\frac{a_1}{a_2}</math> and upper bounding ratio <math>\frac{b_1}{b_2}</math>, we have a mediant of <math>\frac{\phi a_1 + b_1}{\phi a_2 + b_2}</math>. So then the segment from the lower bounding ratio to the mediant has length

<math>

\require{cancel}

\begin{align}

s_1 &= \frac{\~~phia_1~~ + b_1}{\~~phia_2~~ + b_2} - \frac{a_1}{a_2} \\

s_1 &= \frac{\phi a_1 + b_1}{\phi a_2 + b_2} - \frac{a_1}{a_2} \\

&= \frac{a_2(\~~phia_1~~ + b_1)}{a_2(\~~phia_2~~ + b_2)} - \frac{a_1(\~~phia_2~~ + b_2)}{a_2(\~~phia_2~~ + b_2)} \\

&= \frac{a_2(\phi a_1 + b_1)}{a_2(\phi a_2 + b_2)} - \frac{a_1(\phi a_2 + b_2)}{a_2(\phi a_2 + b_2)} \\

&= \frac{a_2(\~~phia_1~~ + b_1) - a_1(\~~phia_2~~ + b_2)}{a_2(\~~phia_2~~ + b_2)} \\

&= \frac{a_2(\phi a_1 + b_1) - a_1(\phi a_2 + b_2)}{a_2(\phi a_2 + b_2)} \\

&= \frac{\~~phia_1a_2~~ + a_2b_1 - \~~phia_1a_2~~ - a_1b_2}{\~~phia_2~~^2 + a_2b_2} \\

&= \frac{\phi a_1a_2 + a_2b_1 - \phi a_1a_2 - a_1b_2}{\phi a_2^2 + a_2b_2} \\

&= \frac{\cancel{\~~phia_1a_2~~} + a_2b_1 - \cancel{\~~phia_1a_2~~} - a_1b_2}{\~~phia_2~~^2 + a_2b_2} \\

&= \frac{\cancel{\phi a_1a_2} + a_2b_1 - \cancel{\phi a_1a_2} - a_1b_2}{\phi a_2^2 + a_2b_2} \\

&= \frac{a_2b_1 - a_1b_2}{\~~phia_2~~^2 + a_2b_2}

&= \frac{a_2b_1 - a_1b_2}{\phi a_2^2 + a_2b_2}

\end{align}

</math>

Line 1,078:

\require{cancel}

\begin{align}

s_2 &= \frac{b_1}{b_2} - \frac{\~~phia_1~~ + b_1}{\~~phia_2~~ + b_2} \\

s_2 &= \frac{b_1}{b_2} - \frac{\phi a_1 + b_1}{\phi a_2 + b_2} \\

&= \frac{b_1(\~~phia_2~~ + b_2)}{b_2(\~~phia_2~~ + b_2)} - \frac{b_2(\~~phia_1~~ + b_1)}{b_2(\~~phia_2~~ + b_2)} \\

&= \frac{b_1(\phi a_2 + b_2)}{b_2(\phi a_2 + b_2)} - \frac{b_2(\phi a_1 + b_1)}{b_2(\phi a_2 + b_2)} \\

&= \frac{b_1(\~~phia_2~~ + b_2) - b_2(\~~phia_1~~ + b_1)}{b_2(\~~phia_2~~ + b_2)} \\

&= \frac{b_1(\phi a_2 + b_2) - b_2(\phi a_1 + b_1)}{b_2(\phi a_2 + b_2)} \\

&= \frac{\~~phia_2b_1~~ + b_1b_2 - \~~phia_1b_2~~ - b_1b_2}{\~~phia_2b_2~~ + b_2^2} \\

&= \frac{\phi a_2b_1 + b_1b_2 - \phi a_1b_2 - b_1b_2}{\phi a_2b_2 + b_2^2} \\

&= \frac{\~~phia_2b_1~~ + \cancel{b_1b_2} - \~~phia_1b_2~~ - \cancel{b_1b_2}}{\~~phia_2b_2~~ + b_2^2} \\

&= \frac{\phi a_2b_1 + \cancel{b_1b_2} - \phi a_1b_2 - \cancel{b_1b_2}}{\phi a_2b_2 + b_2^2} \\

&= \frac{\~~phia_2b_1~~ - \~~phia_1b_2~~}{\~~phia_2b_2~~ + b_2^2}

&= \frac{\phi a_2b_1 - \phi a_1b_2}{\phi a_2b_2 + b_2^2}

\end{align}

</math>

Line 1,092:

\require{cancel}

\begin{align}

r &= \frac{( \~~phia_2b_1~~ - \~~phia_1b_2~~ )(\~~phia_2~~^2 + a_2b_2)}{(\~~phia_2b_2~~ + b_2^2)( a_2b_1 - a_1b_2 )} \\

r &= \frac{( \phi a_2b_1 - \phi a_1b_2 )(\phi a_2^2 + a_2b_2)}{(\phi a_2b_2 + b_2^2)( a_2b_1 - a_1b_2 )} \\

&= \frac{\phi( a_2b_1 - a_1b_2 )a_2(\~~phia_2~~ + b_2)}{b_2(\~~phia_2~~ + b_2)( a_2b_1 - a_1b_2 )} \\

&= \frac{\phi( a_2b_1 - a_1b_2 )a_2(\phi a_2 + b_2)}{b_2(\phi a_2 + b_2)( a_2b_1 - a_1b_2 )} \\

&= \frac{\phi\cancel{( a_2b_1 - a_1b_2 )}a_2\cancel{(\~~phia_2~~ + b_2)}}{b_2\cancel{(\~~phia_2~~ + b_2)}\cancel{( a_2b_1 - a_1b_2 )}} \\

&= \frac{\phi\cancel{( a_2b_1 - a_1b_2 )}a_2\cancel{(\phi a_2 + b_2)}}{b_2\cancel{(\phi a_2 + b_2)}\cancel{( a_2b_1 - a_1b_2 )}} \\

&= \frac{\~~phia_2~~}{b_2}

&= \frac{\phi a_2}{b_2}

\end{align}

</math>

@@ Line 1,059: / Line 1,059: @@
 In particular, we see that the denominator of the weighted ratio finds itself on the same side of the ratio as the weight.
-The full derivation follows. With lower bounding ratio <math>\frac{a_1}{a_2}</math> and upper bounding ratio <math>\frac{b_1}{b_2}</math>, we have a mediant of <math>\frac{\phia_1 + b_1}{\phia_2 + b_2}</math>. So then the segment from the lower bounding ratio to the mediant has length
+The full derivation follows. With lower bounding ratio <math>\frac{a_1}{a_2}</math> and upper bounding ratio <math>\frac{b_1}{b_2}</math>, we have a mediant of <math>\frac{\phi a_1 + b_1}{\phi a_2 + b_2}</math>. So then the segment from the lower bounding ratio to the mediant has length
 <math>
 \require{cancel}
 \begin{align}
-s_1 &= \frac{\phia_1 + b_1}{\phia_2 + b_2} - \frac{a_1}{a_2} \\
+s_1 &= \frac{\phi a_1 + b_1}{\phi a_2 + b_2} - \frac{a_1}{a_2} \\
-&= \frac{a_2(\phia_1 + b_1)}{a_2(\phia_2 + b_2)} - \frac{a_1(\phia_2 + b_2)}{a_2(\phia_2 + b_2)} \\
+&= \frac{a_2(\phi a_1 + b_1)}{a_2(\phi a_2 + b_2)} - \frac{a_1(\phi a_2 + b_2)}{a_2(\phi a_2 + b_2)} \\
-&= \frac{a_2(\phia_1 + b_1) - a_1(\phia_2 + b_2)}{a_2(\phia_2 + b_2)} \\
+&= \frac{a_2(\phi a_1 + b_1) - a_1(\phi a_2 + b_2)}{a_2(\phi a_2 + b_2)} \\
-&= \frac{\phia_1a_2 + a_2b_1 - \phia_1a_2 - a_1b_2}{\phia_2^2 + a_2b_2} \\
+&= \frac{\phi a_1a_2 + a_2b_1 - \phi a_1a_2 - a_1b_2}{\phi a_2^2 + a_2b_2} \\
-&= \frac{\cancel{\phia_1a_2} + a_2b_1 - \cancel{\phia_1a_2} - a_1b_2}{\phia_2^2 + a_2b_2} \\
+&= \frac{\cancel{\phi a_1a_2} + a_2b_1 - \cancel{\phi a_1a_2} - a_1b_2}{\phi a_2^2 + a_2b_2} \\
-&= \frac{a_2b_1 - a_1b_2}{\phia_2^2 + a_2b_2}
+&= \frac{a_2b_1 - a_1b_2}{\phi a_2^2 + a_2b_2}
 \end{align}
 </math>
@@ Line 1,078: / Line 1,078: @@
 \require{cancel}
 \begin{align}
-s_2 &= \frac{b_1}{b_2} - \frac{\phia_1 + b_1}{\phia_2 + b_2} \\
+s_2 &= \frac{b_1}{b_2} - \frac{\phi a_1 + b_1}{\phi a_2 + b_2} \\
-&= \frac{b_1(\phia_2 + b_2)}{b_2(\phia_2 + b_2)} - \frac{b_2(\phia_1 + b_1)}{b_2(\phia_2 + b_2)} \\
+&= \frac{b_1(\phi a_2 + b_2)}{b_2(\phi a_2 + b_2)} - \frac{b_2(\phi a_1 + b_1)}{b_2(\phi a_2 + b_2)} \\
-&= \frac{b_1(\phia_2 + b_2) - b_2(\phia_1 + b_1)}{b_2(\phia_2 + b_2)} \\
+&= \frac{b_1(\phi a_2 + b_2) - b_2(\phi a_1 + b_1)}{b_2(\phi a_2 + b_2)} \\
-&= \frac{\phia_2b_1 + b_1b_2 - \phia_1b_2 - b_1b_2}{\phia_2b_2 + b_2^2} \\
+&= \frac{\phi a_2b_1 + b_1b_2 - \phi a_1b_2 - b_1b_2}{\phi a_2b_2 + b_2^2} \\
-&= \frac{\phia_2b_1 + \cancel{b_1b_2} - \phia_1b_2 - \cancel{b_1b_2}}{\phia_2b_2 + b_2^2} \\
+&= \frac{\phi a_2b_1 + \cancel{b_1b_2} - \phi a_1b_2 - \cancel{b_1b_2}}{\phi a_2b_2 + b_2^2} \\
-&= \frac{\phia_2b_1 - \phia_1b_2}{\phia_2b_2 + b_2^2}
+&= \frac{\phi a_2b_1 - \phi a_1b_2}{\phi a_2b_2 + b_2^2}
 \end{align}
 </math>
@@ Line 1,092: / Line 1,092: @@
 \require{cancel}
 \begin{align}
-r &= \frac{( \phia_2b_1 - \phia_1b_2 )(\phia_2^2 + a_2b_2)}{(\phia_2b_2 + b_2^2)( a_2b_1 - a_1b_2 )} \\
+r &= \frac{( \phi a_2b_1 - \phi a_1b_2 )(\phi a_2^2 + a_2b_2)}{(\phi a_2b_2 + b_2^2)( a_2b_1 - a_1b_2 )} \\
-&= \frac{\phi( a_2b_1 - a_1b_2 )a_2(\phia_2 + b_2)}{b_2(\phia_2 + b_2)( a_2b_1 - a_1b_2 )} \\
+&= \frac{\phi( a_2b_1 - a_1b_2 )a_2(\phi a_2 + b_2)}{b_2(\phi a_2 + b_2)( a_2b_1 - a_1b_2 )} \\
-&= \frac{\phi\cancel{( a_2b_1 - a_1b_2 )}a_2\cancel{(\phia_2 + b_2)}}{b_2\cancel{(\phia_2 + b_2)}\cancel{( a_2b_1 - a_1b_2 )}} \\
+&= \frac{\phi\cancel{( a_2b_1 - a_1b_2 )}a_2\cancel{(\phi a_2 + b_2)}}{b_2\cancel{(\phi a_2 + b_2)}\cancel{( a_2b_1 - a_1b_2 )}} \\
-&= \frac{\phia_2}{b_2}
+&= \frac{\phi a_2}{b_2}
 \end{align}
 </math>