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update formulae

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Motiejus Jakštys 2021-05-19 22:57:49 +03:00 committed by Motiejus Jakštys
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commit 0cd9b36267
1 changed files with 23 additions and 22 deletions
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mj-msc.tex
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@ -588,17 +588,17 @@ table~\ref{table:scale-halfcirlce-diameter}.
\end{table} \end{table}
Sometimes, when working with {\WM}, it is useful to convert between Sometimes, when working with {\WM}, it is useful to convert between
half-circle's diameter and adjusted size. These easily derive from circle's half-circle's diameter $D$ and adjusted size $A_{adj}$. These easily derive
area formula $A = 2\pi r^2$. Diameter: from circle's area formula $A = 2\pi r^2$. Diameter:
\[ \[
D = 2\sqrt{\frac{2 adjsize}{\pi}} D = 2\sqrt{\frac{2 A_{adj}}{\pi}}
\] \]
In reverse, half-circle's area from it's diameter (adjusted size): In reverse, adjusted size $A_{adj}$ from half-circle's diameter:
\[ \[
adjsize = \frac{\pi D^2}{8} A_{adj} = \frac{\pi D^2}{8}
\] \]
\subsection{Definition of a Bend} \subsection{Definition of a Bend}
@ -785,23 +785,24 @@ compactness index is calculated as follows:
\item Construct a polygon by joining first and last vertices of the bend. \item Construct a polygon by joining first and last vertices of the bend.
\item Calculate area of the polygon $A_p$. \item Calculate area of the polygon $A_{p}$.
\item Calculate perimeter $P$ of the polygon. The same value is the \item Calculate perimeter $P$ of the polygon. The same value is the
circumference of the circle: $C = P$. circumference of the circle: $C = P$.
\item Given circle's circumference $C$, circle's area $A_c$ is: \item Given circle's circumference $C$, circle's area $A_{c}$ is:
\[ \[
A_{c} = \frac{C^2}{4\pi} A_{circle} = \frac{C^2}{4\pi}
\] \]
\item Compactness index is $\frac{A_p}{A_c}$: \item Compactness index $c$ is are of the polygon divided by the area of the
circle:
\[ \[
cmp = \frac{A_p}{A_c} = c = \frac{A_{p}}{A_{c}} =
\frac{A_p}{ \frac{C^2}{4\pi} } = \frac{A_{p}}{ \frac{C^2}{4\pi} } =
\frac{4\pi A_p}{C^2} \frac{4\pi A_{p}}{C^2}
\] \]
\end{enumerate} \end{enumerate}
@ -812,11 +813,11 @@ of properties, upon which actions later will be performed.
\subsection{Shape of a Bend} \subsection{Shape of a Bend}
\label{sec:shape-of-a-bend} \label{sec:shape-of-a-bend}
This section introduces \textsc{adjusted size}, which trivially derives from This section introduces \textsc{adjusted size} $A_{adj}$, which trivially
\textsc{compactness index} $cmp$ and shape's area $A$: derives from \textsc{compactness index} $c$ and "polygonized" bend's area $A_{p}$:
\[ \[
adjsize = \frac{0.75 A}{cmp} A_{adj} = \frac{0.75 A_{p}}{c}
\] \]
Adjusted size becomes necessary later to compare bends with each other, and Adjusted size becomes necessary later to compare bends with each other, and
@ -848,19 +849,19 @@ Two conditions must be true to claim that a bend is isolated:
To find out whether two bends are similar, they are compared by 3 components: To find out whether two bends are similar, they are compared by 3 components:
\begin{enumerate} \begin{enumerate}
\item \textsc{adjusted size} \item \textsc{adjusted size} $A_{adj}$
\item \textsc{compactness index} \item \textsc{compactness index} $c$
\item Baseline length \item \textsc{Baseline length} $l$
\end{enumerate} \end{enumerate}
Components 1, 2 and 3 represent a point in a 3-dimensional space, and Euclidean Components 1, 2 and 3 represent a point in a 3-dimensional space, and Euclidean
distance $d$ between those is calculated to differentiate between bends $p$ and distance $d(p,q)$ between those is calculated to differentiate bends $p$ and
$q$: $q$:
\[ \[
d(p,q) = \sqrt{(adjsize_p-adjsize_q)^2 + d(p,q) = \sqrt{(A_{adj(p)}-A_{adj(q)})^2 +
(cmp_p-cmp_q)^2 + (c_p-c_q)^2 +
(baseline_p-baseline_q)^2} (l_p-l_q)^2}
\] \]
The smaller the distance $d$, the more similar the bends are. The smaller the distance $d$, the more similar the bends are.