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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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mj-msc.tex
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@ -588,17 +588,17 @@ table~\ref{table:scale-halfcirlce-diameter}.
\end{table}
Sometimes, when working with {\WM}, it is useful to convert between
half-circle's diameter and adjusted size. These easily derive from circle's
area formula $A = 2\pi r^2$. Diameter:
half-circle's diameter $D$ and adjusted size $A_{adj}$. These easily derive
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}
@ -785,23 +785,24 @@ compactness index is calculated as follows:
\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
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:
\[
cmp = \frac{A_p}{A_c} =
\frac{A_p}{ \frac{C^2}{4\pi} } =
\frac{4\pi A_p}{C^2}
c = \frac{A_{p}}{A_{c}} =
\frac{A_{p}}{ \frac{C^2}{4\pi} } =
\frac{4\pi A_{p}}{C^2}
\]
\end{enumerate}
@ -812,11 +813,11 @@ of properties, upon which actions later will be performed.
\subsection{Shape of a Bend}
\label{sec:shape-of-a-bend}
This section introduces \textsc{adjusted size}, which trivially derives from
\textsc{compactness index} $cmp$ and shape's area $A$:
This section introduces \textsc{adjusted size} $A_{adj}$, which trivially
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
@ -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:
\begin{enumerate}
\item \textsc{adjusted size}
\item \textsc{compactness index}
\item Baseline length
\item \textsc{adjusted size} $A_{adj}$
\item \textsc{compactness index} $c$
\item \textsc{Baseline length} $l$
\end{enumerate}
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
$q$:
\[
d(p,q) = \sqrt{(adjsize_p-adjsize_q)^2 +
(cmp_p-cmp_q)^2 +
(baseline_p-baseline_q)^2}
d(p,q) = \sqrt{(A_{adj(p)}-A_{adj(q)})^2 +
(c_p-c_q)^2 +
(l_p-l_q)^2}
\]
The smaller the distance $d$, the more similar the bends are.