From: ali Date: Thu, 27 Aug 2015 09:26:28 +0000 (+0200) Subject: Update by Ali X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/JournalMultiPeriods.git/commitdiff_plain/473fcf6b4a8be17eb285310e31be480bffcb7138 Update by Ali --- diff --git a/article.tex b/article.tex index 777d52e..ca8c2c3 100644 --- a/article.tex +++ b/article.tex @@ -538,10 +538,64 @@ Zhou~\cite{Zhang05} proved that if the transmission range fulfills the previous hypothesis, a complete coverage of a convex area implies connectivity among the active nodes. -Instead of working with a continuous coverage area, we make it discrete by -considering for each sensor a set of points called primary points. Consequently, -we assume that the sensing disk defined by a sensor is covered if all of its -primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here. +%Instead of working with a continuous coverage area, we make it discrete by considering for each sensor a set of points called primary points. Consequently, we assume that the sensing disk defined by a sensor is covered if all of its primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here. + + +\indent Instead of working with the coverage area, we consider for each sensor a set of points called primary points~\cite{idrees2014coverage}. We also assume that the sensing disk defined by a sensor is covered if all the primary points of this sensor are covered. By knowing the position (point center: ($p_x,p_y$)) of a wireless sensor node and it's sensing range $R_s$, we calculate the primary points directly based on the proposed model. We use these primary points (that can be increased or decreased if necessary) as references to ensure that the monitored region of interest is covered by the selected set of sensors, instead of using all the points in the area. +We can calculate the positions of the selected primary +points in the circle disk of the sensing range of a wireless sensor +node (see Figure~\ref{fig1}) as follows:\\ +Assuming that the point center of a wireless sensor node is located at $(p_x,p_y)$, we can define up to 25 primary points $X_1$ to $X_{25}$.\\ +%$(p_x,p_y)$ = point center of wireless sensor node\\ +$X_1=(p_x,p_y)$ \\ +$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\ +$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\ +$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\ +$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\ +$X_6= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\ +$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\ +$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\ +$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\ +$X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\ +$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\ +$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\ +$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\ +$X_{14}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\ +$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\ +$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\ +$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\ +$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0) $\\ +$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0) $\\ +$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\ +$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\ +$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\ +$X_{23}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\ +$X_{24}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $\\ +$X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $. + + + +\begin{figure} %[h!] +\centering + \begin{multicols}{2} +\centering +\includegraphics[scale=0.28]{fig21.pdf}\\~ (a) +\includegraphics[scale=0.28]{principles13.pdf}\\~(c) +\hfill \hfill +\includegraphics[scale=0.28]{fig25.pdf}\\~(e) +\includegraphics[scale=0.28]{fig22.pdf}\\~(b) +\hfill \hfill +\includegraphics[scale=0.28]{fig24.pdf}\\~(d) +\includegraphics[scale=0.28]{fig26.pdf}\\~(f) +\end{multicols} +\caption{Wireless Sensor Node represented by (a) 5, (b) 9, (c) 13, (d) 17, (e) 21 and (f) 25 primary points respectively} +\label{fig1} +\end{figure} + + + + + %By knowing the position (point center: ($p_x,p_y$)) of a wireless %sensor node and its $R_s$, we calculate the primary points directly diff --git a/fig21.pdf b/fig21.pdf new file mode 100644 index 0000000..1d32fcc Binary files /dev/null and b/fig21.pdf differ diff --git a/fig22.pdf b/fig22.pdf new file mode 100644 index 0000000..7cc4a50 Binary files /dev/null and b/fig22.pdf differ diff --git a/fig24.pdf b/fig24.pdf new file mode 100644 index 0000000..d774bcd Binary files /dev/null and b/fig24.pdf differ diff --git a/fig25.pdf b/fig25.pdf new file mode 100644 index 0000000..5e50035 Binary files /dev/null and b/fig25.pdf differ diff --git a/fig26.pdf b/fig26.pdf new file mode 100644 index 0000000..f369523 Binary files /dev/null and b/fig26.pdf differ diff --git a/principles13.pdf b/principles13.pdf new file mode 100644 index 0000000..9d30005 Binary files /dev/null and b/principles13.pdf differ