From: Manuel Date: Sat, 23 Jun 2012 07:26:47 +0000 (+0200) Subject: J'ai complété le document sur la mise en orbite d'un satellite et ajouté en particuli... X-Git-Url: https://melusine.eu.org/syracuse/G/git/?a=commitdiff_plain;h=a073b16d2ce5dac585504da8489bc1847beee7cc;p=pst-eqdf.git J'ai complété le document sur la mise en orbite d'un satellite et ajouté en particulier l'hodographe : c'est un cercle : intéressant à démontrer(je découvre). J'espère ne pas avoir fait de bêtises, car je vois que le commit rajoute des fichiers sur Lorentz existants déjà ? En prévision l'animation. --- diff --git a/gravitation/eqdf-grav_01.pdf b/gravitation/eqdf-grav_01.pdf deleted file mode 100644 index 3f5a4f4..0000000 Binary files a/gravitation/eqdf-grav_01.pdf and /dev/null differ diff --git a/gravitation/eqdf-grav_01.tex b/gravitation/eqdf-grav_01.tex deleted file mode 100644 index 45c744c..0000000 --- a/gravitation/eqdf-grav_01.tex +++ /dev/null @@ -1,419 +0,0 @@ -\documentclass{article} -\usepackage[a4paper,margin=2cm]{geometry} -\usepackage[latin1]{inputenc} -\usepackage{pstricks-add,pst-eucl,pst-func} -\usepackage{array,amsmath} -\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,linecolor={[rgb]{1 0.5 0}}} -\newpsstyle{vecteurB}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{0 0.5 1}}} -\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0.5 0}}} -%%%%%%%%%%%%%%%%%% -\title{Interaction gravitationnelle avec PSTricks} -\date{19 juin 2\,012} -\begin{document} -\maketitle -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% -\section{Mise en orbite d'un satellite} -\begin{figure}[htbp] - \begin{center} -\psset{unit=2} -\begin{pspicture}(-6,-2)(2,5) -\pstVerb{ - /GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse à l'apogée - /vA GM par div sqrt 1 exc sub mul def -% vitesse au périgée - /vP GM par div sqrt 1 exc add mul def -% coordonnées de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnées de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -% Apogée - /xA par 1 exc sub div theta0 cos mul neg def - /yA par 1 exc sub div theta0 sin mul neg def -% Périgée - /xP par 1 exc add div theta0 cos mul def - /yP par 1 exc add div theta0 sin mul def -% Centre - /xO xP xA add 2 div def - /yO yP yA add 2 div def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,5) -\rput(1,4){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{0000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.05} -\pnode(!xP yP){P} % périgée -\pnode(!xA yA){A} % Apogée -\pnode(!xO yO){O} % centre -\rput(!x0 y0){\psline[style=vecteurB]{->}(!v0x v0y)\uput[ur](!v0x v0y){$\overrightarrow{v_0}$}} -\psline[linestyle=dashed](A)(P) -\pscircle*(A){0.05} -\uput[ul](A){$A$} -\uput[dr](P){$P$} -% position du satellite à un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.05} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurA]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarcn{->}(0,0){0.4}{0}{!theta0} -\uput{0.5}[!theta0 2 div](0,0){$\theta_0$} -\rput(A){\psline[style=vecteurB]{->}(!vAx vAy)} -%\rput(P){\psline[style=vecteurB]{->}(!vPx vPy)} -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,5) -\uput[r](0,4.9){$y$}\uput[u](1.9,0){$x$} -\psdot(O)\uput[r](O){$\Omega$} -\rput{!90 theta0 add}(O){\psline[linestyle=dashed](!b_2 neg 0)(!b_2 0)} -\end{pspicture} -\caption{Mouvement d'un satellite} - \end{center} -\end{figure} -Soit (M) la masse de l'astre et (m) celle du satellite avec $m\ll M$. Le centre de masse du système \{M,m\} est confondu avec le centre de l'astre attracteur. La mise en orbite s'effectue à partir du point $M_0(x_0,y_0)$ avec une vitesse $\overrightarrow{v_0}(v_{0_x},v_{0_x})$. $\theta_0$ est l'angle que fait $\overrightarrow{Ox}$ avec $\overrightarrow{OM_0}$. Le satellite (S), supposé ponctuel, subit de la part de l'astre une force d'attraction gravitationnelle : -\[ -\overrightarrow{F}=-\mathcal{G}\frac{Mm}{r^2}\overrightarrow{u}\qquad \text{avec}\quad \overrightarrow{u}=\frac{\overrightarrow{r}}{r}\quad \text{et}\quad \overrightarrow{r}=\overrightarrow{OS} -\] -Tous les \textit{bons} livres de mécanique\footnote{Comme celui, par exemple, de José-Philippe Pérez, aux éditions Masson.} établissent les relations suivantes : -\[ -r=\frac{p}{1+\mathrm{e}\cos(\theta-\theta_0)} -\] -Paramètres et excentricité ont pour expressions respectives, avec les notations suivantes : $K=\mathcal{G}Mm$, $\mathcal{E}$ l'énergie du système et $L$ le moment cinétique. -\[ -\mathrm{e}=\sqrt{1+\frac{2\mathcal{E}L^2}{mK^2}}\qquad p=\frac{L^2}{mK} -\] -On choisit une vitesse initiale $\overrightarrow{v_0}$ perpendiculaire à $\overrightarrow{OM_0}$, dans ces conditions le moment cinétique et l'énergie, qui restent constants, valent : -\[ -L=mr_0v_0\qquad \mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 -\] -En remplaçant $L$ et $\mathcal{E}$, on obtient pour l'excentricité et le paramètre les expressions suivantes : -\[ -\mathrm{e}=\sqrt{1+\frac{1}{\mathcal{G}^2M^2}\Big(\frac{1}{2}v_0^4r_0^2-\mathcal{G}Mr_0v_0^2\Big)} -\] -\[ -p=\frac{v_0^2r_0^2}{\mathcal{G}M} -\] -On se limite au cas du mouvement elliptique, avec, en conséquence, la condition : -\[ -\mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 < 0 -\] -Le demi-grand axe $a$, le demi-petit axe $b$ sont : -\[ -a=\frac{p}{1-\mathrm{e}^2}\qquad b=\frac{p}{\sqrt{1-\mathrm{e}^2}} -\] -La période $T$ qui obéit à la troisième loi de Képler : -\[ -T^2=\frac{4\pi^2a^3}{\mathcal{G}M} -\] -La vitesse, en un point de l'ellipse, se calcule par : -\[ -v^2=\mathcal{G}M\Big( \frac{2}{r}-\frac{1}{a}\Big) -\] -Sachant que $r_p=\dfrac{p}{1+\mathrm{e}}$ et $r_A=\dfrac{p}{1-\mathrm{e}}$, on en déduit les vitesses au périgée et à l'apogée : -\[ -v_P=\sqrt{\frac{\mathcal{G}M}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}M}{p}}(1-\mathrm{e}) -\] - -\section{L'étude avec PSTricks} -\subsection{La trajectoire} -\def\parametres{ - /GM 1 def % 4e14 def % GxM - /x0 6.5e6 def % position initiale - /y0 0 def - /vx0 0 def % vitesse initiale - /vy0 1e4 def -} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\[ -\left\{ -\begin{array}{rcl} -\ddot{x}&=&-\dfrac{GM}{r^3}x\\[1em] -\ddot{y}&=&-\dfrac{GM}{r^3}y\\ -\end{array} -\right. -\label{eq1} -\qquad r=\sqrt{x^2+y^2} -\] - -On peut dessiner la trajectoire du satellite et de ses caractéristiques de deux façons : -\begin{itemize} - \item par l'utilisation de \verb+\parametricplot+ ; - \item ou celle de \verb+\psplotDiffEqn+. -\end{itemize} -\verb+\parametricplot+ utilise l'expression exacte de l'équation de la trajectoire en coordonnées polaires : -\begin{verbatim} - \parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{% - /radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\end{verbatim} -L'excentricité, la période, demi-grand axe et demi-petit axe sont calculés par quelques lignes de code \textsf{postscript}. Il faut s'assurer que les conditions initiales choisies vérifient bien la condition d'une trajectoire elliptique, pour cela il faut que l'énergie initiale $\mathcal{E}_0<0$, sinon cela entraînera une erreur lors du passage à l'interpréteur \textsf{postscript}. -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul - v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%demi-grand axe - /a_2 par 1 exc dup mul sub div def % demi-grand axe -%demi-petit axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe -% période - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\end{verbatim} -\verb+\psplotDiffEqn+ utilise les équations différentielles du mouvement, en notation algébrique : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,% - linecolor=blue,linewidth=0.1,% - method=rk4,plotpoints=1000,% - algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -Ce qui permet, par ailleurs de vérifier la qualité du tracé par la méthode numérique, en bleu, tandis que le tracé à partir de l'expression exacte est en trait fin en rouge. -% -\begin{center} -\begin{pspicture}(-12,-2)(4,10) -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\psframe*[linecolor=white](-3,-0.2)(0,0.2) -\pscircle[fillcolor=blue!50,fillstyle=solid](0,0){0.5} -\psgrid[unit=2,subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-1)(2,5) -\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,linewidth=0.1,method=rk4,plotpoints=1000,algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\psdot[unit=2,dotsize=0.12](!x0 y0) -\rput(!x0 2 mul y0 2 mul){\psline[style=vecteurC]{->}(!v0x v0y)} -\end{pspicture} -\end{center} -\subsection{La vitesse} -\verb+\psplotDiffEqn+ permet de voir comment varie la vitesse sur l'ellipse : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] - \psplotDiffEqn[xunit=0.2,yunit=5,% - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - linecolor=red,method=rk4,plotpoints=1000, - algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -\begin{center} -\begin{pspicture}(0,-1)(10,7) -\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] -\pstVerb{/GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def}% -\psplotDiffEqn[xunit=0.2,yunit=5, - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - ,linecolor=red,method=rk4,plotpoints=1000,algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% - \multido{\i=1+1,\I=5+5}{9}{\uput[u](\i,0){\I}} -\pnode(! 36.4 5 div 0){P} -\psdot(P)\uput[d](P){Périgée} -\psline[linestyle=dashed](P)(! 36.4 5 div 7) -\pnode(! 36.4 10 div 0){A} -\psdot(A)\uput[d](A){Apogée} -\psline[linestyle=dashed](A)(! 36.4 10 div 7) -\uput[l](0,6.5){$v$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(10,0)(0,0)(0,7) -\uput[u](10,0){$t$(s)} -\uput[ul](0,0){0} -\end{pspicture} -\end{center} -On peut obtenir les caractéristiques de la vitesse en un point quelconque, car le moment cinétique $L=mr\dot{\theta}$ étant constant, pour chaque valeur de $\theta$, on en déduit $r$ puis $\dot{\theta}$. En coordonnées polaires, la vitesse s'exprime par : -\[ -\overrightarrow{v}=\dot{r}\overrightarrow{e_r}+r\dot{\theta}\overrightarrow{e_{\theta}} -\] -$\dot{\theta}$ et $\dot{r}$ s'obtiennent par les relations suivantes : -\[ -\dot{\theta}^2=\frac{r_0v_0}{r} -\] -\[ -\dot{r}=-\frac{p(-\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r^2}{p}\dot{\theta}\sin(\theta-\theta_0) -\] -La chaîne de calculs est la suivante : $\theta\Longrightarrow r\Longrightarrow \dot{\theta}\Longrightarrow \dot{r}\Longrightarrow \overrightarrow{v}$. -\section{Mouvement circulaire} - \begin{center} -\psset{unit=1} -\begin{pspicture}(-5,-5.5)(5,5.5) -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse à l'apogée - /vA GM par div sqrt 1 exc sub mul def -% vitesse au périgée - /vP GM par div sqrt 1 exc add mul def -% coordonnées de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnées de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.75} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(5,5) -\rput(0,-2){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{000000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.1} -%\pnode(!par 1 exc add div theta0 cos mul par 1 exc add div theta0 sin mul){P} % périgée -%\pnode(!par 1 exc sub div theta0 cos mul neg par 1 exc sub div theta0 sin mul neg){A} % Apogée -\rput(!x0 y0){\psline[arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{0 0.5 1}},unit=4]{->}(!v0x v0y)\uput[ur](!v0x 2 mul v0y 2 mul){$\overrightarrow{v_0}$}} -\uput[ur](!x0 y0){$M_0$} -\psline[linestyle=dashed](0,0)(!x0 y0) -% position du satellite à un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.1} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurC]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarc{->}(0,0){1}{0}{!theta0} -\uput{1.1}[!theta0 2 div](0,0){$\theta_0$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(5,0)(0,0)(0,5) -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(5,0)(0,0)(0,5) -\uput[u](0,5){$y$}\uput[r](5,0){$x$} -\end{pspicture} - \end{center} -Il s'obtient très facilement à partir de l'étude précédente si on sait que dans ce cas : -\[ -v_0=\sqrt{\frac{\mathcal{G}M}{r_0}} -\] -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def }% -\end{verbatim} - -\end{document} -\begin{figure}[htbp] - \begin{center} -%\begin{pspicture}[shift=-2,showgrid=true](-3,-1.75)(2,1.5) -\begin{pspicture}(-12,-6)(4,6) -\pstVerb{\parametres}% -\psdot[dotsize=1](0,0) - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,method=rk4,plotpoints=1000,algebraic]{0}{36.5}{1 0 0 1.3}{\eqsatellite}% -% \psplotDiffEqn[whichabs=0,whichord=1,linecolor=red, plotpoints=200,method=varrkiv, varsteptol=.00001,algebraic,showpoints]{0}{35}{5 0 0 %0.25}{\eqsatellite} -\psgrid[unit=2](-6,-3)(2,3) -\end{pspicture} -\caption{Mouvement d'un satellite} - \end{center} -\end{figure} - -\begin{figure}[htbp] - \begin{center} -%\begin{pspicture}[shift=-2,showgrid=true](-3,-1.75)(2,1.5) -\begin{pspicture}(-12,-6)(4,6) -\pstVerb{\parametres}% -\psset{unit=2}% -\psdot[dotsize=1](0,0) - \psplotDiffEqn[whichabs=0,whichord=1,linecolor=blue,method=rk4,plotpoints=1000,algebraic]{0}{50}{1 0 0.2 1.25}{\eqsatellite}% -\psdot(1,0) -\rput(1,0){\psline[style=vecteurA]{->}(0.2,1.25)} -\psgrid(-6,-3)(2,3) -\end{pspicture} -\caption{Mouvement d'un satellite 3} - \end{center} -\end{figure} \ No newline at end of file diff --git a/gravitation/g1.png b/gravitation/g1.png new file mode 100644 index 0000000..a506f23 Binary files /dev/null and b/gravitation/g1.png differ diff --git a/gravitation/g2.png b/gravitation/g2.png new file mode 100644 index 0000000..11d7bf4 Binary files /dev/null and b/gravitation/g2.png differ diff --git a/gravitation/g3.png b/gravitation/g3.png new file mode 100644 index 0000000..7f2b8a6 Binary files /dev/null and b/gravitation/g3.png differ diff --git a/gravitation/g4.png b/gravitation/g4.png new file mode 100644 index 0000000..cac35a4 Binary files /dev/null and b/gravitation/g4.png differ diff --git a/gravitation/gravitation_01.pdf b/gravitation/gravitation_01.pdf index 29ebb92..aa1c235 100644 Binary files a/gravitation/gravitation_01.pdf and b/gravitation/gravitation_01.pdf differ diff --git a/gravitation/gravitation_01.tex b/gravitation/gravitation_01.tex index 93aaa0b..e364ae1 100644 --- a/gravitation/gravitation_01.tex +++ b/gravitation/gravitation_01.tex @@ -1,472 +1,598 @@ -\documentclass{article} -\usepackage[a4paper,margin=2cm]{geometry} -\usepackage[latin1]{inputenc} -\usepackage{pstricks-add,pst-eqdf,pst-func} -\usepackage{array,amsmath} -\usepackage{listings} -\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,linecolor={[rgb]{1 0.5 0}}} -\newpsstyle{vecteurB}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{0 0.5 1}}} -\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0.5 0}}} -%%%%%%%%%%%%%%%%%% -\title{Interaction gravitationnelle avec PSTricks} -\date{19 juin 2\,012} -\begin{document} -\maketitle -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% -\section{Mise en orbite d'un satellite} -\begin{figure}[htbp] - \begin{center} -\psset{unit=2} -\begin{pspicture}(-6,-2)(2,5) -\pstVerb{ - /GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse à l'apogée - /vA GM par div sqrt 1 exc sub mul def -% vitesse au périgée - /vP GM par div sqrt 1 exc add mul def -% coordonnées de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnées de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -% Apogée - /xA par 1 exc sub div theta0 cos mul neg def - /yA par 1 exc sub div theta0 sin mul neg def -% Périgée - /xP par 1 exc add div theta0 cos mul def - /yP par 1 exc add div theta0 sin mul def -% Centre - /xO xP xA add 2 div def - /yO yP yA add 2 div def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,5) -\rput(1,4){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{0000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.05} -\pnode(!xP yP){P} % périgée -\pnode(!xA yA){A} % Apogée -\pnode(!xO yO){O} % centre -\rput(!x0 y0){\psline[style=vecteurB]{->}(!v0x v0y)\uput[ur](!v0x v0y){$\overrightarrow{v_0}$}} -\psline[linestyle=dashed](A)(P) -\pscircle*(A){0.05} -\uput[ul](A){$A$} -\uput[dr](P){$P$} -% position du satellite à un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.05} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurA]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarcn{->}(0,0){0.4}{0}{!theta0} -\uput{0.5}[!theta0 2 div](0,0){$\theta_0$} -\rput(A){\psline[style=vecteurB]{->}(!vAx vAy)} -%\rput(P){\psline[style=vecteurB]{->}(!vPx vPy)} -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,5) -\uput[r](0,4.9){$y$}\uput[u](1.9,0){$x$} -\psdot(O)\uput[r](O){$\Omega$} -\rput{!90 theta0 add}(O){\psline[linestyle=dashed](!b_2 neg 0)(!b_2 0)} -\end{pspicture} -\caption{Mouvement d'un satellite} - \end{center} -\end{figure} -Soit (M) la masse de l'astre et (m) celle du satellite avec $m\ll M$. Le centre de masse du système \{M,m\} est confondu avec le centre de l'astre attracteur. La mise en orbite s'effectue à partir du point $M_0(x_0,y_0)$ avec une vitesse $\overrightarrow{v_0}(v_{0_x},v_{0_x})$. $\theta_0$ est l'angle que fait $\overrightarrow{Ox}$ avec $\overrightarrow{OM_0}$. Le satellite (S), supposé ponctuel, subit de la part de l'astre une force d'attraction gravitationnelle : -\[ -\overrightarrow{F}=-\mathcal{G}\frac{Mm}{r^2}\overrightarrow{u}\qquad \text{avec}\quad \overrightarrow{u}=\frac{\overrightarrow{r}}{r}\quad \text{et}\quad \overrightarrow{r}=\overrightarrow{OS} -\] -Tous les \textit{bons} livres de mécanique\footnote{Comme celui, par exemple, de José-Philippe Pérez, aux éditions Masson.} établissent les relations suivantes : -\[ -r=\frac{p}{1+\mathrm{e}\cos(\theta-\theta_0)} -\] -Paramètres et excentricité ont pour expressions respectives, avec les notations suivantes : $K=\mathcal{G}Mm$, $\mathcal{E}$ l'énergie du système et $L$ le moment cinétique. -\[ -\mathrm{e}=\sqrt{1+\frac{2\mathcal{E}L^2}{mK^2}}\qquad p=\frac{L^2}{mK} -\] -On choisit une vitesse initiale $\overrightarrow{v_0}$ perpendiculaire à $\overrightarrow{OM_0}$, dans ces conditions le moment cinétique et l'énergie, qui restent constants, valent : -\[ -L=mr_0v_0\qquad \mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 -\] -En remplaçant $L$ et $\mathcal{E}$, on obtient pour l'excentricité et le paramètre les expressions suivantes : -\[ -\mathrm{e}=\sqrt{1+\frac{1}{\mathcal{G}^2M^2}\Big(\frac{1}{2}v_0^4r_0^2-\mathcal{G}Mr_0v_0^2\Big)} -\] -\[ -p=\frac{v_0^2r_0^2}{\mathcal{G}M} -\] -On se limite au cas du mouvement elliptique, avec, en conséquence, la condition : -\[ -\mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 < 0 -\] -Le demi-grand axe $a$, le demi-petit axe $b$ sont : -\[ -a=\frac{p}{1-\mathrm{e}^2}\qquad b=\frac{p}{\sqrt{1-\mathrm{e}^2}} -\] -La période $T$ qui obéit à la troisième loi de Képler : -\[ -T^2=\frac{4\pi^2a^3}{\mathcal{G}M} -\] -La vitesse, en un point de l'ellipse, se calcule par : -\[ -v^2=\mathcal{G}M\Big( \frac{2}{r}-\frac{1}{a}\Big) -\] -Sachant que $r_p=\dfrac{p}{1+\mathrm{e}}$ et $r_A=\dfrac{p}{1-\mathrm{e}}$, on en déduit les vitesses au périgée et à l'apogée : -\[ -v_P=\sqrt{\frac{\mathcal{G}M}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}M}{p}}(1-\mathrm{e}) -\] - -\section{L'étude avec PSTricks} -\subsection{La trajectoire} -\def\parametres{ - /GM 1 def % 4e14 def % GxM - /x0 6.5e6 def % position initiale - /y0 0 def - /vx0 0 def % vitesse initiale - /vy0 1e4 def -} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\[ -\left\{ -\begin{array}{rcl} -\ddot{x}&=&-\dfrac{GM}{r^3}x\\[1em] -\ddot{y}&=&-\dfrac{GM}{r^3}y\\ -\end{array} -\right. -\label{eq1} -\qquad r=\sqrt{x^2+y^2} -\] - -On peut dessiner la trajectoire du satellite et de ses caractéristiques de deux façons : -\begin{itemize} - \item par l'utilisation de \verb+\parametricplot+ ; - \item ou celle de \verb+\psplotDiffEqn+. -\end{itemize} -\verb+\parametricplot+ utilise l'expression exacte de l'équation de la trajectoire en coordonnées polaires : -\begin{verbatim} - \parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{% - /radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\end{verbatim} -L'excentricité, la période, demi-grand axe et demi-petit axe sont calculés par quelques lignes de code \textsf{postscript}. Il faut s'assurer que les conditions initiales choisies vérifient bien la condition d'une trajectoire elliptique, pour cela il faut que l'énergie initiale $\mathcal{E}_0<0$, sinon cela entraînera une erreur lors du passage à l'interpréteur \textsf{postscript}. -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul - v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%demi-grand axe - /a_2 par 1 exc dup mul sub div def % demi-grand axe -%demi-petit axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe -% période - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\end{verbatim} -\verb+\psplotDiffEqn+ utilise les équations différentielles du mouvement, en notation algébrique : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] -\def\eqsatellite{% -y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,% - linecolor=blue,linewidth=0.1,% - method=rk4,plotpoints=1000,% - algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -Ce qui permet, par ailleurs de vérifier la qualité du tracé par la méthode numérique, en bleu, tandis que le tracé à partir de l'expression exacte est en trait fin en rouge. -% -\begin{center} -\begin{pspicture}(-12,-2)(4,10) -\pstVerb{ - /GM 1 def - /theta0 -45 def - /r0 0.5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.92 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def -}% -\psframe*[linecolor=white](-3,-0.2)(0,0.2) -\pscircle[fillcolor=blue!50,fillstyle=solid](0,0){0.5} -\psgrid[unit=2,subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-1)(2,5) -\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} - \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,linewidth=0.1,method=rk4,plotpoints=1000,algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% -\parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\psdot[unit=2,dotsize=0.12](!x0 y0) -\rput(!x0 2 mul y0 2 mul){\psline[style=vecteurC]{->}(!v0x v0y)} -\end{pspicture} -\end{center} -\subsection{La vitesse} -\verb+\psplotDiffEqn+ permet de voir comment varie la vitesse sur l'ellipse : -\begin{verbatim} -% x0 y0 x'0 y'0 -% y[0] y[1] y[2] y[3] - \psplotDiffEqn[xunit=0.2,yunit=5,% - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - linecolor=red,method=rk4,plotpoints=1000, - algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -\begin{center} -\begin{pspicture}(0,-1)(10,7) -\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] -\pstVerb{/GM 1 def - /theta0 -35 def - /r0 1 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 1.3 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def}% -\psplotDiffEqn[xunit=0.2,yunit=5, - plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, - ,linecolor=red,method=rk4,plotpoints=1000,algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% - \multido{\i=1+1,\I=5+5}{9}{\uput[u](\i,0){\I}} -\pnode(! 36.4 5 div 0){P} -\psdot(P)\uput[d](P){Périgée} -\psline[linestyle=dashed](P)(! 36.4 5 div 7) -\pnode(! 36.4 10 div 0){A} -\psdot(A)\uput[d](A){Apogée} -\psline[linestyle=dashed](A)(! 36.4 10 div 7) -\uput[l](0,6.5){$v$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(10,0)(0,0)(0,7) -\uput[u](10,0){$t$(s)} -\uput[ul](0,0){0} -\end{pspicture} -\end{center} -On peut obtenir les caractéristiques de la vitesse en un point quelconque, car le moment cinétique $L=mr\dot{\theta}$ étant constant, pour chaque valeur de $\theta$, on en déduit $r$ puis $\dot{\theta}$. En coordonnées polaires, la vitesse s'exprime par : -\[ -\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} -\] -$\dot{\theta}$ et $\dot{r}$ s'obtiennent par les relations suivantes : -\[ -\dot{\theta}=\frac{r_0v_0}{r^2} -\] -\[ -\dot{r}=-\frac{p(-\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\sin(\theta-\theta_0) -\] -La chaîne de calculs est la suivante : $\theta\Longrightarrow r\Longrightarrow \dot{\theta}\Longrightarrow \dot{r}\Longrightarrow \overrightarrow{v}$. - -Le package `\textsf{pst-eqdf}' comprend la commande \verb+\psequadiff+ qui est une version simplifiée de \verb+\psplotDiffEqn+, dont elle ne reprend que la méthode Runge-Kutta~4. Elle permet de sauvegarder sous forme de tableaux et/ou de fichiers toutes les variables et les dérivées de la fonction étudiée, cette possibilité est intéressante pour déterminer les caractéristiques de la vitesse au cours du temps et elle est particulièrement utile pour créer une animation. Nous allons l'utiliser pour dessiner le vecteur-vitesse à quelques instants. - -On sauve successivement le tableau des positions et celui des vitesses. -\begin{verbatim} -\psequadiff[method=rk4,plotpoints=1000, - algebraic, - whichabs=0,whichord=1, - tabname=XiYi]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} -\begin{verbatim} -\psequadiff[method=rk4,plotpoints=1000, - algebraic, - whichabs=2,whichord=3, - tabname=vxvy]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -\end{verbatim} - -Pour ensuite dessiner la trajectoire et les vecteurs-vitesse. -\begin{verbatim} -%\listplot[unit=1]{vxvy aload pop} -% on dessine la vitesse un point sur 100 -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\multido{\i=0+100}{20}{% -\pstVerb{/vX vxvy \i\space get def - /vY vxvy \i\space 1 add get def - /xi XiYi \i\space get def - /yi XiYi \i\space 1 add get def}% -\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} -\end{verbatim} -\begin{center} -\begin{pspicture}(-10,-10)(6,7) -\psset{unit=2}% -\pstVerb{/GM 1 def - /theta0 30 def - /r0 2 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 0.85 def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def}% -\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} -\psequadiff[method=rk4, - plotpoints=1000, - algebraic, - whichabs=0, - whichord=1, - tabname=XiYi -% ,saveData,filename=XiYi.dat -]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -\listplot{XiYi aload pop} -\psequadiff[method=rk4, - plotpoints=1000, - algebraic, - whichabs=2, - whichord=3, - tabname=vxvy -% ,saveData,filename=vxvy.dat -]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% -%\listplot[unit=1]{vxvy aload pop} -% on dessine la vitesse un point sur 100 -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} -\multido{\i=0+100}{20}{% -\pstVerb{/vX vxvy \i\space get def - /vY vxvy \i\space 1 add get def - /xi XiYi \i\space get def - /yi XiYi \i\space 1 add get def}% -\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(3,3) -\end{pspicture} -\end{center} - - - -\section{Mouvement circulaire} - \begin{center} -\psset{unit=1} -\begin{pspicture}(-5,-5.5)(5,5.5) -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def - /Lc r0 v0 mul def % moment cinetique - /par Lc dup mul GM div def % paramètre de l'ellipse -% excentricité - /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def -%%%%%%%%%%%%%% - /a_2 par 1 exc dup mul sub div def % demi-grand axe - /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe - /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def -% vitesse à l'apogée - /vA GM par div sqrt 1 exc sub mul def -% vitesse au périgée - /vP GM par div sqrt 1 exc add mul def -% coordonnées de vA - /vAx vA theta0 90 add cos mul neg def - /vAy vA theta0 90 add sin mul neg def -% coordonnées de vP - /vPx vP theta0 90 add cos mul def - /vPy vP theta0 90 add sin mul def -}% -\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.75} -\psdot[dotstyle=+](0,0) -\uput[d](0,0){O} -\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(5,5) -\rput(0,-2){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{000000}s}} -\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def - radius t cos mul - radius t sin mul} -\pscircle*(!x0 y0){0.1} -%\pnode(!par 1 exc add div theta0 cos mul par 1 exc add div theta0 sin mul){P} % périgée -%\pnode(!par 1 exc sub div theta0 cos mul neg par 1 exc sub div theta0 sin mul neg){A} % Apogée -\rput(!x0 y0){\psline[arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{0 0.5 1}},unit=4]{->}(!v0x v0y)\uput[ur](!v0x 2 mul v0y 2 mul){$\overrightarrow{v_0}$}} -\uput[ur](!x0 y0){$M_0$} -\psline[linestyle=dashed](0,0)(!x0 y0) -% position du satellite à un instant quelconque -\pstVerb{/theta_i 170 def - /radius par 1 exc theta_i theta0 sub cos mul add div def - /xS radius theta_i cos mul def - /yS radius theta_i sin mul def - /ux theta_i cos 1 mul def - /uy theta_i sin 1 mul def - /xi xS ux sub def - /yi yS uy sub def - /xi2 xS ux 2 div sub def - /yi2 yS uy 2 div sub def}% -\pnode(!xi2 yi2){Mi2} -\pnode(!xi yi){Mi} -\pnode(!xS yS){S} -\pscircle*(S){0.1} -\psline[linestyle=dotted](S)(0,0) -\psline[style=vecteurC]{->}(S)(Mi) -\uput[l](S){S} -\uput[u](Mi2){$\overrightarrow{F}$} -\psarc{->}(0,0){1}{0}{!theta0} -\uput{1.1}[!theta0 2 div](0,0){$\theta_0$} -\psline[arrowinset=0.1,arrowsize=0.2]{<->}(5,0)(0,0)(0,5) -\psline[arrowinset=0.05,arrowsize=0.1]{<->}(5,0)(0,0)(0,5) -\uput[u](0,5){$y$}\uput[r](5,0){$x$} -\end{pspicture} - \end{center} -Il s'obtient très facilement à partir de l'étude précédente si on sait que dans ce cas : -\[ -v_0=\sqrt{\frac{\mathcal{G}M}{r_0}} -\] -\begin{verbatim} -\pstVerb{ - /GM 1 def - /theta0 60 def - /r0 5 def - /x0 r0 theta0 cos mul def - /y0 r0 theta0 sin mul def - /v0 GM r0 div sqrt def - /v0x v0 theta0 sin mul neg def - /v0y v0 theta0 cos mul def }% -\end{verbatim} - -\end{document} +\documentclass{article} +\usepackage[a4paper,margin=2cm]{geometry} +\usepackage[latin1]{inputenc} +\usepackage{pstricks-add,pst-eqdf,pst-func} +\usepackage{array,amsmath} +\usepackage{listings} +\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,linecolor={[rgb]{1 0.5 0}}} +\newpsstyle{vecteurB}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{0 0.5 1}}} +\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0.5 0}}} +%%%%%%%%%%%%%%%%%% +\title{Interaction gravitationnelle avec PSTricks} +\date{19 juin 2\,012} +\begin{document} +\maketitle +\def\eqsatellite{% +y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% +\section{Mise en orbite d'un satellite} +\begin{figure}[htbp] + \begin{center} +\psset{unit=2} +\begin{pspicture}(-6,-2)(2,5) +\pstVerb{ + /GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def +% vitesse à l'apogée + /vA GM par div sqrt 1 exc sub mul def +% vitesse au périgée + /vP GM par div sqrt 1 exc add mul def +% coordonnées de vA + /vAx vA theta0 90 add cos mul neg def + /vAy vA theta0 90 add sin mul neg def +% coordonnées de vP + /vPx vP theta0 90 add cos mul def + /vPy vP theta0 90 add sin mul def +% Apogée + /xA par 1 exc sub div theta0 cos mul neg def + /yA par 1 exc sub div theta0 sin mul neg def +% Périgée + /xP par 1 exc add div theta0 cos mul def + /yP par 1 exc add div theta0 sin mul def +% Centre + /xO xP xA add 2 div def + /yO yP yA add 2 div def +}% +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\psdot[dotstyle=+](0,0) +\uput[d](0,0){O} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,5) +\rput(1,4){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{0000}s}} +\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\pscircle*(!x0 y0){0.05} +\pnode(!xP yP){P} % périgée +\pnode(!xA yA){A} % Apogée +\pnode(!xO yO){O} % centre +\rput(!x0 y0){\psline[style=vecteurB]{->}(!v0x v0y)\uput[ur](!v0x v0y){$\overrightarrow{v_0}$}} +\psline[linestyle=dashed](A)(P) +\pscircle*(A){0.05} +\uput[ul](A){$A$} +\uput[dr](P){$P$} +% position du satellite à un instant quelconque +\pstVerb{/theta_i 170 def + /radius par 1 exc theta_i theta0 sub cos mul add div def + /xS radius theta_i cos mul def + /yS radius theta_i sin mul def + /ux theta_i cos 1 mul def + /uy theta_i sin 1 mul def + /xi xS ux sub def + /yi yS uy sub def + /xi2 xS ux 2 div sub def + /yi2 yS uy 2 div sub def}% +\pnode(!xi2 yi2){Mi2} +\pnode(!xi yi){Mi} +\pnode(!xS yS){S} +\pscircle*(S){0.05} +\psline[linestyle=dotted](S)(0,0) +\psline[style=vecteurA]{->}(S)(Mi) +\uput[l](S){S} +\uput[u](Mi2){$\overrightarrow{F}$} +\psarcn{->}(0,0){0.4}{0}{!theta0} +\uput{0.5}[!theta0 2 div](0,0){$\theta_0$} +\rput(A){\psline[style=vecteurB]{->}(!vAx vAy)} +%\rput(P){\psline[style=vecteurB]{->}(!vPx vPy)} +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,5) +\uput[r](0,4.9){$y$}\uput[u](1.9,0){$x$} +\psdot(O)\uput[r](O){$\Omega$} +\rput{!90 theta0 add}(O){\psline[linestyle=dashed](!b_2 neg 0)(!b_2 0)} +\end{pspicture} +\caption{Mouvement d'un satellite} + \end{center} +\end{figure} +Soit (M) la masse de l'astre et (m) celle du satellite avec $m\ll M$. Le centre de masse du système \{M,m\} est confondu avec le centre de l'astre attracteur. La mise en orbite s'effectue à partir du point $M_0(x_0,y_0)$ avec une vitesse $\overrightarrow{v_0}(v_{0_x},v_{0_x})$. $\theta_0$ est l'angle que fait $\overrightarrow{Ox}$ avec $\overrightarrow{OM_0}$. Le satellite (S), supposé ponctuel, subit de la part de l'astre une force d'attraction gravitationnelle : +\[ +\overrightarrow{F}=-\mathcal{G}\frac{Mm}{r^2}\overrightarrow{u}\qquad \text{avec}\quad \overrightarrow{u}=\frac{\overrightarrow{r}}{r}\quad \text{et}\quad \overrightarrow{r}=\overrightarrow{OS} +\] +Tous les \textit{bons} livres de mécanique\footnote{Comme celui, par exemple, de José-Philippe Pérez, aux éditions Masson.} établissent les relations suivantes : +\[ +r=\frac{p}{1+\mathrm{e}\cos(\theta-\theta_0)} +\] +Paramètres et excentricité ont pour expressions respectives, avec les notations suivantes : $K=\mathcal{G}Mm$, $\mathcal{E}$ l'énergie du système et $L$ le moment cinétique. +\[ +\mathrm{e}=\sqrt{1+\frac{2\mathcal{E}L^2}{mK^2}}\qquad p=\frac{L^2}{mK} +\] +On choisit une vitesse initiale $\overrightarrow{v_0}$ perpendiculaire à $\overrightarrow{OM_0}$, dans ces conditions le moment cinétique et l'énergie, qui restent constants, valent : +\[ +L=mr_0v_0\qquad \mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 +\] +En remplaçant $L$ et $\mathcal{E}$, on obtient pour l'excentricité et le paramètre les expressions suivantes : +\[ +\mathrm{e}=\sqrt{1+\frac{1}{\mathcal{G}^2M^2}\Big(\frac{1}{2}v_0^4r_0^2-\mathcal{G}Mr_0v_0^2\Big)} +\] +\[ +p=\frac{v_0^2r_0^2}{\mathcal{G}M} +\] +On se limite au cas du mouvement elliptique, avec, en conséquence, la condition : +\[ +\mathcal{E}=-\frac{K}{r_0}+\frac{1}{2}mv_0^2 < 0 +\] +Le demi-grand axe $a$, le demi-petit axe $b$ sont : +\[ +a=\frac{p}{1-\mathrm{e}^2}\qquad b=\frac{p}{\sqrt{1-\mathrm{e}^2}} +\] +La période $T$ qui obéit à la troisième loi de Képler : +\[ +T^2=\frac{4\pi^2a^3}{\mathcal{G}M} +\] +La vitesse, en un point de l'ellipse, se calcule par : +\[ +v^2=\mathcal{G}M\Big( \frac{2}{r}-\frac{1}{a}\Big) +\] +Sachant que $r_p=\dfrac{p}{1+\mathrm{e}}$ et $r_A=\dfrac{p}{1-\mathrm{e}}$, on en déduit les vitesses au périgée et à l'apogée : +\[ +v_P=\sqrt{\frac{\mathcal{G}M}{p}}(1+\mathrm{e})\qquad v_A=\sqrt{\frac{\mathcal{G}M}{p}}(1-\mathrm{e}) +\] + +\section{L'étude avec PSTricks} +\subsection{La trajectoire} +\def\parametres{ + /GM 1 def % 4e14 def % GxM + /x0 6.5e6 def % position initiale + /y0 0 def + /vx0 0 def % vitesse initiale + /vy0 1e4 def +} +% x0 y0 x'0 y'0 +% y[0] y[1] y[2] y[3] +\[ +\left\{ +\begin{array}{rcl} +\ddot{x}&=&-\dfrac{GM}{r^3}x\\[1em] +\ddot{y}&=&-\dfrac{GM}{r^3}y\\ +\end{array} +\right. +\label{eq1} +\qquad r=\sqrt{x^2+y^2} +\] + +On peut dessiner la trajectoire du satellite et de ses caractéristiques de deux façons : +\begin{itemize} + \item par l'utilisation de \verb+\parametricplot+ ; + \item ou celle de \verb+\psplotDiffEqn+. +\end{itemize} +\verb+\parametricplot+ utilise l'expression exacte de l'équation de la trajectoire en coordonnées polaires : +\begin{verbatim} + \parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{% + /radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\end{verbatim} +L'excentricité, la période, demi-grand axe et demi-petit axe sont calculés par quelques lignes de code \textsf{postscript}. Il faut s'assurer que les conditions initiales choisies vérifient bien la condition d'une trajectoire elliptique, pour cela il faut que l'énergie initiale $\mathcal{E}_0<0$, sinon cela entraînera une erreur lors du passage à l'interpréteur \textsf{postscript}. +\begin{verbatim} +\pstVerb{ + /GM 1 def + /theta0 -45 def + /r0 0.5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.92 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul + v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%demi-grand axe + /a_2 par 1 exc dup mul sub div def % demi-grand axe +%demi-petit axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe +% période + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +}% +\end{verbatim} +\verb+\psplotDiffEqn+ utilise les équations différentielles du mouvement, en notation algébrique : +\begin{verbatim} +% x0 y0 x'0 y'0 +% y[0] y[1] y[2] y[3] +\def\eqsatellite{% +y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)} + \psplotDiffEqn[unit=2,whichabs=0,whichord=1,% + linecolor=blue,linewidth=0.1,% + method=rk4,plotpoints=1000,% + algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} +Ce qui permet, par ailleurs de vérifier la qualité du tracé par la méthode numérique, en bleu, tandis que le tracé à partir de l'expression exacte est en trait fin en rouge. +% +\begin{center} +\begin{pspicture}(-12,-2)(4,10) +\pstVerb{ + /GM 1 def + /theta0 -45 def + /r0 0.5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.92 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +}% +\psframe*[linecolor=white](-3,-0.2)(0,0.2) +\pscircle[fillcolor=blue!50,fillstyle=solid](0,0){0.5} +\psgrid[unit=2,subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-1)(2,5) +\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} + \psplotDiffEqn[unit=2,whichabs=0,whichord=1,linecolor=blue,linewidth=0.1,method=rk4,plotpoints=1000,algebraic]{0}{37.8}{x0 y0 v0x v0y}{\eqsatellite}% +\parametricplot[linecolor=red,unit=2,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\psdot[unit=2,dotsize=0.12](!x0 y0) +\rput(!x0 2 mul y0 2 mul){\psline[style=vecteurC]{->}(!v0x v0y)} +\end{pspicture} +\end{center} +\subsection{La vitesse} +\verb+\psplotDiffEqn+ permet de voir comment varie la vitesse sur l'ellipse : +\begin{verbatim} +% x0 y0 x'0 y'0 +% y[0] y[1] y[2] y[3] + \psplotDiffEqn[xunit=0.2,yunit=5,% + plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, + linecolor=red,method=rk4,plotpoints=1000, + algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} +\begin{center} +\begin{pspicture}(0,-1)(10,7) +\psgrid[subgriddiv=0,gridcolor=lightgray,griddots=10,gridlabels=0pt] +\pstVerb{/GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def}% +\psplotDiffEqn[xunit=0.2,yunit=5, + plotfuncy=dup 2 get dup mul exch 3 get dup mul add sqrt, + linecolor=red,method=rk4,plotpoints=1000,algebraic]{0}{50}{x0 y0 v0x v0y}{\eqsatellite}% + \multido{\i=1+1,\I=5+5}{9}{\uput[u](\i,0){\I}} +\pnode(! 36.4 5 div 0){P} +\psdot(P)\uput[d](P){Périgée} +\psline[linestyle=dashed](P)(! 36.4 5 div 7) +\pnode(! 36.4 10 div 0){A} +\psdot(A)\uput[d](A){Apogée} +\psline[linestyle=dashed](A)(! 36.4 10 div 7) +\uput[l](0,6.5){$v$} +\psline[arrowinset=0.1,arrowsize=0.2]{<->}(10,0)(0,0)(0,7) +\uput[u](10,0){$t$(s)} +\uput[ul](0,0){0} +\end{pspicture} +\end{center} +On peut obtenir les caractéristiques de la vitesse en un point quelconque, car le moment cinétique $L=mr\dot{\theta}$ étant constant, pour chaque valeur de $\theta$, on en déduit $r$ puis $\dot{\theta}$. En coordonnées polaires, la vitesse s'exprime par : +\[ +\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} +\] +$\dot{\theta}$ et $\dot{r}$ s'obtiennent par les relations suivantes : +\[ +\dot{\theta}=\frac{r_0v_0}{r^2} +\] +\[ +\dot{r}=-\frac{p(-\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\sin(\theta-\theta_0) +\] +La chaîne de calculs est la suivante : $\theta\Longrightarrow r\Longrightarrow \dot{\theta}\Longrightarrow \dot{r}\Longrightarrow \overrightarrow{v}$. + +Le package `\textsf{pst-eqdf}' comprend la commande \verb+\psequadiff+ qui est une version simplifiée de \verb+\psplotDiffEqn+, dont elle ne reprend que la méthode Runge-Kutta~4. Elle permet de sauvegarder sous forme de tableaux et/ou de fichiers toutes les variables et les dérivées de la fonction étudiée, cette possibilité est intéressante pour déterminer les caractéristiques de la vitesse au cours du temps et elle est particulièrement utile pour créer une animation. Nous allons l'utiliser pour dessiner le vecteur-vitesse à quelques instants. + +On sauve successivement le tableau des positions et celui des vitesses. +\begin{verbatim} +\psequadiff[method=rk4,plotpoints=1000, + algebraic, + whichabs=0,whichord=1, + tabname=XiYi]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} +\begin{verbatim} +\psequadiff[method=rk4,plotpoints=1000, + algebraic, + whichabs=2,whichord=3, + tabname=vxvy]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\end{verbatim} + +Pour ensuite dessiner la trajectoire et les vecteurs-vitesse. +\begin{verbatim} +%\listplot[unit=1]{vxvy aload pop} +% on dessine la vitesse un point sur 100 +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\multido{\i=0+100}{20}{% +\pstVerb{/vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} +\end{verbatim} +\begin{center} +\begin{pspicture}(-10,-10)(6,7) +\psset{unit=2}% +\pstVerb{/GM 1 def + /theta0 30 def + /r0 2 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 0.85 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def}% +\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi +% ,saveData,filename=XiYi.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy +% ,saveData,filename=vxvy.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +%\listplot[unit=1]{vxvy aload pop} +% on dessine la vitesse un point sur 100 +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\multido{\i=0+100}{20}{% +\pstVerb{/vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(3,3) +\end{pspicture} +\end{center} +% +\section{Mouvement circulaire} + \begin{center} +\psset{unit=1} +\begin{pspicture}(-5,-5.5)(5,5.5) +\pstVerb{ + /GM 1 def + /theta0 60 def + /r0 5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 GM r0 div sqrt def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 mul a_2 3 exp GM div sqrt mul def +% vitesse à l'apogée + /vA GM par div sqrt 1 exc sub mul def +% vitesse au périgée + /vP GM par div sqrt 1 exc add mul def +% coordonnées de vA + /vAx vA theta0 90 add cos mul neg def + /vAy vA theta0 90 add sin mul neg def +% coordonnées de vP + /vPx vP theta0 90 add cos mul def + /vPy vP theta0 90 add sin mul def +}% +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.75} +\psdot[dotstyle=+](0,0) +\uput[d](0,0){O} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(5,5) +\rput(0,-2){\psframebox[linestyle=none,fillstyle=solid,fillcolor=white]{T=\psPrintValue[decimals=2]{periode}\hphantom{000000}s}} +\parametricplot[linecolor=red,plotpoints=360]{0}{360}{/radius par 1 exc t theta0 sub cos mul add div def + radius t cos mul + radius t sin mul} +\pscircle*(!x0 y0){0.1} +%\pnode(!par 1 exc add div theta0 cos mul par 1 exc add div theta0 sin mul){P} % périgée +%\pnode(!par 1 exc sub div theta0 cos mul neg par 1 exc sub div theta0 sin mul neg){A} % Apogée +\rput(!x0 y0){\psline[arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{0 0.5 1}},unit=4]{->}(!v0x v0y)\uput[ur](!v0x 2 mul v0y 2 mul){$\overrightarrow{v_0}$}} +\uput[ur](!x0 y0){$M_0$} +\psline[linestyle=dashed](0,0)(!x0 y0) +% position du satellite à un instant quelconque +\pstVerb{/theta_i 170 def + /radius par 1 exc theta_i theta0 sub cos mul add div def + /xS radius theta_i cos mul def + /yS radius theta_i sin mul def + /ux theta_i cos 1 mul def + /uy theta_i sin 1 mul def + /xi xS ux sub def + /yi yS uy sub def + /xi2 xS ux 2 div sub def + /yi2 yS uy 2 div sub def}% +\pnode(!xi2 yi2){Mi2} +\pnode(!xi yi){Mi} +\pnode(!xS yS){S} +\pscircle*(S){0.1} +\psline[linestyle=dotted](S)(0,0) +\psline[style=vecteurC]{->}(S)(Mi) +\uput[l](S){S} +\uput[u](Mi2){$\overrightarrow{F}$} +\psarc{->}(0,0){1}{0}{!theta0} +\uput{1.1}[!theta0 2 div](0,0){$\theta_0$} +\psline[arrowinset=0.1,arrowsize=0.2]{<->}(5,0)(0,0)(0,5) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(5,0)(0,0)(0,5) +\uput[u](0,5){$y$}\uput[r](5,0){$x$} +\end{pspicture} + \end{center} +Il s'obtient très facilement à partir de l'étude précédente si on sait que dans ce cas : +\[ +v_0=\sqrt{\frac{\mathcal{G}M}{r_0}} +\] +\begin{verbatim} +\pstVerb{ + /GM 1 def + /theta0 60 def + /r0 5 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 GM r0 div sqrt def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def }% +\end{verbatim} +\section{L'hodographe du mouvement du satellite} +On rappelle qu'en coordonnées polaires le vecteur-vitesse s'écrit : +\[ +\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} +\] +\[ +\dot{\theta}=\frac{r_0v_0}{r^2} +\] +\[ +\dot{r}=-\frac{p(-\mathrm{e}\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0) +\] +On peut exprimer $v$ uniquement en fonction de $\theta$ : +\[ +\overrightarrow{v}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\overrightarrow{u_r}+\frac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\overrightarrow{u_\theta} +\] +Ses composantes dans la base $(\overrightarrow{u_r},u_\theta)$ sont : +\[ +\left\{ +\begin{array}{rcl} +v_r&=&\dfrac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\\[1em] +v_\theta&=&\dfrac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big) +\end{array} +\right. +\label{eqv1} +\] +On repasse aux coordonnées cartésiennes par une rotation d'angle $(-\theta)$. +\[ +\left\{ +\begin{array}{rcl} +\dot{x}&=&v_r\cos\theta-v_\theta\sin\theta\\[1em] +\dot{y}&=&v_r\sin\theta+v_\theta\cos\theta +\end{array} +\right. +\label{eqv2} +\] +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&\mathrm{e}\sin(\theta-\theta_0)\cos\theta-\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\mathrm{e}\sin(\theta-\theta_0)\sin\theta+\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\cos\theta +\end{array} +\right. +\label{eqv3} +\] +En utilisant les relations trigonométriques de soustraction : +\[ +\sin\alpha\cos\beta-\cos\alpha\sin\beta=\sin(\alpha-\beta)\qquad \cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta) +\] +On obtient : +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&-\mathrm{e}\sin\theta_0-\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\hphantom{-}\mathrm{e}\cos\theta_0+\cos\theta +\end{array} +\right. +\label{eqv4} +\] +\[ +\left\{ +\begin{array}{rcl} +\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0&=&-\dfrac{r_0v_0}{p}\sin\theta\\[1em] +\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0&=&\dfrac{r_0v_0}{p}\cos\theta +\end{array} +\right. +\label{eqv5} +\] +L'équation de l'hodographe : +\[ +\left(\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0\right)^2+\left(\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\right)^2=\left(\dfrac{r_0v_0}{p}\right)^2 +\] +est celle d'un cercle centré en $\Big(\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0,\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\Big)$, de rayon $R=\dfrac{r_0v_0}{p}$. Ce qu'on vérifie sur le graphe suivant : en rouge l'hodographe a été obtenu à partir de l'équation différentielle et en noir avec un trait plus fin par l'expression exacte de l'équation du cercle. +\begin{center} +\begin{pspicture}(-12,-2)(4,8) +\psset{unit=2}% +\uput[r](0,4){$y$}\uput[u](2,0){$x$} +\pstVerb{/GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +% rayon de l'hodographe + /rH r0 v0 mul par div def +% les coordonnées du centre + /xCH rH exc mul theta0 sin mul neg def + /yCH rH exc mul theta0 cos mul def}% +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi, +% saveData,filename=XiYi.dat +]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy, +% saveData,filename=vxvy.dat +]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot[unit=1,linecolor=red,linewidth=0.075]{vxvy aload pop} +% on dessine la vitesse un point sur 50 +\multido{\i=0+50}{40}{% +\pstVerb{ /vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[unit=1,linecolor=blue]{->}(!vX vY)}} +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.2} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-6,-2)(2,4) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,4) +\pscircle(!xCH yCH){!rH} +\psdot[dotstyle=+](!xCH yCH) +\end{pspicture} +\end{center} +\end{document} diff --git a/gravitation/gravitation_02.pdf b/gravitation/gravitation_02.pdf index 8b08656..ccab032 100644 Binary files a/gravitation/gravitation_02.pdf and b/gravitation/gravitation_02.pdf differ diff --git a/gravitation/gravitation_02.tex b/gravitation/gravitation_02.tex new file mode 100644 index 0000000..770e71e --- /dev/null +++ b/gravitation/gravitation_02.tex @@ -0,0 +1,194 @@ +\documentclass{article} +\usepackage[a4paper,margin=2cm]{geometry} +\usepackage[latin1]{inputenc} +\usepackage{pst-eqdf,pst-func} +\usepackage{array,amsmath} +\newpsstyle{vecteurA}{arrowinset=0.05,arrowsize=0.125,linecolor={[rgb]{1 0.5 0}}} +\newpsstyle{vecteurB}{arrowinset=0.05,arrowsize=0.1,linecolor={[rgb]{0 0.5 1}}} +\newpsstyle{vecteurC}{arrowinset=0.1,arrowsize=0.2,linecolor={[rgb]{1 0.5 0}}} +%%%%%%%%%%%%%%%%%% +\title{L'hodographe du mouvement d'un satellite} +\date{23 juin 2\,012} +\begin{document} +\maketitle +\def\eqsatellite{% +y[2]|y[3]|-GM*y[0]/((sqrt(y[0]^2+y[1]^2))^3)|-GM*y[1]/((sqrt(y[0]^2+y[1]^2))^3)}% +On rappelle qu'en coordonnées polaires le vecteur-vitesse s'écrit : +\[ +\overrightarrow{v}=\dot{r}\overrightarrow{u_r}+r\dot{\theta}\overrightarrow{u_{\theta}} +\] +\[ +\dot{\theta}=\frac{r_0v_0}{r^2} +\] +\[ +\dot{r}=-\frac{p(-\mathrm{e}\dot{\theta}\sin(\theta-\theta_0)}{(1+\mathrm{e}\cos(\theta-\theta_0))^2}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0) +\] +On peut exprimer $v$ uniquement en fonction de $\theta$ : +\[ +\overrightarrow{v}=\frac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\overrightarrow{u_r}+\frac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\overrightarrow{u_\theta} +\] +Ses composantes dans la base $(\overrightarrow{u_r},u_\theta)$ sont : +\[ +\left\{ +\begin{array}{rcl} +v_r&=&\dfrac{r_0v_0}{p}\mathrm{e}\sin(\theta-\theta_0)\\[1em] +v_\theta&=&\dfrac{r_0v_0}{p}\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big) +\end{array} +\right. +\label{eqv1} +\] +On repasse aux coordonnées cartésiennes par une rotation d'angle $(-\theta)$. +\[ +\left\{ +\begin{array}{rcl} +\dot{x}&=&v_r\cos\theta-v_\theta\sin\theta\\[1em] +\dot{y}&=&v_r\sin\theta+v_\theta\cos\theta +\end{array} +\right. +\label{eqv2} +\] +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&\mathrm{e}\sin(\theta-\theta_0)\cos\theta-\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\mathrm{e}\sin(\theta-\theta_0)\sin\theta+\Big(1+\mathrm{e}\cos(\theta-\theta_0)\Big)\cos\theta +\end{array} +\right. +\label{eqv3} +\] +En utilisant les relations trigonométriques de soustraction : +\[ +\sin\alpha\cos\beta-\cos\alpha\sin\beta=\sin(\alpha-\beta)\qquad \cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha-\beta) +\] +On obtient : +\[ +\left\{ +\begin{array}{rcl} +\dfrac{p}{r_0v_0}\dot{x}&=&-\mathrm{e}\sin\theta_0-\sin\theta\\[1em] +\dfrac{p}{r_0v_0}\dot{y}&=&\hphantom{-}\mathrm{e}\cos\theta_0+\cos\theta +\end{array} +\right. +\label{eqv4} +\] +\[ +\left\{ +\begin{array}{rcl} +\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0&=&-\dfrac{r_0v_0}{p}\sin\theta\\[1em] +\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0&=&\dfrac{r_0v_0}{p}\cos\theta +\end{array} +\right. +\label{eqv5} +\] +L'équation de l'hodographe : +\[ +\left(\dot{x}+\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0\right)^2+\left(\dot{y}-\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\right)^2=\left(\dfrac{r_0v_0}{p}\right)^2 +\] +est celle d'un cercle centré en $\Big(\dfrac{r_0v_0}{p}\mathrm{e}\sin\theta_0,\dfrac{r_0v_0}{p}\mathrm{e}\cos\theta_0\Big)$, de rayon $R=\dfrac{r_0v_0}{p}$. Ce qu'on vérifie sur le graphe suivant : en rouge l'hodographe a été obtenu à partir de l'équation différentielle et en noir avec un trait plus fin par l'expression exacte de l'équation du cercle. +\begin{center} +\begin{pspicture}(-10,-2)(4,8) +\psset{unit=2}% +\uput[r](0,4){$y$}\uput[u](2,0){$x$} +\pstVerb{/GM 1 def + /theta0 -35 def + /r0 1 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 1.3 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def +% rayon de l'hodographe + /rH r0 v0 mul par div def +% les coordonnées du centre + /xCH rH exc mul theta0 sin mul neg def + /yCH rH exc mul theta0 cos mul def}% +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi, + saveData,filename=XiYi.dat]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy, + saveData,filename=vxvy.dat]{0}{36.5}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot[unit=1,linecolor=red,linewidth=0.075]{vxvy aload pop} +% on dessine la vitesse un point sur 50 +\multido{\i=0+50}{40}{% +\pstVerb{ /vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[unit=1,linecolor=blue]{->}(!vX vY)}} +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.2} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-2)(2,4) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(2,0)(0,0)(0,4) +\pscircle(!xCH yCH){!rH} +\psdot[dotstyle=+](!xCH yCH) +\end{pspicture} +\end{center} +\newpage +\begin{center} +\begin{pspicture}(-10,-10)(6,6) +\psset{unit=2}% +\uput[r](0,3){$y$}\uput[u](3,0){$x$} +\pstVerb{/GM 1 def + /theta0 30 def + /r0 2 def + /x0 r0 theta0 cos mul def + /y0 r0 theta0 sin mul def + /v0 0.85 def + /v0x v0 theta0 sin mul neg def + /v0y v0 theta0 cos mul def + /Lc r0 v0 mul def % moment cinetique + /par Lc dup mul GM div def % paramètre de l'ellipse +% excentricité + /exc 1 0.5 v0 4 exp mul r0 dup mul mul GM r0 mul v0 dup mul mul sub GM dup mul div 2 mul add sqrt def +%%%%%%%%%%%%%% + /a_2 par 1 exc dup mul sub div def % demi-grand axe + /b_2 par 1 exc dup mul sub sqrt div def % demi-petit axe + /periode 2 3.1416 dup mul a_2 3 exp mul GM div sqrt mul def}% +\rput(-2,0){T=\psPrintValue[decimals=2]{periode}\hphantom{00000}s} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=0, + whichord=1, + tabname=XiYi, +% saveData,filename=XiYi.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot{XiYi aload pop} +\psequadiff[method=rk4, + plotpoints=1000, + algebraic, + whichabs=2, + whichord=3, + tabname=vxvy, +% saveData,filename=vxvy.dat +]{0}{43}{x0 y0 v0x v0y}{\eqsatellite}% +\listplot[unit=2,linecolor=red]{vxvy aload pop} +% on dessine la vitesse un point sur 100 +\pscircle[fillcolor=gray!70,fillstyle=solid](0,0){0.3} +\multido{\i=0+100}{20}{% +\pstVerb{/vX vxvy \i\space get def + /vY vxvy \i\space 1 add get def + /xi XiYi \i\space get def + /yi XiYi \i\space 1 add get def}% +\rput(!xi yi){\psline[style=vecteurA]{->}(! vX 2 mul vY 2 mul)}} +\psgrid[subgriddiv=2,gridcolor=lightgray,gridlabels=8pt](-5,-5)(3,3) +\psline[arrowinset=0.05,arrowsize=0.1]{<->}(3,0)(0,0)(0,3) +\end{pspicture} +\end{center} +\end{document} diff --git a/gravitation/h1.png b/gravitation/h1.png new file mode 100644 index 0000000..7d3b31b Binary files /dev/null and b/gravitation/h1.png differ diff --git a/gravitation/h2.png b/gravitation/h2.png new file mode 100644 index 0000000..43cc423 Binary files /dev/null and b/gravitation/h2.png differ diff --git a/gravitation/pst-eqdf.tex b/gravitation/pst-eqdf.tex index 0585839..803270e 100644 --- a/gravitation/pst-eqdf.tex +++ b/gravitation/pst-eqdf.tex @@ -41,7 +41,7 @@ \define@key[psset]{pst-eqd}{plotfuncx}{\edef\psk@plotfuncx{#1}}% \define@key[psset]{pst-eqd}{plotfuncy}{\edef\psk@plotfuncy{#1}}% \define@key[psset]{pst-eqd}{tabname}{\edef\psk@tabname{#1}}% -\define@key[psset]{pst-eqd}{filename}{\edef\psk@filename{\datRoot #1}}% +\define@key[psset]{pst-eqd}{filename}{\edef\psk@filename{\datRoot#1}}% \newif\ifPst@buildvector% \define@key[psset]{pst-eqd}{buildvector}[true]{\@nameuse{Pst@buildvector#1}}% \newif\ifPst@saveData% @@ -62,7 +62,6 @@ \pst@killglue \begingroup \use@par -% \addto@pscode{% \pstVerb{ /x #1 def /x1 #2 def diff --git a/lorenz/LorenzXY.dat b/lorenz/LorenzXY.dat index f703118..5cd616d 100644 --- a/lorenz/LorenzXY.dat +++ b/lorenz/LorenzXY.dat @@ -1,2501 +1,2501 @@ -10 10 -9.98523 9.69224 -9.94194 9.37201 -9.87188 9.04414 -9.77712 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36.0605 -8.82258 35.2275 -8.04178 34.3526 -7.28631 33.4569 -6.56497 32.5568 -5.88418 31.6641 -5.24827 30.7873 -4.65974 29.9315 -4.11953 29.0999 -3.62735 28.294 -3.18195 27.5141 -2.78137 26.7599 -2.42311 26.0305 -2.10437 25.3249 -1.82215 24.6418 -1.57337 23.9801 -1.35499 23.3386 -1.16404 22.7164 -0.997693 22.1124 -0.853283 21.5258 -0.728331 20.9558 -0.620558 20.4016 -0.527884 19.8628 -0.448429 19.3386 -0.3805 18.8287 -0.322591 18.3324 -0.273359 17.8494 -0.231624 17.3793 -0.196347 16.9217 -0.166617 16.4762 -0.141645 16.0425 -0.120744 15.6203 -0.103322 15.2092 -0.088868 14.8089 -0.0769463 14.4192 -0.0671837 14.0397 -0.0592631 13.6703 -0.0529161 13.3106 -0.0479168 12.9603 -0.0440759 12.6193 -0.0412359 12.2872 -0.039267 11.9639 -0.038063 11.6491 -0.0375384 11.3425 -0.0376258 11.0441 -0.0382735 10.7535 -0.0394435 10.4705 -0.04111 10.195 -0.0432581 9.92679 -0.0458831 9.6656 -0.048989 9.41129 -0.0525888 9.16369 -0.0567034 8.9226 -0.0613618 8.68787 -0.0666009 8.45933 -0.0724658 8.23682 -0.0790096 8.02018 -0.0862944 7.80926 -0.0943915 7.60392 -0.103382 7.40401 -0.113358 7.2094 -0.124423 7.01996 -0.136695 6.83556 -0.150303 6.65609 -0.165394 6.48142 -0.182134 6.31145 -0.200704 6.14608 -0.221312 5.98522 -0.244185 5.82878 -0.269581 5.6767 -0.297785 5.5289 -0.329118 5.38535 -0.363937 5.24602 -0.402641 5.11089 -0.445676 4.97997 -0.493539 4.85332 -0.546788 4.73101 -0.606043 4.61317 -0.671999 4.49997 -0.745432 4.39164 -0.827206 4.28853 -0.918288 4.19104 -1.01976 4.09972 -1.13281 4.01528 -1.2588 3.9386 -1.3992 3.8708 -1.55569 3.8133 -1.7301 3.76786 -1.92448 3.73671 -2.14109 3.72261 -2.38242 3.72899 -2.6512 3.76013 -2.95044 3.8213 -3.28338 3.91902 -3.65352 4.0613 -4.06458 4.25793 -4.52049 4.52083 -5.02524 4.86442 -5.58286 5.30592 -6.19717 5.86575 -6.87157 6.56768 -7.60868 7.43889 -8.40987 8.50958 -9.27469 9.8121 -10.2001 11.3792 -11.1795 13.2413 -12.202 15.422 -13.2508 17.933 -14.3032 20.7668 -15.3293 23.8894 -16.2924 27.2337 -17.1505 30.6964 -17.8582 34.14 -18.3709 37.4025 -18.6493 40.3146 -18.664 42.7228 -18.4003 44.5126 -17.8602 45.6256 -17.0628 46.0658 -16.0417 45.8944 -14.8411 45.2142 -13.5103 44.1489 -12.0991 42.8243 -10.6539 41.3538 -9.21441 39.8295 -7.8128 38.3201 -6.47326 36.8717 -5.21253 35.5121 -4.04091 34.2545 -2.96339 33.1024 -1.98081 32.0525 -1.09097 31.098 -0.289462 30.2304 --0.429536 29.4405 --1.07273 28.7195 --1.64724 28.0596 --2.16032 27.4539 --2.61906 26.8964 --3.03032 26.3825 --3.40055 25.9081 --3.73575 25.4703 --4.04148 25.0669 --4.32278 24.6963 --4.58422 24.3572 --4.82993 24.0493 --5.06355 23.7721 --5.28832 23.5259 --5.50709 23.311 --5.72232 23.128 --5.93612 22.9778 --6.15026 22.8614 --6.3662 22.7797 --6.58508 22.734 --6.80776 22.7255 --7.03478 22.7553 --7.26642 22.8246 --7.50264 22.9343 --7.74311 23.0852 --7.9872 23.2777 --8.23398 23.512 --8.4822 23.7875 --8.73032 24.1033 --8.97648 24.4578 --9.21854 24.8484 --9.4541 25.272 --9.68053 25.7243 --9.89499 26.2001 --10.0945 26.6936 --10.2762 27.1978 --10.4369 27.7052 --10.5738 28.2077 --10.6844 28.6967 --10.7662 29.1638 --10.8174 29.6007 --10.8367 29.9996 --10.8232 30.3536 --10.7767 30.6568 --10.6978 30.9046 --10.5876 31.0938 --10.4479 31.2226 --10.281 31.2907 --10.0898 31.2991 --9.87754 31.2501 --9.64766 31.147 --9.40389 30.9938 --9.15001 30.7951 --8.88973 30.556 --8.62668 30.2815 --8.36425 29.9768 --8.10562 29.6469 --7.85364 29.2966 --7.61085 28.9302 --7.37946 28.552 --7.16134 28.1656 --6.95806 27.7745 --6.77085 27.3816 --6.60072 26.9898 --6.44838 26.6013 --6.31435 26.2186 --6.19897 25.8435 --6.1024 25.478 --6.0247 25.1237 --5.96581 24.7823 --5.92559 24.4552 --5.90385 24.1441 --5.90035 23.8503 --5.91482 23.5754 --5.94698 23.3208 --5.99652 23.0881 --6.06313 22.8788 --6.14646 22.6946 --6.24619 22.5372 --6.36192 22.4082 --6.49327 22.3094 --6.63978 22.2427 --6.80094 22.2099 --6.97616 22.2128 --7.16473 22.2531 --7.36586 22.3326 --7.57857 22.4527 --7.80173 22.6148 --8.03401 22.8199 --8.27386 23.0685 --8.51948 23.3608 --8.76881 23.6963 --9.01954 24.0735 --9.26909 24.4906 --9.51459 24.9443 --9.75296 25.4306 --9.98091 25.9444 --10.195 26.4794 --10.3918 27.0284 --10.5677 27.5833 --10.7194 28.1351 --10.8437 28.6746 --10.9378 29.1922 --10.9994 29.6784 --11.0267 30.1245 --11.0185 30.5224 --10.9744 30.8652 --10.8948 31.1475 --10.7808 31.3654 --10.6341 31.5169 --10.4573 31.6013 --10.2534 31.6199 --10.026 31.5751 --9.77885 31.4706 --9.51614 31.3109 --9.242 31.1014 --8.96058 30.8477 --8.67587 30.5555 --8.39165 30.2307 --8.11139 29.8788 --7.83825 29.5051 --7.57498 29.1144 --7.32399 28.7112 --7.0873 28.2996 --6.86655 27.8831 --6.66307 27.4649 --6.47786 27.0479 --6.31166 26.6346 --6.16496 26.2273 --6.03807 25.8279 --5.9311 25.4383 --5.84406 25.0603 --5.77683 24.6952 --5.72922 24.3448 --5.70099 24.0104 --5.69184 23.6935 --5.70149 23.3955 --5.7296 23.118 --5.77586 22.8623 --5.83994 22.6302 --5.92152 22.4232 --6.02028 22.243 --6.13588 22.0915 --6.26797 21.9705 --6.41616 21.8819 --6.58 21.8277 --6.75901 21.8101 --6.95258 21.8309 --7.16001 21.8922 --7.38045 21.9958 --7.61288 22.1435 --7.85607 22.3368 --8.10859 22.5767 --8.36872 22.864 --8.63447 23.1986 --8.90356 23.58 --9.17339 24.0065 --9.44103 24.4756 --9.70328 24.9837 --9.95665 25.5258 --10.1974 26.096 --10.4218 26.6868 --10.6258 27.2897 --10.8055 27.8953 --10.9573 28.4932 --11.0777 29.0727 --11.1638 29.6232 --11.2132 30.1341 --11.2242 30.5959 --11.1958 31.0001 --11.1281 31.3397 --11.0217 31.6096 --10.8783 31.8066 --10.7004 31.9294 --10.4912 31.9788 --10.2544 31.9569 --9.9942 31.8678 --9.71511 31.7163 --9.42177 31.5083 --9.11882 31.2502 --8.81078 30.9484 --8.50193 30.6096 --8.19623 30.2401 --7.89726 29.8458 --7.60817 29.4322 --7.3317 29.0044 --7.07012 28.5668 --6.82532 28.1234 --6.59877 27.6776 --6.39159 27.2326 --6.20459 26.791 --6.03831 26.3552 --5.89305 25.9271 --5.76891 25.5088 --5.66586 25.1018 --5.58374 24.7077 --5.5223 24.328 --5.48125 23.964 --5.46024 23.6173 --5.45893 23.2891 --5.47695 22.9809 --5.51395 22.6941 --5.56959 22.4302 --5.64355 22.191 --5.73552 21.978 --5.8452 21.7931 --5.97228 21.6381 --6.11646 21.5152 --6.2774 21.4263 --6.4547 21.3737 --6.64793 21.3597 --6.85653 21.3864 --7.07984 21.4564 --7.31704 21.5716 --7.56712 21.7341 --7.82885 21.9458 --8.10075 22.2081 --8.38103 22.5219 --8.6676 22.8874 --8.958 23.3043 --9.24942 23.7709 --9.53868 24.2846 --9.82224 24.8414 --10.0962 25.4362 --10.3565 26.0619 --10.5988 26.7104 --10.8187 27.372 --11.0118 28.0359 --11.1738 28.6905 --11.3011 29.3234 --11.3902 29.9225 --11.4385 30.4758 --11.4441 30.9726 --11.4061 31.4035 --11.3245 31.761 --11.2005 32.0397 --11.036 32.2365 --10.834 32.3507 --10.5982 32.3837 --10.3331 32.3389 --10.0434 32.2212 --9.73428 32.0369 --9.41103 31.7929 --9.0788 31.4966 --8.74258 31.1557 --8.40702 30.7775 --8.07637 30.3689 --7.75444 29.9364 --7.4445 29.486 --7.14934 29.0228 --6.87125 28.5514 --6.61204 28.076 --6.3731 27.5999 --6.15544 27.1261 --5.95973 26.6572 --5.78634 26.1954 --5.63543 25.7427 --5.50696 25.3007 --5.40074 24.8711 --5.31648 24.4553 --5.25383 24.0546 --5.21236 23.6702 --5.19166 23.3036 --5.1913 22.956 --5.21088 22.6288 --5.25 22.3234 --5.30831 22.0412 --5.38549 21.7839 --5.48124 21.5532 --5.59531 21.351 --5.72744 21.1791 --5.87739 21.0398 --6.0449 20.9353 --6.22968 20.8679 --6.4314 20.8402 --6.64961 20.8548 --6.88379 20.9142 --7.13323 21.021 --7.39705 21.1779 --7.67414 21.387 --7.96312 21.6505 --8.26229 21.9698 --8.56959 22.3459 --8.88257 22.779 --9.19837 23.268 --9.51368 23.8109 --9.82478 24.404 --10.1275 25.0423 --10.4174 25.7189 --10.6895 26.4251 --10.9391 27.1507 --11.161 27.8837 --11.3503 28.6111 --11.5025 29.319 --11.6136 29.9931 --11.6802 30.6198 --11.6997 31.1859 --11.6709 31.6803 --11.5935 32.0937 --11.4682 32.4194 --11.2972 32.6535 --11.0837 32.7947 --10.8317 32.8446 --10.5461 32.8069 --10.2323 32.6874 --9.89624 32.4932 --9.54377 32.2326 --9.18079 31.9141 --8.81296 31.5465 --8.44556 31.1383 --8.08336 30.6975 --7.73063 30.2313 --7.391 29.7463 --7.06755 29.2484 --6.76276 28.7424 --6.47857 28.2328 --6.21642 27.723 --5.97733 27.2163 --5.76193 26.7151 --5.57052 26.2217 --5.40317 25.738 --5.25971 25.2655 --5.13984 24.8058 --5.04315 24.3601 --4.96914 23.9296 --4.91731 23.5156 --4.88711 23.1192 --4.87803 22.7416 --4.88958 22.3839 --4.92132 22.0476 --4.97285 21.734 --5.04382 21.4445 --5.13396 21.1809 --5.24301 20.9449 --5.37079 20.7385 --5.51712 20.5638 --5.68184 20.423 --5.86481 20.3187 --6.06583 20.2534 --6.28467 20.2301 --6.52101 20.2516 --6.7744 20.3209 --7.04422 20.4412 --7.32967 20.6155 --7.62967 20.8465 --7.94283 21.1369 --8.26742 21.4887 --8.60128 21.9032 --8.94179 22.381 --9.28584 22.9213 --9.62981 23.5222 --9.96955 24.1798 --10.3004 24.8884 --10.6172 25.6404 --10.9146 26.4259 --11.1868 27.233 --11.4281 28.048 --11.6328 28.8556 --11.7956 29.6394 --11.9119 30.383 --11.978 31.0703 --11.991 31.6863 --11.9495 32.2183 --11.8533 32.6562 --11.7038 32.9928 --11.5035 33.2247 --11.2564 33.3515 --10.9674 33.3761 --10.6425 33.3038 --10.288 33.1425 --9.91065 32.9011 --9.51733 32.5896 --9.11468 32.2184 --8.709 31.7975 --8.30605 31.3367 --7.91101 30.8447 --7.52834 30.3295 --7.16181 29.7979 --6.81451 29.256 --6.48883 28.7088 --6.18658 28.1605 --5.90902 27.6145 --5.65694 27.0738 --5.43072 26.5407 --5.23043 26.0171 --5.05588 25.5048 --4.90667 25.0049 --4.78229 24.5189 --4.6821 24.0477 --4.60545 23.5924 --4.55165 23.154 --4.52005 22.7335 --4.51002 22.3318 --4.52099 21.9501 --4.55245 21.5896 --4.60397 21.2515 --4.67521 20.9372 --4.7659 20.6483 --4.87582 20.3864 --5.00485 20.1536 --5.15291 19.952 --5.31995 19.7838 --5.50597 19.6516 --5.71094 19.5583 --5.93481 19.5068 --6.17749 19.5004 --6.43876 19.5425 --6.71828 19.6368 --7.01552 19.7868 --7.32969 19.9962 --7.65971 20.2684 --8.00414 20.6066 --8.36108 21.0134 --8.72816 21.4906 --9.10244 22.0389 --9.48039 22.6576 --9.85781 23.3445 --10.2299 24.0948 --10.5912 24.902 --10.9358 25.7565 --11.2572 26.6464 --11.5487 27.5571 --11.8037 28.4719 --12.0155 29.3722 --12.1782 30.2385 --12.2865 31.0512 --12.3365 31.7915 --12.3252 32.4428 --12.2517 32.9911 --12.1165 33.4262 --11.9217 33.742 --11.6712 33.9366 --11.3703 34.012 --11.0254 33.9738 --10.6438 33.8306 --10.2332 33.593 --9.80154 33.2728 --9.35674 32.8825 --8.90622 32.4341 --8.45686 31.9394 --8.01479 31.4089 --7.58534 30.8521 --7.17297 30.2772 --6.78131 29.691 --6.41317 29.0993 --6.07063 28.5068 --5.75511 27.9174 --5.46746 27.334 --5.20804 26.759 --4.97683 26.1944 --4.7735 25.6416 --4.59749 25.102 --4.44809 24.5764 --4.32446 24.0659 --4.22573 23.5712 --4.15102 23.093 --4.09946 22.6321 --4.07024 22.1893 --4.06262 21.7654 --4.07595 21.3613 --4.10968 20.978 --4.16334 20.6166 --4.23661 20.2784 --4.32923 19.9649 --4.44108 19.6776 --4.57209 19.4184 --4.7223 19.1893 --4.89182 18.9928 --5.0808 18.8312 --5.28941 18.7077 --5.51786 18.6253 --5.76628 18.5876 --6.03477 18.5984 --6.32331 18.6616 --6.63172 18.7817 --6.9596 18.9631 --7.30628 19.2102 --7.67069 19.5275 --8.05137 19.9188 --8.44631 20.3877 --8.85291 20.9368 --9.26787 21.5671 --9.68714 22.2784 --10.1059 23.068 --10.5185 23.9307 --10.9185 24.8583 --11.2988 25.8395 --11.6517 26.8594 --11.9692 27.9 --12.2433 28.9406 --12.4662 29.9581 --12.6309 30.9288 --12.7314 31.8288 --12.7632 32.6361 --12.7237 33.3313 --12.6121 33.8995 --12.4298 34.3305 --12.1805 34.6195 --11.8694 34.7672 --11.5035 34.779 --11.0913 34.6643 --10.6416 34.4356 --10.164 34.1074 --9.66778 33.6949 --9.16209 33.2132 --8.65523 32.6769 --8.15469 32.0989 --7.66695 31.4909 --7.19742 30.8629 --6.75049 30.2229 --6.32951 29.5778 --5.93692 28.9329 --5.57436 28.2923 --5.24274 27.6593 --4.94238 27.0362 --4.67313 26.425 --4.43447 25.8268 --4.2256 25.2428 --4.04551 24.6736 --3.8931 24.12 --3.76718 23.5823 --3.66658 23.0611 --3.59014 22.5566 --3.53678 22.0695 --3.50551 21.6001 --3.49543 21.1491 --3.50579 20.7169 --3.53595 20.3045 --3.58541 19.9126 --3.6538 19.5423 --3.7409 19.1946 --3.8466 18.871 --3.97093 18.5731 --4.11403 18.3027 --4.27614 18.0618 --4.45758 17.8528 --4.65877 17.6785 --4.88015 17.5419 --5.12221 17.4465 --5.38542 17.3962 --5.6702 17.3953 --5.97691 17.4485 --6.30574 17.5608 --6.65668 17.7377 --7.02943 17.9848 --7.42333 18.3078 --7.83726 18.7123 --8.2695 19.2036 --8.71768 19.7859 --9.1786 20.4627 --9.64818 21.2351 --10.1213 22.1025 --10.592 23.0608 --11.0528 24.1026 --11.4958 25.2162 --11.9118 26.3857 --12.2912 27.5906 --12.6239 28.8062 --12.9003 30.0047 --13.111 31.1562 --13.2481 32.2303 --13.3053 33.1981 --13.2786 34.0345 --13.1663 34.7191 --12.9696 35.2385 --12.6925 35.5863 --12.3412 35.7632 --11.9244 35.7767 --11.4521 35.6394 --10.9355 35.3681 --10.3862 34.9815 --9.81575 34.4995 --9.23512 33.9415 --8.65434 33.3253 --8.08226 32.6669 --7.52643 31.9802 --6.99301 31.2765 --6.48683 30.5649 --6.01147 29.8527 --5.56938 29.1452 --5.16201 28.4465 --4.78999 27.7594 --4.45326 27.0859 --4.15123 26.4274 --3.8829 25.7847 --3.64698 25.1583 --3.442 24.5484 --3.26635 23.9552 --3.11842 23.3787 --2.99659 22.8189 --2.89932 22.2758 --2.82516 21.7494 --2.77279 21.2398 --2.741 20.7472 --2.72878 20.2717 --2.73523 19.8136 --2.75964 19.3732 --2.80146 18.9511 --2.8603 18.5479 --2.93593 18.1643 --3.02827 17.8012 --3.1374 17.4597 --3.26355 17.1411 --3.40705 16.847 --3.5684 16.5792 --3.74819 16.3399 --3.94713 16.1315 --4.16601 15.957 --4.4057 15.8197 --4.6671 15.7235 --4.95114 15.6727 --5.25874 15.6724 --5.59074 15.7281 --5.94784 15.846 --6.33058 16.0329 --6.7392 16.2959 --7.17354 16.6428 --7.63295 17.0815 --8.11613 17.6195 --8.62097 18.264 --9.14441 19.0211 --9.68224 19.8949 --10.229 20.887 --10.7777 21.9954 --11.3201 23.2134 --11.8463 24.5288 --12.3451 25.9235 --12.8042 27.3726 --13.2106 28.8451 --13.5514 30.3046 --13.8142 31.7112 --13.9878 33.0238 --14.0636 34.203 --14.0356 35.2141 --13.9016 36.0303 --13.663 36.6339 --13.3248 37.0182 --12.8958 37.1867 --12.3874 37.1522 --11.8131 36.9347 --11.1878 36.5591 --10.5265 36.0524 --9.84402 35.4418 --9.15403 34.753 --8.46877 34.0089 --7.79871 33.229 --7.15245 32.4293 --6.53673 31.6223 --5.95651 30.8174 --5.41518 30.0216 --4.9147 29.2396 --4.45591 28.4745 --4.03865 27.7282 --3.66205 27.0017 --3.32465 26.2951 --3.02459 25.6085 --2.75976 24.9415 --2.52788 24.2936 --2.32663 23.6642 --2.15368 23.0527 --2.00679 22.4585 --1.88383 21.8813 --1.78281 21.3203 --1.7019 20.7753 --1.63946 20.2458 --1.59402 19.7315 --1.56429 19.2323 --1.54918 18.7477 --1.54776 18.2778 --1.55928 17.8224 --1.58317 17.3815 --1.61901 16.955 --1.66654 16.5432 --1.72564 16.1461 --1.79634 15.7641 --1.87882 15.3975 --1.97338 15.0469 --2.08045 14.7129 --2.20062 14.3964 --2.33456 14.0982 --2.48312 13.8198 --2.64724 13.5626 --2.82799 13.3285 --3.02657 13.1197 --3.2443 12.9388 --3.4826 12.789 --3.74301 12.6741 --4.02712 12.5985 --4.33661 12.5672 --4.67319 12.5864 --5.03854 12.663 --5.43429 12.8049 --5.8619 13.0211 --6.32258 13.3217 --6.81717 13.7178 --7.34596 14.2213 --7.90851 14.8447 --8.50338 15.6004 --9.12793 16.5004 --9.77798 17.5548 --10.4475 18.7706 --11.1284 20.1506 --11.8101 21.6909 --12.4797 23.3792 --13.1219 25.1936 --13.7191 27.1006 --14.2524 29.0557 --14.702 31.004 --15.0487 32.8834 --15.2754 34.6287 --15.3681 36.1773 --15.3179 37.4754 --15.1213 38.4828 --14.7812 39.1771 --14.3066 39.5549 --13.7118 39.6307 --13.0155 39.4337 --12.2389 39.0032 --11.4043 38.3834 --10.534 37.6194 --9.64851 36.7531 --8.76623 35.8212 --7.90266 34.854 --7.07024 33.875 --6.27846 32.9017 --5.53401 31.9462 --4.84112 31.0164 --4.20194 30.1167 --3.61687 29.2493 --3.08499 28.4145 --2.60433 27.6117 --2.17218 26.8396 --1.78534 26.0964 --1.44029 25.3805 --1.13336 24.69 --0.86085 24.0233 --0.619134 23.3789 --0.4047 22.7555 --0.214212 22.1519 --0.0445337 21.567 -0.107255 21.0 -0.243855 20.4501 -0.367752 19.9166 -0.481222 19.3991 -0.586349 18.897 -0.685037 18.4099 -0.779031 17.9376 -0.869934 17.4797 -0.959225 17.036 -1.04828 16.6065 -1.1384 16.191 -1.2308 15.7896 -1.32666 15.4024 -1.42713 15.0295 -1.53334 14.6711 -1.64641 14.3277 -1.76749 13.9999 -1.89775 13.6882 -2.0384 13.3935 -2.1907 13.117 -2.35598 12.8599 -2.53564 12.6239 -2.73115 12.4109 -2.94407 12.2234 -3.17607 12.0642 -3.42888 11.9368 -3.70433 11.8451 -4.0043 11.7941 -4.33075 11.7894 -4.68563 11.8376 -5.07088 11.9464 -5.48833 12.1246 -5.93965 12.3823 -6.42619 12.7306 -6.94888 13.182 -7.50802 13.7496 -8.10305 14.4475 -8.7323 15.2895 -9.39266 16.2885 -10.0793 17.4553 -10.7851 18.7972 -11.5009 20.3156 -12.2144 22.0044 -12.9108 23.8474 -13.5728 25.8168 -14.1804 27.8717 -14.7126 29.9587 -15.1477 32.0138 -15.4654 33.9662 -15.6478 35.744 -15.6819 37.281 -15.5601 38.5234 -15.2816 39.4358 -14.8527 40.0036 -14.2862 40.2337 -13.6002 40.1518 -12.8167 39.7975 -11.9599 39.2187 -11.0543 38.4659 -10.1232 37.5879 -9.18762 36.6278 -8.26579 35.6219 -7.37249 34.5987 -6.51914 33.5795 -5.714 32.5791 -4.96247 31.6072 -4.2675 30.6691 -3.63002 29.7675 -3.04937 28.9029 -2.52365 28.0744 -2.05007 27.2804 -1.62523 26.5189 -1.24535 25.7876 -0.906428 25.0845 -0.604425 24.4076 -0.335336 23.7551 -0.0952844 23.1254 --0.119435 22.5172 --0.312314 21.9293 --0.48662 21.3607 --0.645386 20.8107 --0.791413 20.2786 --0.927284 19.7638 --1.05537 19.266 --1.17787 18.7848 --1.29678 18.3202 --1.41399 17.8719 --1.53123 17.4401 --1.65015 17.0248 --1.77228 16.6264 --1.89914 16.2451 --2.03215 15.8816 --2.17273 15.5364 --2.32228 15.2106 --2.48222 14.9052 --2.65397 14.6216 --2.83898 14.3614 --3.03874 14.1267 --3.25478 13.9198 --3.48868 13.7435 --3.74206 13.6012 --4.01657 13.4969 --4.31391 13.4352 --4.63573 13.4213 --4.98369 13.4616 --5.35933 13.563 --5.76407 13.7335 --6.19909 13.982 --6.66521 14.3182 --7.16281 14.7528 --7.69159 15.2966 --8.25046 15.9608 --8.83724 16.7561 --9.44844 17.6918 --10.079 18.775 --10.7222 20.009 --11.3689 21.3922 --12.0083 22.9159 --12.6271 24.5635 --13.2102 26.3087 --13.7407 28.1155 --14.2011 29.9385 --14.5737 31.7246 --14.8421 33.4166 --14.9923 34.957 --15.0141 36.2934 --14.9016 37.383 --14.6548 38.1971 --14.2787 38.7227 --13.7837 38.9633 --13.1846 38.9369 --12.4992 38.6726 --11.7474 38.2071 --10.9495 37.5799 --10.1252 36.8301 --9.29261 35.9938 --8.46766 35.1021 --7.66369 34.1806 --6.89133 33.2491 --6.15861 32.3225 --5.47113 31.4111 --4.83235 30.5216 --4.24395 29.6581 --3.7061 28.8225 --3.21783 28.0155 --2.77729 27.2366 --2.38198 26.4849 --2.02897 25.7591 --1.71509 25.0578 --1.43704 24.3797 --1.1915 23.7234 --0.975207 23.0875 --0.785047 22.4711 --0.618048 21.8731 --0.471435 21.2926 --0.342632 20.7288 --0.229273 20.1809 --0.129195 19.6484 --0.0404337 19.1307 -0.0387866 18.6273 -0.110069 18.1377 -0.174855 17.6616 -0.234446 17.1985 -0.290013 16.748 -0.342613 16.3099 -0.393211 15.8839 -0.442685 15.4697 -0.491849 15.0671 -0.541459 14.6758 -0.592231 14.2956 -0.644852 13.9264 -0.699988 13.5681 -0.758299 13.2205 -0.820446 12.8836 -0.887106 12.5574 -0.958975 12.2419 -1.03678 11.9372 -1.12129 11.6436 -1.21334 11.3613 -1.31379 11.0906 -1.42361 10.8321 -1.54381 10.5863 -1.67554 10.3543 -1.81999 10.1369 -1.97851 9.93541 -2.15253 9.75154 -2.34365 9.5872 -2.55356 9.44481 -2.78412 9.32728 -3.03733 9.23817 -3.31532 9.18177 -3.62037 9.16322 -3.95487 9.18869 -4.32132 9.26552 -4.72224 9.40237 -5.16016 9.60944 -5.63751 9.89854 -6.15647 10.2833 -6.71885 10.779 -7.32585 11.4027 -7.97776 12.1728 -8.67367 13.1085 -9.41099 14.2286 -10.1851 15.5504 -10.9885 17.0873 -11.811 18.8461 -12.6384 20.8242 -13.4529 23.0057 -14.2327 25.3585 -14.9526 27.8317 -15.5852 30.3561 -16.1018 32.8455 -16.4752 35.2034 -16.6816 37.3313 -16.7036 39.1393 -16.5318 40.5571 -16.1663 41.5422 -15.6171 42.0837 -14.9028 42.202 -14.0491 41.9426 -13.0861 41.3683 -12.0458 40.5495 -10.9596 39.5564 -9.85634 38.4522 -8.76118 37.29 -7.6947 36.1112 -6.67285 34.9461 -5.70713 33.8149 -4.80506 32.73 -3.97078 31.6976 -3.20563 30.7198 -2.50881 29.7957 -1.87784 28.9227 -1.30911 28.0974 -0.798193 27.316 -0.340204 26.5749 --0.0699892 25.8704 --0.437571 25.1997 --0.767648 24.5599 --1.06515 23.949 --1.33477 23.365 --1.58093 22.8067 --1.80775 22.2729 --2.01906 21.763 --2.2184 21.2764 --2.40902 20.8129 --2.59393 20.3727 --2.77588 19.956 --2.95742 19.5634 --3.14091 19.1956 --3.32853 18.8537 --3.5223 18.5389 --3.72415 18.2528 --3.93585 17.9972 --4.15908 17.7745 --4.39543 17.5871 --4.64637 17.4379 --4.91327 17.3304 --5.19738 17.2683 --5.49982 17.2558 --5.82152 17.2977 --6.1632 17.3989 --6.52532 17.565 --6.90802 17.8017 --7.31099 18.1148 --7.73345 18.5102 --8.17401 18.9933 --8.63056 19.5689 --9.10016 20.2407 --9.57895 21.0104 --10.062 21.8777 --10.5434 22.8394 --11.0159 23.8883 --11.4714 25.0134 --11.9008 26.1989 --12.2943 27.4244 --12.6416 28.6653 --12.9326 29.893 --13.1578 31.077 --13.3086 32.1858 --13.3784 33.1893 --13.3625 34.0607 --13.2591 34.7786 --13.069 35.328 --12.7958 35.7015 --12.4459 35.8993 --12.0278 35.9284 --11.5517 35.8017 --11.0292 35.536 --10.4721 35.1509 --9.8924 34.6669 --9.30134 34.1039 --8.70934 33.4808 --8.12561 32.814 --7.55797 32.118 --7.01284 31.4046 --6.49524 30.6832 --6.00889 29.9613 --5.55634 29.2445 --5.1391 28.5367 --4.75783 27.8411 --4.41247 27.1595 --4.10241 26.4932 --3.8266 25.8431 --3.5837 25.2096 --3.37217 24.5929 --3.19036 23.993 --3.03657 23.4099 --2.90913 22.8436 --2.80641 22.294 --2.72691 21.7611 --2.66923 21.2449 --2.63211 20.7454 --2.61446 20.2627 --2.61535 19.7972 --2.634 19.349 --2.66982 18.9185 --2.72235 18.5063 --2.79134 18.1131 --2.87666 17.7395 --2.97834 17.3866 --3.09658 17.0555 --3.2317 16.7477 --3.38415 16.4647 --3.55452 16.2085 --3.74351 15.9814 --3.95191 15.786 --4.18061 15.6255 --4.43056 15.5035 --4.70274 15.4239 --4.99817 15.3916 --5.31779 15.4118 --5.66251 15.4905 --6.03305 15.6341 --6.42993 15.85 --6.85336 16.1458 --7.30309 16.5295 --7.77834 17.0094 --8.27759 17.5935 --8.79846 18.2889 --9.3375 19.1017 --9.89004 20.0357 --10.45 21.0916 --11.0099 22.2661 --11.5604 23.5508 --12.0911 24.9314 --12.5898 26.3866 --13.0436 27.8886 --13.4389 29.4029 --13.7621 30.8902 --14.0009 32.3078 --14.1444 33.6132 --14.1843 34.7666 --14.116 35.7343 --13.9382 36.4916 --13.654 37.0245 --13.2701 37.3304 --12.7967 37.417 --12.2467 37.3012 --11.6348 37.0066 --10.9767 36.5606 --10.2878 35.9919 --9.5833 35.3285 --8.87666 34.5961 --8.17982 33.817 --7.50277 33.01 --6.85355 32.1898 --6.23826 31.3681 --5.66123 30.553 --5.12523 29.7507 --4.63167 28.9649 --4.18088 28.1981 --3.77227 27.4517 --3.40458 26.7261 --3.07606 26.0213 --2.78458 25.3369 --2.52784 24.6724 --2.3034 24.0272 --2.10881 23.4005 --1.94167 22.7917 --1.79968 22.2003 --1.68066 21.6255 --1.58258 21.0669 --1.50362 20.5239 --1.44211 19.9963 --1.39657 19.4836 --1.36573 18.9855 --1.34847 18.5017 --1.34388 18.0321 --1.35118 17.5765 --1.3698 17.1347 --1.39929 16.7067 --1.43937 16.2926 --1.48989 15.8923 --1.55083 15.5061 --1.62234 15.1341 --1.70464 14.7766 --1.79814 14.4342 --1.90332 14.1074 --2.02082 13.7969 --2.15138 13.5036 --2.29587 13.2288 --2.4553 12.9737 --2.63078 12.7402 --2.82355 12.5302 --3.03498 12.3462 --3.26656 12.1913 --3.51988 12.0689 --3.79665 11.9833 --4.09863 11.9394 --4.42767 11.943 --4.78562 12.0009 --5.17431 12.1208 --5.59547 12.3118 --6.05061 12.584 --6.54096 12.9487 --7.06726 13.4183 --7.62959 14.0059 --8.22714 14.7254 --8.85795 15.59 --9.51857 16.6122 --10.2037 17.8018 --10.9061 19.1649 --11.6158 20.7015 --12.3205 22.4034 --13.0048 24.2526 --13.6511 26.219 --14.2397 28.2602 --14.7496 30.3214 --15.1596 32.3382 --15.4501 34.2404 --15.6043 35.9582 --15.6103 37.4282 --15.4619 38.6009 --15.1595 39.4447 --14.7105 39.949 --14.1285 40.1239 --13.432 39.997 --12.6432 39.6087 --11.786 39.0068 --10.8844 38.2406 --9.96093 37.3572 --9.03608 36.398 --8.12718 35.3974 --7.24831 34.3826 --6.41026 33.3733 --5.62076 32.3836 --4.88478 31.4222 --4.20494 30.4943 --3.58192 29.602 --3.01492 28.7459 --2.50194 27.9249 --2.04015 27.1375 --1.62615 26.3817 --1.25619 25.6554 --0.926336 24.9566 --0.632602 24.2833 --0.371066 23.6339 --0.137933 23.0068 -0.0704143 22.4008 -0.257385 21.8146 -0.426161 21.2475 -0.579695 20.6985 -0.720711 20.1671 -0.851718 19.6527 -0.97502 19.1549 -1.09274 18.6735 -1.20683 18.2081 -1.3191 17.7587 -1.43124 17.3253 -1.54484 16.908 -1.66139 16.5069 -1.78233 16.1223 -1.90906 15.7548 -2.04295 15.4048 -2.18536 15.0733 -2.33764 14.7611 -2.5012 14.4695 -2.67744 14.1999 -2.86782 13.9542 -3.07384 13.7345 -3.29706 13.5434 -3.53909 13.3839 -3.8016 13.2598 -4.08628 13.1751 -4.39485 13.135 -4.72904 13.1452 -5.09051 13.2123 -5.48082 13.3439 -5.90139 13.5486 -6.35334 13.8359 -6.8374 14.2162 -7.35379 14.7006 -7.90198 15.3007 -8.48051 16.028 -9.08674 16.8933 -9.7166 17.9058 -10.3643 19.0716 -11.022 20.3927 -11.6797 21.8651 -12.3254 23.4771 -12.9444 25.2078 -13.5205 27.0262 -14.0358 28.8909 -14.4719 30.7512 -14.811 32.5498 -15.0367 34.2263 -15.1361 35.7224 -15.1003 36.9873 -14.9257 37.9824 -14.6145 38.6848 -14.1745 39.0889 -13.6185 39.2056 -12.9635 39.0597 -12.229 38.686 -11.436 38.1248 -10.6054 37.4175 -9.7568 36.6037 -8.90796 35.7185 -8.07403 34.7915 -7.26739 33.8461 -6.49762 32.9 -5.77169 31.9662 -5.09418 31.053 -4.46764 30.1657 -3.89291 29.3072 -3.36949 28.4785 -2.89583 27.6795 -2.4696 26.9093 -2.08794 26.1666 -1.74766 25.4499 -1.44536 24.7577 -1.17759 24.0886 -0.940941 23.441 -0.732097 22.8139 -0.547894 22.206 -0.385353 21.6163 -0.241695 21.044 -0.114355 20.4884 -0.000974216 19.9486 --0.100598 19.4242 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45.2142 -13.5103 44.1489 -12.0991 42.8243 -10.6539 41.3538 -9.21441 39.8295 -7.8128 38.3201 -6.47326 36.8717 -5.21253 35.5121 -4.04091 34.2545 -2.96339 33.1024 -1.98081 32.0525 -1.09097 31.098 -0.289462 30.2304 --0.429536 29.4405 --1.07273 28.7195 --1.64724 28.0596 --2.16032 27.4539 --2.61906 26.8964 --3.03032 26.3825 --3.40055 25.9081 --3.73575 25.4703 --4.04148 25.0669 --4.32278 24.6963 --4.58422 24.3572 --4.82993 24.0493 --5.06355 23.7721 --5.28832 23.5259 --5.50709 23.311 --5.72232 23.128 --5.93612 22.9778 --6.15026 22.8614 --6.3662 22.7797 --6.58508 22.734 --6.80776 22.7255 --7.03478 22.7553 --7.26642 22.8246 --7.50264 22.9343 --7.74311 23.0852 --7.9872 23.2777 --8.23398 23.512 --8.4822 23.7875 --8.73032 24.1033 --8.97648 24.4578 --9.21854 24.8484 --9.4541 25.272 --9.68053 25.7243 --9.89499 26.2001 --10.0945 26.6936 --10.2762 27.1978 --10.4369 27.7052 --10.5738 28.2077 --10.6844 28.6967 --10.7662 29.1638 --10.8174 29.6007 --10.8367 29.9996 --10.8232 30.3536 --10.7767 30.6568 --10.6978 30.9046 --10.5876 31.0938 --10.4479 31.2226 --10.281 31.2907 --10.0898 31.2991 --9.87754 31.2501 --9.64766 31.147 --9.40389 30.9938 --9.15001 30.7951 --8.88973 30.556 --8.62668 30.2815 --8.36425 29.9768 --8.10562 29.6469 --7.85364 29.2966 --7.61085 28.9302 --7.37946 28.552 --7.16134 28.1656 --6.95806 27.7745 --6.77085 27.3816 --6.60072 26.9898 --6.44838 26.6013 --6.31435 26.2186 --6.19897 25.8435 --6.1024 25.478 --6.0247 25.1237 --5.96581 24.7823 --5.92559 24.4552 --5.90385 24.1441 --5.90035 23.8503 --5.91482 23.5754 --5.94698 23.3208 --5.99652 23.0881 --6.06313 22.8788 --6.14646 22.6946 --6.24619 22.5372 --6.36192 22.4082 --6.49327 22.3094 --6.63978 22.2427 --6.80094 22.2099 --6.97616 22.2128 --7.16473 22.2531 --7.36586 22.3326 --7.57857 22.4527 --7.80173 22.6148 --8.03401 22.8199 --8.27386 23.0685 --8.51948 23.3608 --8.76881 23.6963 --9.01954 24.0735 --9.26909 24.4906 --9.51459 24.9443 --9.75296 25.4306 --9.98091 25.9444 --10.195 26.4794 --10.3918 27.0284 --10.5677 27.5833 --10.7194 28.1351 --10.8437 28.6746 --10.9378 29.1922 --10.9994 29.6784 --11.0267 30.1245 --11.0185 30.5224 --10.9744 30.8652 --10.8948 31.1475 --10.7808 31.3654 --10.6341 31.5169 --10.4573 31.6013 --10.2534 31.6199 --10.026 31.5751 --9.77885 31.4706 --9.51614 31.3109 --9.242 31.1014 --8.96058 30.8477 --8.67587 30.5555 --8.39165 30.2307 --8.11139 29.8788 --7.83825 29.5051 --7.57498 29.1144 --7.32399 28.7112 --7.0873 28.2996 --6.86655 27.8831 --6.66307 27.4649 --6.47786 27.0479 --6.31166 26.6346 --6.16496 26.2273 --6.03807 25.8279 --5.9311 25.4383 --5.84406 25.0603 --5.77683 24.6952 --5.72922 24.3448 --5.70099 24.0104 --5.69184 23.6935 --5.70149 23.3955 --5.7296 23.118 --5.77586 22.8623 --5.83994 22.6302 --5.92152 22.4232 --6.02028 22.243 --6.13588 22.0915 --6.26797 21.9705 --6.41616 21.8819 --6.58 21.8277 --6.75901 21.8101 --6.95258 21.8309 --7.16001 21.8922 --7.38045 21.9958 --7.61288 22.1435 --7.85607 22.3368 --8.10859 22.5767 --8.36872 22.864 --8.63447 23.1986 --8.90356 23.58 --9.17339 24.0065 --9.44103 24.4756 --9.70328 24.9837 --9.95665 25.5258 --10.1974 26.096 --10.4218 26.6868 --10.6258 27.2897 --10.8055 27.8953 --10.9573 28.4932 --11.0777 29.0727 --11.1638 29.6232 --11.2132 30.1341 --11.2242 30.5959 --11.1958 31.0001 --11.1281 31.3397 --11.0217 31.6096 --10.8783 31.8066 --10.7004 31.9294 --10.4912 31.9788 --10.2544 31.9569 --9.9942 31.8678 --9.71511 31.7163 --9.42177 31.5083 --9.11882 31.2502 --8.81078 30.9484 --8.50193 30.6096 --8.19623 30.2401 --7.89726 29.8458 --7.60817 29.4322 --7.3317 29.0044 --7.07012 28.5668 --6.82532 28.1234 --6.59877 27.6776 --6.39159 27.2326 --6.20459 26.791 --6.03831 26.3552 --5.89305 25.9271 --5.76891 25.5088 --5.66586 25.1018 --5.58374 24.7077 --5.5223 24.328 --5.48125 23.964 --5.46024 23.6173 --5.45893 23.2891 --5.47695 22.9809 --5.51395 22.6941 --5.56959 22.4302 --5.64355 22.191 --5.73552 21.978 --5.8452 21.7931 --5.97228 21.6381 --6.11646 21.5152 --6.2774 21.4263 --6.4547 21.3737 --6.64793 21.3597 --6.85653 21.3864 --7.07984 21.4564 --7.31704 21.5716 --7.56712 21.7341 --7.82885 21.9458 --8.10075 22.2081 --8.38103 22.5219 --8.6676 22.8874 --8.958 23.3043 --9.24942 23.7709 --9.53868 24.2846 --9.82224 24.8414 --10.0962 25.4362 --10.3565 26.0619 --10.5988 26.7104 --10.8187 27.372 --11.0118 28.0359 --11.1738 28.6905 --11.3011 29.3234 --11.3902 29.9225 --11.4385 30.4758 --11.4441 30.9726 --11.4061 31.4035 --11.3245 31.761 --11.2005 32.0397 --11.036 32.2365 --10.834 32.3507 --10.5982 32.3837 --10.3331 32.3389 --10.0434 32.2212 --9.73428 32.0369 --9.41103 31.7929 --9.0788 31.4966 --8.74258 31.1557 --8.40702 30.7775 --8.07637 30.3689 --7.75444 29.9364 --7.4445 29.486 --7.14934 29.0228 --6.87125 28.5514 --6.61204 28.076 --6.3731 27.5999 --6.15544 27.1261 --5.95973 26.6572 --5.78634 26.1954 --5.63543 25.7427 --5.50696 25.3007 --5.40074 24.8711 --5.31648 24.4553 --5.25383 24.0546 --5.21236 23.6702 --5.19166 23.3036 --5.1913 22.956 --5.21088 22.6288 --5.25 22.3234 --5.30831 22.0412 --5.38549 21.7839 --5.48124 21.5532 --5.59531 21.351 --5.72744 21.1791 --5.87739 21.0398 --6.0449 20.9353 --6.22968 20.8679 --6.4314 20.8402 --6.64961 20.8548 --6.88379 20.9142 --7.13323 21.021 --7.39705 21.1779 --7.67414 21.387 --7.96312 21.6505 --8.26229 21.9698 --8.56959 22.3459 --8.88257 22.779 --9.19837 23.268 --9.51368 23.8109 --9.82478 24.404 --10.1275 25.0423 --10.4174 25.7189 --10.6895 26.4251 --10.9391 27.1507 --11.161 27.8837 --11.3503 28.6111 --11.5025 29.319 --11.6136 29.9931 --11.6802 30.6198 --11.6997 31.1859 --11.6709 31.6803 --11.5935 32.0937 --11.4682 32.4194 --11.2972 32.6535 --11.0837 32.7947 --10.8317 32.8446 --10.5461 32.8069 --10.2323 32.6874 --9.89624 32.4932 --9.54377 32.2326 --9.18079 31.9141 --8.81296 31.5465 --8.44556 31.1383 --8.08336 30.6975 --7.73063 30.2313 --7.391 29.7463 --7.06755 29.2484 --6.76276 28.7424 --6.47857 28.2328 --6.21642 27.723 --5.97733 27.2163 --5.76193 26.7151 --5.57052 26.2217 --5.40317 25.738 --5.25971 25.2655 --5.13984 24.8058 --5.04315 24.3601 --4.96914 23.9296 --4.91731 23.5156 --4.88711 23.1192 --4.87803 22.7416 --4.88958 22.3839 --4.92132 22.0476 --4.97285 21.734 --5.04382 21.4445 --5.13396 21.1809 --5.24301 20.9449 --5.37079 20.7385 --5.51712 20.5638 --5.68184 20.423 --5.86481 20.3187 --6.06583 20.2534 --6.28467 20.2301 --6.52101 20.2516 --6.7744 20.3209 --7.04422 20.4412 --7.32967 20.6155 --7.62967 20.8465 --7.94283 21.1369 --8.26742 21.4887 --8.60128 21.9032 --8.94179 22.381 --9.28584 22.9213 --9.62981 23.5222 --9.96955 24.1798 --10.3004 24.8884 --10.6172 25.6404 --10.9146 26.4259 --11.1868 27.233 --11.4281 28.048 --11.6328 28.8556 --11.7956 29.6394 --11.9119 30.383 --11.978 31.0703 --11.991 31.6863 --11.9495 32.2183 --11.8533 32.6562 --11.7038 32.9928 --11.5035 33.2247 --11.2564 33.3515 --10.9674 33.3761 --10.6425 33.3038 --10.288 33.1425 --9.91065 32.9011 --9.51733 32.5896 --9.11468 32.2184 --8.709 31.7975 --8.30605 31.3367 --7.91101 30.8447 --7.52834 30.3295 --7.16181 29.7979 --6.81451 29.256 --6.48883 28.7088 --6.18658 28.1605 --5.90902 27.6145 --5.65694 27.0738 --5.43072 26.5407 --5.23043 26.0171 --5.05588 25.5048 --4.90667 25.0049 --4.78229 24.5189 --4.6821 24.0477 --4.60545 23.5924 --4.55165 23.154 --4.52005 22.7335 --4.51002 22.3318 --4.52099 21.9501 --4.55245 21.5896 --4.60397 21.2515 --4.67521 20.9372 --4.7659 20.6483 --4.87582 20.3864 --5.00485 20.1536 --5.15291 19.952 --5.31995 19.7838 --5.50597 19.6516 --5.71094 19.5583 --5.93481 19.5068 --6.17749 19.5004 --6.43876 19.5425 --6.71828 19.6368 --7.01552 19.7868 --7.32969 19.9962 --7.65971 20.2684 --8.00414 20.6066 --8.36108 21.0134 --8.72816 21.4906 --9.10244 22.0389 --9.48039 22.6576 --9.85781 23.3445 --10.2299 24.0948 --10.5912 24.902 --10.9358 25.7565 --11.2572 26.6464 --11.5487 27.5571 --11.8037 28.4719 --12.0155 29.3722 --12.1782 30.2385 --12.2865 31.0512 --12.3365 31.7915 --12.3252 32.4428 --12.2517 32.9911 --12.1165 33.4262 --11.9217 33.742 --11.6712 33.9366 --11.3703 34.012 --11.0254 33.9738 --10.6438 33.8306 --10.2332 33.593 --9.80154 33.2728 --9.35674 32.8825 --8.90622 32.4341 --8.45686 31.9394 --8.01479 31.4089 --7.58534 30.8521 --7.17297 30.2772 --6.78131 29.691 --6.41317 29.0993 --6.07063 28.5068 --5.75511 27.9174 --5.46746 27.334 --5.20804 26.759 --4.97683 26.1944 --4.7735 25.6416 --4.59749 25.102 --4.44809 24.5764 --4.32446 24.0659 --4.22573 23.5712 --4.15102 23.093 --4.09946 22.6321 --4.07024 22.1893 --4.06262 21.7654 --4.07595 21.3613 --4.10968 20.978 --4.16334 20.6166 --4.23661 20.2784 --4.32923 19.9649 --4.44108 19.6776 --4.57209 19.4184 --4.7223 19.1893 --4.89182 18.9928 --5.0808 18.8312 --5.28941 18.7077 --5.51786 18.6253 --5.76628 18.5876 --6.03477 18.5984 --6.32331 18.6616 --6.63172 18.7817 --6.9596 18.9631 --7.30628 19.2102 --7.67069 19.5275 --8.05137 19.9188 --8.44631 20.3877 --8.85291 20.9368 --9.26787 21.5671 --9.68714 22.2784 --10.1059 23.068 --10.5185 23.9307 --10.9185 24.8583 --11.2988 25.8395 --11.6517 26.8594 --11.9692 27.9 --12.2433 28.9406 --12.4662 29.9581 --12.6309 30.9288 --12.7314 31.8288 --12.7632 32.6361 --12.7237 33.3313 --12.6121 33.8995 --12.4298 34.3305 --12.1805 34.6195 --11.8694 34.7672 --11.5035 34.779 --11.0913 34.6643 --10.6416 34.4356 --10.164 34.1074 --9.66778 33.6949 --9.16209 33.2132 --8.65523 32.6769 --8.15469 32.0989 --7.66695 31.4909 --7.19742 30.8629 --6.75049 30.2229 --6.32951 29.5778 --5.93692 28.9329 --5.57436 28.2923 --5.24274 27.6593 --4.94238 27.0362 --4.67313 26.425 --4.43447 25.8268 --4.2256 25.2428 --4.04551 24.6736 --3.8931 24.12 --3.76718 23.5823 --3.66658 23.0611 --3.59014 22.5566 --3.53678 22.0695 --3.50551 21.6001 --3.49543 21.1491 --3.50579 20.7169 --3.53595 20.3045 --3.58541 19.9126 --3.6538 19.5423 --3.7409 19.1946 --3.8466 18.871 --3.97093 18.5731 --4.11403 18.3027 --4.27614 18.0618 --4.45758 17.8528 --4.65877 17.6785 --4.88015 17.5419 --5.12221 17.4465 --5.38542 17.3962 --5.6702 17.3953 --5.97691 17.4485 --6.30574 17.5608 --6.65668 17.7377 --7.02943 17.9848 --7.42333 18.3078 --7.83726 18.7123 --8.2695 19.2036 --8.71768 19.7859 --9.1786 20.4627 --9.64818 21.2351 --10.1213 22.1025 --10.592 23.0608 --11.0528 24.1026 --11.4958 25.2162 --11.9118 26.3857 --12.2912 27.5906 --12.6239 28.8062 --12.9003 30.0047 --13.111 31.1562 --13.2481 32.2303 --13.3053 33.1981 --13.2786 34.0345 --13.1663 34.7191 --12.9696 35.2385 --12.6925 35.5863 --12.3412 35.7632 --11.9244 35.7767 --11.4521 35.6394 --10.9355 35.3681 --10.3862 34.9815 --9.81575 34.4995 --9.23512 33.9415 --8.65434 33.3253 --8.08226 32.6669 --7.52643 31.9802 --6.99301 31.2765 --6.48683 30.5649 --6.01147 29.8527 --5.56938 29.1452 --5.16201 28.4465 --4.78999 27.7594 --4.45326 27.0859 --4.15123 26.4274 --3.8829 25.7847 --3.64698 25.1583 --3.442 24.5484 --3.26635 23.9552 --3.11842 23.3787 --2.99659 22.8189 --2.89932 22.2758 --2.82516 21.7494 --2.77279 21.2398 --2.741 20.7472 --2.72878 20.2717 --2.73523 19.8136 --2.75964 19.3732 --2.80146 18.9511 --2.8603 18.5479 --2.93593 18.1643 --3.02827 17.8012 --3.1374 17.4597 --3.26355 17.1411 --3.40705 16.847 --3.5684 16.5792 --3.74819 16.3399 --3.94713 16.1315 --4.16601 15.957 --4.4057 15.8197 --4.6671 15.7235 --4.95114 15.6727 --5.25874 15.6724 --5.59074 15.7281 --5.94784 15.846 --6.33058 16.0329 --6.7392 16.2959 --7.17354 16.6428 --7.63295 17.0815 --8.11613 17.6195 --8.62097 18.264 --9.14441 19.0211 --9.68224 19.8949 --10.229 20.887 --10.7777 21.9954 --11.3201 23.2134 --11.8463 24.5288 --12.3451 25.9235 --12.8042 27.3726 --13.2106 28.8451 --13.5514 30.3046 --13.8142 31.7112 --13.9878 33.0238 --14.0636 34.203 --14.0356 35.2141 --13.9016 36.0303 --13.663 36.6339 --13.3248 37.0182 --12.8958 37.1867 --12.3874 37.1522 --11.8131 36.9347 --11.1878 36.5591 --10.5265 36.0524 --9.84402 35.4418 --9.15403 34.753 --8.46877 34.0089 --7.79871 33.229 --7.15245 32.4293 --6.53673 31.6223 --5.95651 30.8174 --5.41518 30.0216 --4.9147 29.2396 --4.45591 28.4745 --4.03865 27.7282 --3.66205 27.0017 --3.32465 26.2951 --3.02459 25.6085 --2.75976 24.9415 --2.52788 24.2936 --2.32663 23.6642 --2.15368 23.0527 --2.00679 22.4585 --1.88383 21.8813 --1.78281 21.3203 --1.7019 20.7753 --1.63946 20.2458 --1.59402 19.7315 --1.56429 19.2323 --1.54918 18.7477 --1.54776 18.2778 --1.55928 17.8224 --1.58317 17.3815 --1.61901 16.955 --1.66654 16.5432 --1.72564 16.1461 --1.79634 15.7641 --1.87882 15.3975 --1.97338 15.0469 --2.08045 14.7129 --2.20062 14.3964 --2.33456 14.0982 --2.48312 13.8198 --2.64724 13.5626 --2.82799 13.3285 --3.02657 13.1197 --3.2443 12.9388 --3.4826 12.789 --3.74301 12.6741 --4.02712 12.5985 --4.33661 12.5672 --4.67319 12.5864 --5.03854 12.663 --5.43429 12.8049 --5.8619 13.0211 --6.32258 13.3217 --6.81717 13.7178 --7.34596 14.2213 --7.90851 14.8447 --8.50338 15.6004 --9.12793 16.5004 --9.77798 17.5548 --10.4475 18.7706 --11.1284 20.1506 --11.8101 21.6909 --12.4797 23.3792 --13.1219 25.1936 --13.7191 27.1006 --14.2524 29.0557 --14.702 31.004 --15.0487 32.8834 --15.2754 34.6287 --15.3681 36.1773 --15.3179 37.4754 --15.1213 38.4828 --14.7812 39.1771 --14.3066 39.5549 --13.7118 39.6307 --13.0155 39.4337 --12.2389 39.0032 --11.4043 38.3834 --10.534 37.6194 --9.64851 36.7531 --8.76623 35.8212 --7.90266 34.854 --7.07024 33.875 --6.27846 32.9017 --5.53401 31.9462 --4.84112 31.0164 --4.20194 30.1167 --3.61687 29.2493 --3.08499 28.4145 --2.60433 27.6117 --2.17218 26.8396 --1.78534 26.0964 --1.44029 25.3805 --1.13336 24.69 --0.86085 24.0233 --0.619134 23.3789 --0.4047 22.7555 --0.214212 22.1519 --0.0445337 21.567 -0.107255 21.0 -0.243855 20.4501 -0.367752 19.9166 -0.481222 19.3991 -0.586349 18.897 -0.685037 18.4099 -0.779031 17.9376 -0.869934 17.4797 -0.959225 17.036 -1.04828 16.6065 -1.1384 16.191 -1.2308 15.7896 -1.32666 15.4024 -1.42713 15.0295 -1.53334 14.6711 -1.64641 14.3277 -1.76749 13.9999 -1.89775 13.6882 -2.0384 13.3935 -2.1907 13.117 -2.35598 12.8599 -2.53564 12.6239 -2.73115 12.4109 -2.94407 12.2234 -3.17607 12.0642 -3.42888 11.9368 -3.70433 11.8451 -4.0043 11.7941 -4.33075 11.7894 -4.68563 11.8376 -5.07088 11.9464 -5.48833 12.1246 -5.93965 12.3823 -6.42619 12.7306 -6.94888 13.182 -7.50802 13.7496 -8.10305 14.4475 -8.7323 15.2895 -9.39266 16.2885 -10.0793 17.4553 -10.7851 18.7972 -11.5009 20.3156 -12.2144 22.0044 -12.9108 23.8474 -13.5728 25.8168 -14.1804 27.8717 -14.7126 29.9587 -15.1477 32.0138 -15.4654 33.9662 -15.6478 35.744 -15.6819 37.281 -15.5601 38.5234 -15.2816 39.4358 -14.8527 40.0036 -14.2862 40.2337 -13.6002 40.1518 -12.8167 39.7975 -11.9599 39.2187 -11.0543 38.4659 -10.1232 37.5879 -9.18762 36.6278 -8.26579 35.6219 -7.37249 34.5987 -6.51914 33.5795 -5.714 32.5791 -4.96247 31.6072 -4.2675 30.6691 -3.63002 29.7675 -3.04937 28.9029 -2.52365 28.0744 -2.05007 27.2804 -1.62523 26.5189 -1.24535 25.7876 -0.906428 25.0845 -0.604425 24.4076 -0.335336 23.7551 -0.0952844 23.1254 --0.119435 22.5172 --0.312314 21.9293 --0.48662 21.3607 --0.645386 20.8107 --0.791413 20.2786 --0.927284 19.7638 --1.05537 19.266 --1.17787 18.7848 --1.29678 18.3202 --1.41399 17.8719 --1.53123 17.4401 --1.65015 17.0248 --1.77228 16.6264 --1.89914 16.2451 --2.03215 15.8816 --2.17273 15.5364 --2.32228 15.2106 --2.48222 14.9052 --2.65397 14.6216 --2.83898 14.3614 --3.03874 14.1267 --3.25478 13.9198 --3.48868 13.7435 --3.74206 13.6012 --4.01657 13.4969 --4.31391 13.4352 --4.63573 13.4213 --4.98369 13.4616 --5.35933 13.563 --5.76407 13.7335 --6.19909 13.982 --6.66521 14.3182 --7.16281 14.7528 --7.69159 15.2966 --8.25046 15.9608 --8.83724 16.7561 --9.44844 17.6918 --10.079 18.775 --10.7222 20.009 --11.3689 21.3922 --12.0083 22.9159 --12.6271 24.5635 --13.2102 26.3087 --13.7407 28.1155 --14.2011 29.9385 --14.5737 31.7246 --14.8421 33.4166 --14.9923 34.957 --15.0141 36.2934 --14.9016 37.383 --14.6548 38.1971 --14.2787 38.7227 --13.7837 38.9633 --13.1846 38.9369 --12.4992 38.6726 --11.7474 38.2071 --10.9495 37.5799 --10.1252 36.8301 --9.29261 35.9938 --8.46766 35.1021 --7.66369 34.1806 --6.89133 33.2491 --6.15861 32.3225 --5.47113 31.4111 --4.83235 30.5216 --4.24395 29.6581 --3.7061 28.8225 --3.21783 28.0155 --2.77729 27.2366 --2.38198 26.4849 --2.02897 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-2.96339 -7.33565 33.1024 -1.98081 -7.37549 32.0525 -1.09097 -7.35703 31.098 -0.289462 -7.30244 30.2304 --0.429536 -7.22885 29.4405 --1.07273 -7.14921 28.7195 --1.64724 -7.07316 28.0596 --2.16032 -7.00775 27.4539 --2.61906 -6.95799 26.8964 --3.03032 -6.92734 26.3825 --3.40055 -6.91807 25.9081 --3.73575 -6.93159 25.4703 --4.04148 -6.96859 25.0669 --4.32278 -7.02929 24.6963 --4.58422 -7.11352 24.3572 --4.82993 -7.22081 24.0493 --5.06355 -7.35048 23.7721 --5.28832 -7.50165 23.5259 --5.50709 -7.67329 23.311 --5.72232 -7.86422 23.128 --5.93612 -8.07312 22.9778 --6.15026 -8.29848 22.8614 --6.3662 -8.53864 22.7797 --6.58508 -8.79171 22.734 --6.80776 -9.0556 22.7255 --7.03478 -9.32792 22.7553 --7.26642 -9.606 22.8246 --7.50264 -9.88688 22.9343 --7.74311 -10.1672 23.0852 --7.9872 -10.4434 23.2777 --8.23398 -10.7113 23.512 --8.4822 -10.9667 23.7875 --8.73032 -11.205 24.1033 --8.97648 -11.4214 24.4578 --9.21854 -11.6109 24.8484 --9.4541 -11.7686 25.272 --9.68053 -11.8898 25.7243 --9.89499 -11.97 26.2001 --10.0945 -12.0054 26.6936 --10.2762 -11.9927 27.1978 --10.4369 -11.9298 27.7052 --10.5738 -11.8155 28.2077 --10.6844 -11.6499 28.6967 --10.7662 -11.4343 29.1638 --10.8174 -11.1717 29.6007 --10.8367 -10.8662 29.9996 --10.8232 -10.523 30.3536 --10.7767 -10.1485 30.6568 --10.6978 -9.74987 30.9046 --10.5876 -9.33465 31.0938 --10.4479 -8.91066 31.2226 --10.281 -8.48556 31.2907 --10.0898 -8.06665 31.2991 --9.87754 -7.66057 31.2501 --9.64766 -7.27317 31.147 --9.40389 -6.90936 30.9938 --9.15001 -6.57306 30.7951 --8.88973 -6.26721 30.556 --8.62668 -5.99379 30.2815 --8.36425 -5.75394 29.9768 --8.10562 -5.54804 29.6469 --7.85364 -5.37582 29.2966 --7.61085 -5.23654 28.9302 --7.37946 -5.12907 28.552 --7.16134 -5.05199 28.1656 --6.95806 -5.00376 27.7745 --6.77085 -4.98275 27.3816 --6.60072 -4.98731 26.9898 --6.44838 -5.01585 26.6013 --6.31435 -5.06689 26.2186 --6.19897 -5.13905 25.8435 --6.1024 -5.23108 25.478 --6.0247 -5.34189 25.1237 --5.96581 -5.47052 24.7823 --5.92559 -5.61615 24.4552 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-9.47358 31.3654 --10.6341 -9.01202 31.5169 --10.4573 -8.54823 31.6013 --10.2534 -8.09052 31.6199 --10.026 -7.64648 31.5751 --9.77885 -7.22277 31.4706 --9.51614 -6.82493 31.3109 --9.242 -6.4574 31.1014 --8.96058 -6.12342 30.8477 --8.67587 -5.82514 30.5555 --8.39165 -5.56372 30.2307 --8.11139 -5.33944 29.8788 --7.83825 -5.15189 29.5051 --7.57498 -5.00005 29.1144 --7.32399 -4.88253 28.7112 --7.0873 -4.79762 28.2996 --6.86655 -4.74347 27.8831 --6.66307 -4.71816 27.4649 --6.47786 -4.71979 27.0479 --6.31166 -4.74652 26.6346 --6.16496 -4.79667 26.2273 --6.03807 -4.86868 25.8279 --5.9311 -4.96117 25.4383 --5.84406 -5.07295 25.0603 --5.77683 -5.203 24.6952 --5.72922 -5.35045 24.3448 --5.70099 -5.5146 24.0104 --5.69184 -5.69488 23.6935 --5.70149 -5.8908 23.3955 --5.7296 -6.10199 23.118 --5.77586 -6.3281 22.8623 --5.83994 -6.56881 22.6302 --5.92152 -6.82379 22.4232 --6.02028 -7.09266 22.243 --6.13588 -7.37493 22.0915 --6.26797 -7.67001 21.9705 --6.41616 -7.97709 21.8819 --6.58 -8.29518 21.8277 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23.7709 --9.53868 -12.409 24.2846 --9.82224 -12.6167 24.8414 --10.0962 -12.7749 25.4362 --10.3565 -12.8769 26.0619 --10.5988 -12.917 26.7104 --10.8187 -12.8905 27.372 --11.0118 -12.7942 28.0359 --11.1738 -12.6265 28.6905 --11.3011 -12.3879 29.3234 --11.3902 -12.0809 29.9225 --11.4385 -11.7103 30.4758 --11.4441 -11.2829 30.9726 --11.4061 -10.8074 31.4035 --11.3245 -10.2939 31.761 --11.2005 -9.75351 32.0397 --11.036 -9.19779 32.2365 --10.834 -8.63825 32.3507 --10.5982 -8.08583 32.3837 --10.3331 -7.55047 32.3389 --10.0434 -7.04078 32.2212 --9.73428 -6.56385 32.0369 --9.41103 -6.12513 31.7929 --9.0788 -5.72847 31.4966 --8.74258 -5.37619 31.1557 --8.40702 -5.06929 30.7775 --8.07637 -4.80759 30.3689 --7.75444 -4.59001 29.9364 --7.4445 -4.41479 29.486 --7.14934 -4.27966 29.0228 --6.87125 -4.18207 28.5514 --6.61204 -4.11933 28.076 --6.3731 -4.08874 27.5999 --6.15544 -4.08771 27.1261 --5.95973 -4.1138 26.6572 --5.78634 -4.16479 26.1954 --5.63543 -4.23873 25.7427 --5.50696 -4.33392 25.3007 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30.6198 --11.6997 -11.6551 31.1859 --11.6709 -11.1391 31.6803 --11.5935 -10.5774 32.0937 --11.4682 -9.98274 32.4194 --11.2972 -9.36859 32.6535 --11.0837 -8.74836 32.7947 --10.8317 -8.13487 32.8446 --10.5461 -7.53981 32.8069 --10.2323 -6.97327 32.6874 --9.89624 -6.44351 32.4932 --9.54377 -5.95684 32.2326 --9.18079 -5.51761 31.9141 --8.81296 -5.12836 31.5465 --8.44556 -4.79001 31.1383 --8.08336 -4.50213 30.6975 --7.73063 -4.26322 30.2313 --7.391 -4.07101 29.7463 --7.06755 -3.92265 29.2484 --6.76276 -3.81502 28.7424 --6.47857 -3.74485 28.2328 --6.21642 -3.70891 27.723 --5.97733 -3.70411 27.2163 --5.76193 -3.72761 26.7151 --5.57052 -3.77682 26.2217 --5.40317 -3.84949 25.738 --5.25971 -3.94369 25.2655 --5.13984 -4.05779 24.8058 --5.04315 -4.1905 24.3601 --4.96914 -4.34082 23.9296 --4.91731 -4.50801 23.5156 --4.88711 -4.69155 23.1192 --4.87803 -4.89117 22.7416 --4.88958 -5.10673 22.3839 --4.92132 -5.33827 22.0476 --4.97285 -5.5859 21.734 --5.04382 -5.84982 21.4445 --5.13396 -6.13027 21.1809 --5.24301 -6.42746 20.9449 --5.37079 -6.74156 20.7385 --5.51712 -7.07263 20.5638 --5.68184 -7.42058 20.423 --5.86481 -7.7851 20.3187 --6.06583 -8.16562 20.2534 --6.28467 -8.5612 20.2301 --6.52101 -8.97049 20.2516 --6.7744 -9.39163 20.3209 --7.04422 -9.82216 20.4412 --7.32967 -10.2589 20.6155 --7.62967 -10.6981 20.8465 --7.94283 -11.1348 21.1369 --8.26742 -11.5636 21.4887 --8.60128 -11.9779 21.9032 --8.94179 -12.3702 22.381 --9.28584 -12.7325 22.9213 --9.62981 -13.0557 23.5222 --9.96955 -13.3306 24.1798 --10.3004 -13.5477 24.8884 --10.6172 -13.6977 25.6404 --10.9146 -13.7721 26.4259 --11.1868 -13.7637 27.233 --11.4281 -13.667 28.048 --11.6328 -13.4788 28.8556 --11.7956 -13.1987 29.6394 --11.9119 -12.8294 30.383 --11.978 -12.3769 31.0703 --11.991 -11.8499 31.6863 --11.9495 -11.2601 32.2183 --11.8533 -10.6214 32.6562 --11.7038 -9.94912 32.9928 --11.5035 -9.25908 33.2247 --11.2564 -8.56701 33.3515 --10.9674 -7.88764 33.3761 --10.6425 -7.23407 33.3038 --10.288 -6.61733 33.1425 --9.91065 -6.04611 32.9011 --9.51733 -5.52665 32.5896 --9.11468 -5.06291 32.2184 --8.709 -4.6567 31.7975 --8.30605 -4.30806 31.3367 --7.91101 -4.01556 30.8447 --7.52834 -3.77667 30.3295 --7.16181 -3.5881 29.7979 --6.81451 -3.44608 29.256 --6.48883 -3.34666 28.7088 --6.18658 -3.28586 28.1605 --5.90902 -3.25985 27.6145 --5.65694 -3.26508 27.0738 --5.43072 -3.29832 26.5407 --5.23043 -3.35673 26.0171 --5.05588 -3.43787 25.5048 --4.90667 -3.53967 25.0049 --4.78229 -3.6605 24.5189 --4.6821 -3.79904 24.0477 --4.60545 -3.95435 23.5924 --4.55165 -4.12576 23.154 --4.52005 -4.31289 22.7335 --4.51002 -4.51558 22.3318 --4.52099 -4.73389 21.9501 --4.55245 -4.96803 21.5896 --4.60397 -5.21833 21.2515 --4.67521 -5.48522 20.9372 --4.7659 -5.76919 20.6483 --4.87582 -6.07075 20.3864 --5.00485 -6.39038 20.1536 --5.15291 -6.72846 19.952 --5.31995 -7.08529 19.7838 --5.50597 -7.46095 19.6516 --5.71094 -7.85529 19.5583 --5.93481 -8.26782 19.5068 --6.17749 -8.69767 19.5004 --6.43876 -9.14344 19.5425 --6.71828 -9.60314 19.6368 --7.01552 -10.0741 19.7868 --7.32969 -10.5527 19.9962 --7.65971 -11.0347 20.2684 --8.00414 -11.5145 20.6066 --8.36108 -11.9856 21.0134 --8.72816 -12.4404 21.4906 --9.10244 -12.87 22.0389 --9.48039 -13.2648 22.6576 --9.85781 -13.614 23.3445 --10.2299 -13.9066 24.0948 --10.5912 -14.1311 24.902 --10.9358 -14.2766 25.7565 --11.2572 -14.3332 26.6464 --11.5487 -14.2923 27.5571 --11.8037 -14.1481 28.4719 --12.0155 -13.8975 29.3722 --12.1782 -13.5412 30.2385 --12.2865 -13.0836 31.0512 --12.3365 -12.5332 31.7915 --12.3252 -11.9021 32.4428 --12.2517 -11.2056 32.9911 --12.1165 -10.4612 33.4262 --11.9217 -9.68782 33.742 --11.6712 -8.90462 33.9366 --11.3703 -8.12984 34.012 --11.0254 -7.38 33.9738 --10.6438 -6.6692 33.8306 --10.2332 -6.00866 33.593 --9.80154 -5.40657 33.2728 --9.35674 -4.86816 32.8825 --8.90622 -4.39595 32.4341 --8.45686 -3.99014 31.9394 --8.01479 -3.64902 31.4089 --7.58534 -3.3695 30.8521 --7.17297 -3.14746 30.2772 --6.78131 -2.97825 29.691 --6.41317 -2.8569 29.0993 --6.07063 -2.77848 28.5068 --5.75511 -2.73826 27.9174 --5.46746 -2.73182 27.334 --5.20804 -2.7552 26.759 --4.97683 -2.8049 26.1944 --4.7735 -2.87791 25.6416 --4.59749 -2.97174 25.102 --4.44809 -3.08435 24.5764 --4.32446 -3.21416 24.0659 --4.22573 -3.36 23.5712 --4.15102 -3.52105 23.093 --4.09946 -3.69685 22.6321 --4.07024 -3.88721 22.1893 --4.06262 -4.09219 21.7654 --4.07595 -4.31207 21.3613 --4.10968 -4.5473 20.978 --4.16334 -4.7985 20.6166 --4.23661 -5.06638 20.2784 --4.32923 -5.35171 19.9649 --4.44108 -5.65534 19.6776 --4.57209 -5.97809 19.4184 --4.7223 -6.32075 19.1893 --4.89182 -6.68402 18.9928 --5.0808 -7.06846 18.8312 --5.28941 -7.47441 18.7077 --5.51786 -7.90195 18.6253 --5.76628 -8.35076 18.5876 --6.03477 -8.82008 18.5984 --6.32331 -9.30854 18.6616 --6.63172 -9.81408 18.7817 --6.9596 -10.3338 18.9631 --7.30628 -10.8638 19.2102 --7.67069 -11.399 19.5275 --8.05137 -11.9333 19.9188 --8.44631 -12.4589 20.3877 --8.85291 -12.9668 20.9368 --9.26787 -13.4464 21.5671 --9.68714 -13.8861 22.2784 --10.1059 -14.2728 23.068 --10.5185 -14.5928 23.9307 --10.9185 -14.8324 24.8583 --11.2988 -14.9779 25.8395 --11.6517 -15.0173 26.8594 --11.9692 -14.9405 27.9 --12.2433 -14.7407 28.9406 --12.4662 -14.4152 29.9581 --12.6309 -13.9657 30.9288 --12.7314 -13.3991 31.8288 --12.7632 -12.7276 32.6361 --12.7237 -11.9674 33.3313 --12.6121 -11.1387 33.8995 --12.4298 -10.2643 34.3305 --12.1805 -9.36761 34.6195 --11.8694 -8.47187 34.7672 --11.5035 -7.59844 34.779 --11.0913 -6.76589 34.6643 --10.6416 -5.98919 34.4356 --10.164 -5.27945 34.1074 --9.66778 -4.64388 33.6949 --9.16209 -4.08609 33.2132 --8.65523 -3.60658 32.6769 --8.15469 -3.20334 32.0989 --7.66695 -2.87245 31.4909 --7.19742 -2.6087 30.8629 --6.75049 -2.4061 30.2229 --6.32951 -2.25834 29.5778 --5.93692 -2.15913 28.9329 --5.57436 -2.10245 28.2923 --5.24274 -2.08272 27.6593 --4.94238 -2.09495 27.0362 --4.67313 -2.13473 26.425 --4.43447 -2.19831 25.8268 --4.2256 -2.28256 25.2428 --4.04551 -2.38493 24.6736 --3.8931 -2.50342 24.12 --3.76718 -2.63654 23.5823 --3.66658 -2.78325 23.0611 --3.59014 -2.9429 22.5566 --3.53678 -3.1152 22.0695 --3.50551 -3.30015 21.6001 --3.49543 -3.49802 21.1491 --3.50579 -3.70933 20.7169 --3.53595 -3.93475 20.3045 --3.58541 -4.17516 19.9126 --3.6538 -4.43153 19.5423 --3.7409 -4.70497 19.1946 --3.8466 -4.99663 18.871 --3.97093 -5.30775 18.5731 --4.11403 -5.63956 18.3027 --4.27614 -5.99326 18.0618 --4.45758 -6.37001 17.8528 --4.65877 -6.77082 17.6785 --4.88015 -7.19655 17.5419 --5.12221 -7.64778 17.4465 --5.38542 -8.12474 17.3962 --5.6702 -8.62723 17.3953 --5.97691 -9.15443 17.4485 --6.30574 -9.7048 17.5608 --6.65668 -10.2759 17.7377 --7.02943 -10.8642 17.9848 --7.42333 -11.4648 18.3078 --7.83726 -12.0715 18.7123 --8.2695 -12.6762 19.2036 --8.71768 -13.2692 19.7859 --9.1786 -13.8387 20.4627 --9.64818 -14.3712 21.2351 --10.1213 -14.8513 22.1025 --10.592 -15.2623 23.0608 --11.0528 -15.5865 24.1026 --11.4958 -15.8063 25.2162 --11.9118 -15.9051 26.3857 --12.2912 -15.8685 27.5906 --12.6239 -15.6855 28.8062 --12.9003 -15.3501 30.0047 --13.111 -14.8623 31.1562 --13.2481 -14.2286 32.2303 --13.3053 -13.4625 33.1981 --13.2786 -12.5837 34.0345 --13.1663 -11.6173 34.7191 --12.9696 -10.592 35.2385 --12.6925 -9.53798 35.5863 --12.3412 -8.48509 35.7632 --11.9244 -7.4607 35.7767 --11.4521 -6.4883 35.6394 --10.9355 -5.58651 35.3681 --10.3862 -4.76865 34.9815 --9.81575 -4.04278 34.4995 --9.23512 -3.41223 33.9415 --8.65434 -2.87633 33.3253 --8.08226 -2.43127 32.6669 --7.52643 -2.07101 31.9802 --6.99301 -1.78806 31.2765 --6.48683 -1.5742 30.5649 --6.01147 -1.42103 29.8527 --5.56938 -1.32038 29.1452 --5.16201 -1.26462 28.4465 --4.78999 -1.24682 27.7594 --4.45326 -1.26083 27.0859 --4.15123 -1.30139 26.4274 --3.8829 -1.36401 25.7847 --3.64698 -1.44504 25.1583 --3.442 -1.54152 24.5484 --3.26635 -1.65116 23.9552 --3.11842 -1.77225 23.3787 --2.99659 -1.90362 22.8189 --2.89932 -2.04452 22.2758 --2.82516 -2.1946 21.7494 --2.77279 -2.35385 21.2398 --2.741 -2.52255 20.7472 --2.72878 -2.70121 20.2717 --2.73523 -2.89058 19.8136 --2.75964 -3.09157 19.3732 --2.80146 -3.30527 18.9511 --2.8603 -3.53289 18.5479 --2.93593 -3.77579 18.1643 --3.02827 -4.03542 17.8012 --3.1374 -4.31331 17.4597 --3.26355 -4.61108 17.1411 --3.40705 -4.93042 16.847 --3.5684 -5.27302 16.5792 --3.74819 -5.6406 16.3399 --3.94713 -6.03484 16.1315 --4.16601 -6.45735 15.957 --4.4057 -6.90958 15.8197 --4.6671 -7.39278 15.7235 --4.95114 -7.90788 15.6727 --5.25874 -8.45538 15.6724 --5.59074 -9.03518 15.7281 --5.94784 -9.64642 15.846 --6.33058 -10.2872 16.0329 --6.7392 -10.9546 16.2959 --7.17354 -11.6439 16.6428 --7.63295 -12.3487 17.0815 --8.11613 -13.0605 17.6195 --8.62097 -13.7684 18.264 --9.14441 -14.4588 19.0211 --9.68224 -15.1154 19.8949 --10.229 -15.7192 20.887 --10.7777 -16.2487 21.9954 --11.3201 -16.6806 23.2134 --11.8463 -16.9907 24.5288 --12.3451 -17.1556 25.9235 --12.8042 -17.1538 27.3726 --13.2106 -16.9684 28.8451 --13.5514 -16.5889 30.3046 --13.8142 -16.0128 31.7112 --13.9878 -15.2473 33.0238 --14.0636 -14.3098 34.203 --14.0356 -13.2268 35.2141 --13.9016 -12.0327 36.0303 --13.663 -10.7672 36.6339 --13.3248 -9.47198 37.0182 --12.8958 -8.18746 37.1867 --12.3874 -6.95005 37.1522 --11.8131 -5.78987 36.9347 --11.1878 -4.72953 36.5591 --10.5265 -3.78377 36.0524 --9.84402 -2.95988 35.4418 --9.15403 -2.25878 34.753 --8.46877 -1.67625 34.0089 --7.79871 -1.20437 33.229 --7.15245 -0.832858 32.4293 --6.53673 -0.55015 31.6223 --5.95651 -0.344349 30.8174 --5.41518 -0.203867 30.0216 --4.9147 -0.117877 29.2396 --4.45591 -0.0765892 28.4745 --4.03865 -0.0713823 27.7282 --3.66205 -0.0948386 27.0017 --3.32465 -0.14071 26.2951 --3.02459 -0.20384 25.6085 --2.75976 -0.280063 24.9415 --2.52788 -0.366091 24.2936 --2.32663 -0.459404 23.6642 --2.15368 -0.558138 23.0527 --2.00679 -0.660988 22.4585 --1.88383 -0.76712 21.8813 --1.78281 -0.876089 21.3203 --1.7019 -0.987779 20.7753 --1.63946 -1.10234 20.2458 --1.59402 -1.22014 19.7315 --1.56429 -1.34175 19.2323 --1.54918 -1.46788 18.7477 --1.54776 -1.59936 18.2778 --1.55928 -1.73718 17.8224 --1.58317 -1.88239 17.3815 --1.61901 -2.03615 16.955 --1.66654 -2.19974 16.5432 --1.72564 -2.37449 16.1461 --1.79634 -2.56185 15.7641 --1.87882 -2.76338 15.3975 --1.97338 -2.98071 15.0469 --2.08045 -3.21559 14.7129 --2.20062 -3.4699 14.3964 --2.33456 -3.7456 14.0982 --2.48312 -4.04479 13.8198 --2.64724 -4.36968 13.5626 --2.82799 -4.72257 13.3285 --3.02657 -5.10586 13.1197 --3.2443 -5.52201 12.9388 --3.4826 -5.97351 12.789 --3.74301 -6.46282 12.6741 --4.02712 -6.99231 12.5985 --4.33661 -7.56412 12.5672 --4.67319 -8.18007 12.5864 --5.03854 -8.84145 12.663 --5.43429 -9.5488 12.8049 --5.8619 -10.3016 13.0211 --6.32258 -11.098 13.3217 --6.81717 -11.9341 13.7178 --7.34596 -12.8041 14.2213 --7.90851 -13.6989 14.8447 --8.50338 -14.6061 15.6004 --9.12793 -15.5093 16.5004 --9.77798 -16.3875 17.5548 --10.4475 -17.2148 18.7706 --11.1284 -17.9607 20.1506 --11.8101 -18.5904 21.6909 --12.4797 -19.0663 23.3792 --13.1219 -19.3497 25.1936 --13.7191 -19.4043 27.1006 --14.2524 -19.1996 29.0557 --14.702 -18.7145 31.004 --15.0487 -17.9422 32.8834 --15.2754 -16.8921 34.6287 --15.3681 -15.5913 36.1773 --15.3179 -14.0841 37.4754 --15.1213 -12.4277 38.4828 --14.7812 -10.6879 39.1771 --14.3066 -8.93187 39.5549 --13.7118 -7.22268 39.6307 --13.0155 -5.61365 39.4337 --12.2389 -4.14526 39.0032 --11.4043 -2.84387 38.3834 --10.534 -1.72224 37.6194 --9.64851 -0.78149 36.7531 --8.76623 -0.0136414 35.8212 --7.90266 0.595567 34.854 --7.07024 1.06413 33.875 --6.27846 1.41167 32.9017 --5.53401 1.65783 31.9462 --4.84112 1.82123 31.0164 --4.20194 1.91878 30.1167 --3.61687 1.96536 29.2493 --3.08499 1.97376 28.4145 --2.60433 1.95471 27.6117 --2.17218 1.9171 26.8396 --1.78534 1.8681 26.0964 --1.44029 1.81346 25.3805 --1.13336 1.7577 24.69 --0.86085 1.70429 24.0233 --0.619134 1.65591 23.3789 --0.4047 1.61452 22.7555 --0.214212 1.58158 22.1519 --0.0445337 1.55812 21.567 -0.107255 1.54489 21.0 -0.243855 1.54238 20.4501 -0.367752 1.55096 19.9166 -0.481222 1.57089 19.3991 -0.586349 1.60239 18.897 -0.685037 1.64567 18.4099 -0.779031 1.70096 17.9376 -0.869934 1.76853 17.4797 -0.959225 1.84876 17.036 -1.04828 1.94206 16.6065 -1.1384 2.04899 16.191 -1.2308 2.17018 15.7896 -1.32666 2.30643 15.4024 -1.42713 2.45864 15.0295 -1.53334 2.62785 14.6711 -1.64641 2.81526 14.3277 -1.76749 3.02222 13.9999 -1.89775 3.25025 13.6882 -2.0384 3.50101 13.3935 -2.1907 3.77634 13.117 -2.35598 4.07826 12.8599 -2.53564 4.40892 12.6239 -2.73115 4.77064 12.4109 -2.94407 5.16586 12.2234 -3.17607 5.59713 12.0642 -3.42888 6.06706 11.9368 -3.70433 6.57823 11.8451 -4.0043 7.13316 11.7941 -4.33075 7.73415 11.7894 -4.68563 8.38312 11.8376 -5.07088 9.08143 11.9464 -5.48833 9.8296 12.1246 -5.93965 10.627 12.3823 -6.42619 11.4713 12.7306 -6.94888 12.3582 13.182 -7.50802 13.2806 13.7496 -8.10305 14.2283 14.4475 -8.7323 15.1869 15.2895 -9.39266 16.1372 16.2885 -10.0793 17.0552 17.4553 -10.7851 17.911 18.7972 -11.5009 18.6695 20.3156 -12.2144 19.2913 22.0044 -12.9108 19.734 23.8474 -13.5728 19.955 25.8168 -14.1804 19.9156 27.8717 -14.7126 19.5846 29.9587 -15.1477 18.944 32.0138 -15.4654 17.9925 33.9662 -15.6478 16.7483 35.744 -15.6819 15.25 37.281 -15.5601 13.5538 38.5234 -15.2816 11.7286 39.4358 -14.8527 9.8493 40.0036 -14.2862 7.98936 40.2337 -13.6002 6.21382 40.1518 -12.8167 4.57452 39.7975 -11.9599 3.10755 39.2187 -11.0543 1.83305 38.4659 -10.1232 0.756842 37.5879 -9.18762 -0.126762 36.6278 -8.26579 -0.831704 35.6219 -7.37249 -1.37709 34.5987 -6.51914 -1.78454 33.5795 -5.714 -2.07616 32.5791 -4.96247 -2.27313 31.6072 -4.2675 -2.39481 30.6691 -3.63002 -2.45835 29.7675 -3.04937 -2.47848 28.9029 -2.52365 -2.46761 28.0744 -2.05007 -2.43596 27.2804 -1.62523 -2.39186 26.5189 -1.24535 -2.34195 25.7876 -0.906428 -2.29146 25.0845 -0.604425 -2.24445 24.4076 -0.335336 -2.20404 23.7551 -0.0952844 -2.17257 23.1254 --0.119435 -2.1518 22.5172 --0.312314 -2.143 21.9293 --0.48662 -2.14712 21.3607 --0.645386 -2.16487 20.8107 --0.791413 -2.19678 20.2786 --0.927284 -2.24329 19.7638 --1.05537 -2.30479 19.266 --1.17787 -2.38169 18.7848 --1.29678 -2.4744 18.3202 --1.41399 -2.58343 17.8719 --1.53123 -2.70936 17.4401 --1.65015 -2.85289 17.0248 --1.77228 -3.01481 16.6264 --1.89914 -3.19607 16.2451 --2.03215 -3.39775 15.8816 --2.17273 -3.62108 15.5364 --2.32228 -3.86741 15.2106 --2.48222 -4.13827 14.9052 --2.65397 -4.4353 14.6216 --2.83898 -4.76029 14.3614 --3.03874 -5.11514 14.1267 --3.25478 -5.50181 13.9198 --3.48868 -5.92234 13.7435 --3.74206 -6.37876 13.6012 --4.01657 -6.87302 13.4969 --4.31391 -7.40691 13.4352 --4.63573 -7.98195 13.4213 --4.98369 -8.59919 13.4616 --5.35933 -9.25908 13.563 --5.76407 -9.96114 13.7335 --6.19909 -10.7037 13.982 --6.66521 -11.4837 14.3182 --7.16281 -12.2958 14.7528 --7.69159 -13.1325 15.2966 --8.25046 -13.9833 15.9608 --8.83724 -14.8342 16.7561 --9.44844 -15.6673 17.6918 --10.079 -16.4607 18.775 --10.7222 -17.1882 20.009 --11.3689 -17.82 21.3922 --12.0083 -18.323 22.9159 --12.6271 -18.6631 24.5635 --13.2102 -18.8069 26.3087 --13.7407 -18.7249 28.1155 --14.2011 -18.3948 29.9385 --14.5737 -17.8046 31.7246 --14.8421 -16.9561 33.4166 --14.9923 -15.8658 34.957 --15.0141 -14.5658 36.2934 --14.9016 -13.1011 37.383 --14.6548 -11.5267 38.1971 --14.2787 -9.90194 38.7227 --13.7837 -8.28547 38.9633 --13.1846 -6.73019 38.9369 --12.4992 -5.27952 38.6726 --11.7474 -3.96515 38.2071 --10.9495 -2.80661 37.5799 --10.1252 -1.81214 36.8301 --9.29261 -0.980455 35.9938 --8.46766 -0.303063 35.1021 --7.66369 0.233442 34.1806 --6.89133 0.645289 33.2491 --6.15861 0.949842 32.3225 --5.47113 1.16433 31.4111 --4.83235 1.30499 30.5216 --4.24395 1.38656 29.6581 --3.7061 1.42202 28.8225 --3.21783 1.42251 28.0155 --2.77729 1.39741 27.2366 --2.38198 1.35448 26.4849 --2.02897 1.30001 25.7591 --1.71509 1.23903 25.0578 --1.43704 1.17546 24.3797 --1.1915 1.11234 23.7234 --0.975207 1.05195 23.0875 --0.785047 0.995961 22.4711 --0.618048 0.945552 21.8731 --0.471435 0.901531 21.2926 --0.342632 0.86441 20.7288 --0.229273 0.834485 20.1809 --0.129195 0.811893 19.6484 --0.0404337 0.796665 19.1307 -0.0387866 0.788769 18.6273 -0.110069 0.788136 18.1377 -0.174855 0.794694 17.6616 -0.234446 0.808382 17.1985 -0.290013 0.829171 16.748 -0.342613 0.857076 16.3099 -0.393211 0.892166 15.8839 -0.442685 0.934572 15.4697 -0.491849 0.984494 15.0671 -0.541459 1.04221 14.6758 -0.592231 1.10807 14.2956 -0.644852 1.18253 13.9264 -0.699988 1.26613 13.5681 -0.758299 1.35948 13.2205 -0.820446 1.46335 12.8836 -0.887106 1.57856 12.5574 -0.958975 1.7061 12.2419 -1.03678 1.84706 11.9372 -1.12129 2.00267 11.6436 -1.21334 2.1743 11.3613 -1.31379 2.36349 11.0906 -1.42361 2.57193 10.8321 -1.54381 2.80151 10.5863 -1.67554 3.05429 10.3543 -1.81999 3.33254 10.1369 -1.97851 3.63872 9.93541 -2.15253 3.97551 9.75154 -2.34365 4.34581 9.5872 -2.55356 4.75272 9.44481 -2.78412 5.19952 9.32728 -3.03733 5.68964 9.23817 -3.31532 6.22663 9.18177 -3.62037 6.81406 9.16322 -3.95487 7.45541 9.18869 -4.32132 8.1539 9.26552 -4.72224 8.9123 9.40237 -5.16016 9.7326 9.60944 -5.63751 10.6156 9.89854 -6.15647 11.5606 10.2833 -6.71885 12.5644 10.779 -7.32585 13.6208 11.4027 -7.97776 14.7196 12.1728 -8.67367 15.8457 13.1085 -9.41099 16.9778 14.2286 -10.1851 18.0875 15.5504 -10.9885 19.1386 17.0873 -11.811 20.0869 18.8461 -12.6384 20.8809 20.8242 -13.4529 21.4635 23.0057 -14.2327 21.7759 25.3585 -14.9526 21.7626 27.8317 -15.5852 21.3783 30.3561 -16.1018 20.5951 32.8455 -16.4752 19.4095 35.2034 -16.6816 17.8469 37.3313 -16.7036 15.9624 39.1393 -16.5318 13.8369 40.5571 -16.1663 11.5685 41.5422 -15.6171 9.26164 42.0837 -14.9028 7.01529 42.202 -14.0491 4.91287 41.9426 -13.0861 3.01602 41.3683 -12.0458 1.36238 40.5495 -10.9596 -0.0330236 39.5564 -9.85634 -1.17387 38.4522 -8.76118 -2.07747 37.29 -7.6947 -2.76973 36.1112 -6.67285 -3.2807 34.9461 -5.70713 -3.6413 33.8149 -4.80506 -3.88103 32.73 -3.97078 -4.0266 31.6976 -3.20563 -4.10132 30.7198 -2.50881 -4.12495 29.7957 -1.87784 -4.11384 28.9227 -1.30911 -4.08128 28.0974 -0.798193 -4.0379 27.316 -0.340204 -3.99206 26.5749 --0.0699892 -3.95026 25.8704 --0.437571 -3.91748 25.1997 --0.767648 -3.8975 24.5599 --1.06515 -3.89316 23.949 --1.33477 -3.90654 23.365 --1.58093 -3.9392 22.8067 --1.80775 -3.99229 22.2729 --2.01906 -4.06667 21.763 --2.2184 -4.16302 21.2764 --2.40902 -4.2819 20.8129 --2.59393 -4.42382 20.3727 --2.77588 -4.58928 19.956 --2.95742 -4.7788 19.5634 --3.14091 -4.99291 19.1956 --3.32853 -5.23222 18.8537 --3.5223 -5.49736 18.5389 --3.72415 -5.789 18.2528 --3.93585 -6.10781 17.9972 --4.15908 -6.45444 17.7745 --4.39543 -6.82947 17.5871 --4.64637 -7.23336 17.4379 --4.91327 -7.66636 17.3304 --5.19738 -8.12841 17.2683 --5.49982 -8.61909 17.2558 --5.82152 -9.13742 17.2977 --6.1632 -9.68173 17.3989 --6.52532 -10.2495 17.565 --6.90802 -10.8372 17.8017 --7.31099 -11.4399 18.1148 --7.73345 -12.0514 18.5102 --8.17401 -12.6637 18.9933 --8.63056 -13.267 19.5689 --9.10016 -13.8494 20.2407 --9.57895 -14.3973 21.0104 --10.062 -14.8951 21.8777 --10.5434 -15.3255 22.8394 --11.0159 -15.6706 23.8883 --11.4714 -15.9119 25.0134 --11.9008 -16.032 26.1989 --12.2943 -16.0153 27.4244 --12.6416 -15.8499 28.6653 --12.9326 -15.5287 29.893 --13.1578 -15.0505 31.077 --13.3086 -14.4211 32.1858 --13.3784 -13.6534 33.1893 --13.3625 -12.7671 34.0607 --13.2591 -11.7876 34.7786 --13.069 -10.7441 35.328 --12.7958 -9.66783 35.7015 --12.4459 -8.58984 35.8993 --12.0278 -7.53876 35.9284 --11.5517 -6.53931 35.8017 --11.0292 -5.61118 35.536 --10.4721 -4.76855 35.1509 --9.8924 -4.02013 34.6669 --9.30134 -3.3696 34.1039 --8.70934 -2.81646 33.4808 --8.12561 -2.35686 32.814 --7.55797 -1.9846 32.118 --7.01284 -1.69194 31.4046 --6.49524 -1.47034 30.6832 --6.00889 -1.31107 29.9613 --5.55634 -1.20565 29.2445 --5.1391 -1.14613 28.5367 --4.75783 -1.12529 27.8411 --4.41247 -1.13676 27.1595 --4.10241 -1.17504 26.4932 --3.8266 -1.23548 25.8431 --3.5837 -1.31426 25.2096 --3.37217 -1.40829 24.5929 --3.19036 -1.51519 23.993 --3.03657 -1.63316 23.4099 --2.90913 -1.76095 22.8436 --2.80641 -1.89776 22.294 --2.72691 -2.04319 21.7611 --2.66923 -2.19718 21.2449 --2.63211 -2.35999 20.7454 --2.61446 -2.53209 20.2627 --2.61535 -2.71419 19.7972 --2.634 -2.9072 19.349 --2.66982 -3.11216 18.9185 --2.72235 -3.33029 18.5063 --2.79134 -3.56292 18.1131 --2.87666 -3.81149 17.7395 --2.97834 -4.07754 17.3866 --3.09658 -4.3627 17.0555 --3.2317 -4.66867 16.7477 --3.38415 -4.9972 16.4647 --3.55452 -5.35006 16.2085 --3.74351 -5.72905 15.9814 --3.95191 -6.13589 15.786 --4.18061 -6.57223 15.6255 --4.43056 -7.03956 15.5035 --4.70274 -7.53913 15.4239 --4.99817 -8.07185 15.3916 --5.31779 -8.63812 15.4118 --5.66251 -9.23771 15.4905 --6.03305 -9.86955 15.6341 --6.42993 -10.5315 15.85 --6.85336 -11.22 16.1458 --7.30309 -11.9299 16.5295 --7.77834 -12.654 17.0094 --8.27759 -13.383 17.5935 --8.79846 -14.1047 18.2889 --9.3375 -14.8043 19.1017 --9.89004 -15.464 20.0357 --10.45 -16.063 21.0916 --11.0099 -16.5785 22.2661 --11.5604 -16.9856 23.5508 --12.0911 -17.2591 24.9314 --12.5898 -17.3748 26.3866 --13.0436 -17.3116 27.8886 --13.4389 -17.0535 29.4029 --13.7621 -16.592 30.8902 --14.0009 -15.9281 32.3078 --14.1444 -15.0727 33.6132 --14.1843 -14.0476 34.7666 --14.116 -12.884 35.7343 --13.9382 -11.6202 36.4916 --13.654 -10.299 37.0245 --13.2701 -8.9641 37.3304 --12.7967 -7.65641 37.417 --12.2467 -6.41163 37.3012 --11.6348 -5.2581 37.0066 --10.9767 -4.21596 36.5606 --10.2878 -3.29711 35.9919 --9.5833 -2.506 35.3285 --8.87666 -1.84091 34.5961 --8.17982 -1.29539 33.817 --7.50277 -0.859733 33.01 --6.85355 -0.522345 32.1898 --6.23826 -0.270789 31.3681 --5.66123 -0.09264 30.553 --5.12523 0.0239268 29.7507 --4.63167 0.0897606 28.9649 --4.18088 0.114527 28.1981 --3.77227 0.106627 27.4517 --3.40458 0.0732056 26.7261 --3.07606 0.0202152 26.0213 --2.78458 -0.0474829 25.3369 --2.52784 -0.126003 24.6724 --2.3034 -0.212315 24.0272 --2.10881 -0.304123 23.4005 --1.94167 -0.399759 22.7917 --1.79968 -0.49808 22.2003 --1.68066 -0.59838 21.6255 --1.58258 -0.700318 21.0669 --1.50362 -0.803847 20.5239 --1.44211 -0.909166 19.9963 --1.39657 -1.01667 19.4836 --1.36573 -1.12692 18.9855 --1.34847 -1.24061 18.5017 --1.34388 -1.35856 18.0321 --1.35118 -1.48167 17.5765 --1.3698 -1.61096 17.1347 --1.39929 -1.74749 16.7067 --1.43937 -1.89246 16.2926 --1.48989 -2.0471 15.8923 --1.55083 -2.21276 15.5061 --1.62234 -2.39086 15.1341 --1.70464 -2.58293 14.7766 --1.79814 -2.79058 14.4342 --1.90332 -3.01558 14.1074 --2.02082 -3.25976 13.7969 --2.15138 -3.5251 13.5036 --2.29587 -3.81372 13.2288 --2.4553 -4.12785 12.9737 --2.63078 -4.46985 12.7402 --2.82355 -4.8422 12.5302 --3.03498 -5.24748 12.3462 --3.26656 -5.68832 12.1913 --3.51988 -6.1674 12.0689 --3.79665 -6.68734 11.9833 --4.09863 -7.25064 11.9394 --4.42767 -7.85955 11.943 --4.78562 -8.51587 12.0009 --5.17431 -9.22081 12.1208 --5.59547 -9.97464 12.3118 --6.05061 -10.7764 12.584 --6.54096 -11.6234 12.9487 --7.06726 -12.5109 13.4183 --7.62959 -13.4312 14.0059 --8.22714 -14.3733 14.7254 --8.85795 -15.3222 15.59 --9.51857 -16.258 16.6122 --10.2037 -17.1557 17.8018 --10.9061 -17.9851 19.1649 --11.6158 -18.7106 20.7015 --12.3205 -19.2927 22.4034 --13.0048 -19.6897 24.2526 --13.6511 -19.8603 26.219 --14.2397 -19.7676 28.2602 --14.7496 -19.3834 30.3214 --15.1596 -18.6928 32.3382 --15.4501 -17.6982 34.2404 --15.6043 -16.4211 35.9582 --15.6103 -14.9029 37.4282 --15.4619 -13.2011 38.6009 --15.1595 -11.385 39.4447 --14.7105 -9.5281 39.949 --14.1285 -7.70143 40.1239 --13.432 -5.96687 39.997 --12.6432 -4.37287 39.6087 --11.786 -2.95232 39.0068 --10.8844 -1.7227 38.2406 --9.96093 -0.687914 37.3572 --9.03608 0.158948 36.398 --8.12718 0.832398 35.3974 --7.24831 1.3516 34.3826 --6.41026 1.73788 33.3733 --5.62076 2.01283 32.3836 --4.88478 2.19699 31.4222 --4.20494 2.30911 30.4943 --3.58192 2.36569 29.602 --3.01492 2.3809 28.7459 --2.50194 2.36665 27.9249 --2.04015 2.33276 27.1375 --1.62615 2.28717 26.3817 --1.25619 2.23626 25.6554 --0.926336 2.18501 24.9566 --0.632602 2.1373 24.2833 --0.371066 2.09609 23.6339 --0.137933 2.06361 23.0068 -0.0704143 2.04152 22.4008 -0.257385 2.03101 21.8146 -0.426161 2.03299 21.2475 -0.579695 2.04809 20.6985 -0.720711 2.07683 20.1671 -0.851718 2.11959 19.6527 -0.97502 2.17674 19.1549 -1.09274 2.24866 18.6735 -1.20683 2.33574 18.2081 -1.3191 2.43846 17.7587 -1.43124 2.55736 17.3253 -1.54484 2.69311 16.908 -1.66139 2.84649 16.5069 -1.78233 3.0184 16.1223 -1.90906 3.20989 15.7548 -2.04295 3.42216 15.4048 -2.18536 3.65655 15.0733 -2.33764 3.91454 14.7611 -2.5012 4.19778 14.4695 -2.67744 4.50805 14.1999 -2.86782 4.84726 13.9542 -3.07384 5.21741 13.7345 -3.29706 5.62061 13.5434 -3.53909 6.059 13.3839 -3.8016 6.53468 13.2598 -4.08628 7.04968 13.1751 -4.39485 7.60582 13.135 -4.72904 8.20457 13.1452 -5.09051 8.84692 13.2123 -5.48082 9.5331 13.3439 -5.90139 10.2624 13.5486 -6.35334 11.0326 13.8359 -6.8374 11.8399 14.2162 -7.35379 12.6782 14.7006 -7.90198 13.5389 15.3007 -8.48051 14.4098 16.028 -9.08674 15.2753 16.8933 -9.7166 16.1154 17.9058 -10.3643 16.9058 19.0716 -11.022 17.6178 20.3927 -11.6797 18.2191 21.8651 -12.3254 18.6746 23.4771 -12.9444 18.9488 25.2078 -13.5205 19.0079 27.0262 -14.0358 18.8237 28.8909 -14.4719 18.3768 30.7512 -14.811 17.6599 32.5498 -15.0367 16.6811 34.2263 -15.1361 15.4646 35.7224 -15.1003 14.0496 36.9873 -14.9257 12.4884 37.9824 -14.6145 10.8408 38.6848 -14.1745 9.1691 39.0889 -13.6185 7.53242 39.2056 -12.9635 5.98166 39.0597 -12.229 4.55648 38.686 -11.436 3.28377 38.1248 -10.6054 2.17789 37.4175 -9.7568 1.24217 36.6037 -8.90796 0.471104 35.7185 -8.07403 -0.147134 34.7915 -7.26739 -0.628361 33.8461 -6.49762 -0.990362 32.9 -5.77169 -1.25134 31.9662 -5.09418 -1.42881 31.053 -4.46764 -1.53891 30.1657 -3.89291 -1.59602 29.3072 -3.36949 -1.61261 28.4785 -2.89583 -1.59924 27.6795 -2.4696 -1.56471 26.9093 -2.08794 -1.51622 26.1666 -1.74766 -1.45954 25.4499 -1.44536 -1.39924 24.7577 -1.17759 -1.33886 24.0886 -0.940941 -1.28112 23.441 -0.732097 -1.22802 22.8139 -0.547894 -1.18103 22.206 -0.385353 -1.14119 21.6163 -0.241695 -1.1092 21.044 -0.114355 -1.0855 20.4884 -0.000974216 -1.07037 19.9486 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Luque - Paquet pst-eqdf - PSTricks  Niveau supérieur - + + M. Luque + Paquet pst-eqdf + PSTricks  Niveau supérieur + \ No newline at end of file