doc/par-courbeR3.tex

   1 \section {Courbes de fonctions de $R$ vers $R^3$}
   2
   3 %% \section{Fonctions R --> R\textsuperscript{3}}
   4
   5 Le tracé de la fonction ainsi définie fait appel à l'objet
   6 \Cadre{courbe} et à l'option \Cadre{function}.
   7
   8 On pourra réaliser le tracé de l'hélice :
   9
  10 %% \[
  11 %% \left\{
  12 %% \begin{array}{lcl}
  13 %% x&=&3\cos(4t)\\
  14 %% y&=&3\sin(4t)\\
  15 %% z&=&t
  16 %% \end{array}
  17 %% \right.
  18 %% \]
  19 en notation algébrique avec la fonction :
  20
  21 %%\begin{boxedverbatim}
  22 \begin{verbatim}
  23 \defFunction[algebraic]{helice}(t){3*cos(4*t)}{3*sin(4*t)}{t}
  24 \end{verbatim}
  25 %%\end{boxedverbatim}
  26
  27 %\defFunction[algebraic]{f}(t){t}{(t^2)/10-1}{}
  28 \begin{minipage}{0.4\linewidth}
  29 \psset{unit=0.5}
  30 \psset{lightsrc=10 -20 50,viewpoint=50 -20 20 rtp2xyz,Decran=50}
  31 \begin{pspicture}(-6,-3)(6,8)
  32 \psframe*[linecolor=blue!50](-6,-3)(6,8)
  33 \psSolid[object=grille,base=-4 4 -4 4,linecolor=red,linewidth=0.5\pslinewidth]%
  34 \axesIIID(0,0,0)(4,4,7)
  35 \defFunction[algebraic]{helice}(t){3*cos(4*t)}{3*sin(4*t)}{t}
  36 \psSolid[object=courbe,
  37         r=0,
  38         range=0 6,
  39         linecolor=blue,linewidth=0.1,
  40         resolution=360,
  41         function=helice]%
  42 \end{pspicture}
  43 \end{minipage}
  44 \hfill
  45 \begin{minipage}{0.55\linewidth}
  46 %%\begin{boxedverbatim}
  47 \begin{verbatim}
  48 \defFunction[algebraic]%
  49    {helice}(t){cos(3*t)}{sin(3*t)}{t}
  50 \psSolid[object=courbe,
  51         range=0 6,
  52         r=0,
  53         linecolor=blue,
  54         resolution=360,
  55         function=helice]%
  56 \end{verbatim}
  57 %%\end{boxedverbatim}
  58 \end{minipage}
  59
  60 \begin{minipage}{0.4\linewidth}
  61 \psset{unit=0.5}
  62 \begin{pspicture}(-6,-3)(6,8)
  63 \psframe*[linecolor=blue!50](-6,-3)(6,8)
  64 \psset{lightsrc=10 -20 50,viewpoint=50 -20 30 rtp2xyz,Decran=50}
  65 \psSolid[object=grille,base=-4 4 -4 4,linecolor=red,linewidth=0.5\pslinewidth]%
  66 \axesIIID(0,0,0)(4,4,7)
  67 \psset{range=-4 4}
  68 \defFunction{CosRad}(t){t 2 mul Cos 4 mul}{t}{0}
  69 \psSolid[object=courbe,linewidth=0.1,
  70         r=0,
  71         linecolor=red,
  72         resolution=360,
  73         function=CosRad]%
  74 \psSolid[object=grille,base=-4 4 -4 4,linecolor=blue,linewidth=0.5\pslinewidth](0,0,3)
  75 \psPoint(0,0,3){O1}\psPoint(0,0,7){Z1}\psline(O1)(Z1)\psline[linestyle=dashed](O1)(O)
  76 \pstVerb{/tmin -4 def /tmax 4 def}%
  77 \defFunction{sinRad}(t){t}{t Sin 3 mul}{3}
  78 \psSolid[object=courbe,linewidth=0.1,
  79         r=0,
  80         linecolor=blue,
  81         resolution=30,
  82         function=sinRad]%
  83 \end{pspicture}
  84 \end{minipage}
  85 \hfill
  86 \begin{minipage}{0.55\linewidth}
  87 %%\begin{boxedverbatim}
  88 \begin{verbatim}
  89 \psset{range=-4 4}
  90 \psSolid[object=grille,base=-4 4 -4 4]%
  91 \defFunction{CosRad}(t){t 2 mul Cos 4 mul}{t}{0}
  92 \psSolid[object=courbe,linewidth=0.1,
  93         r=0,
  94         linecolor=red,
  95         resolution=360,
  96         function=CosRad]%
  97 \defFunction{sinRad}(t){t}{t Sin 3 mul}{3}
  98 \psSolid[object=grille,base=-4 4 -4 4](0,0,3)
  99 \psSolid[object=courbe,
 100         r=0,
 101         linecolor=blue,
 102         resolution=360,
 103         function=sinRad]
 104 \end{verbatim}
 105 %%\end{boxedverbatim}
 106 \end{minipage}
 107
 108 \begin{minipage}{0.4\linewidth}
 109 \psset{unit=0.5}
 110 \begin{pspicture}(-5,0)(7,9)
 111 \psset{lightsrc=10 -20 50,viewpoint=50 -20 20 rtp2xyz,Decran=50}
 112 \psSolid[object=grille,base=-4 4 -4 4,linecolor=lightgray,linewidth=0.5\pslinewidth]%
 113 \psSolid[object=grille,base=-4 4 0 8,linecolor=lightgray,RotX=90,linewidth=0.5\pslinewidth](0,4,0)
 114 \psSolid[object=grille,base=-4 4 -4 4,linecolor=lightgray,RotY=90,linewidth=0.5\pslinewidth](-4,0,4)
 115 \defFunction[algebraic]{helice}%
 116 (t){1.3*(1-cos(2.5*t))*cos(6*t)}{1.3*(1-cos(2.5*t))*sin(6*t)}{t}
 117 \defFunction[algebraic]{helice_xy}(t){1.3*(1-cos(2.5*t))*cos(6*t)}{1.3*(1-cos(2.5*t))*sin(6*t)}{0}
 118 \defFunction[algebraic]{helice_xz}(t){1.3*(1-cos(2.5*t))*cos(6*t)}{4}{t}
 119 \defFunction[algebraic]{helice_yz}(t){-4}{1.3*(1-cos(2.5*t))*sin(6*t)}{t}
 120 \psset{range=0 8}
 121 \psSolid[object=courbe,
 122         r=0,
 123         linecolor=blue,
 124         linewidth=0.05,
 125         resolution=360,
 126         normal=0 0 1,
 127         function=helice_xy]%
 128 \psSolid[object=courbe,
 129         r=0,
 130         linecolor=green,
 131         linewidth=0.05,
 132         resolution=360,
 133         normal=0 0 1,
 134         function=helice_xz]%
 135 \psSolid[object=courbe,
 136         r=0,
 137         linewidth=0.05,
 138         resolution=360,
 139         normal=0 0 1,
 140         function=helice_yz]%
 141 \psSolid[object=courbe,
 142         r=0,
 143         linecolor=red,
 144         linewidth=0.1,
 145         resolution=360,
 146         function=helice]%
 147  \end{pspicture}
 148 \end{minipage}
 149 \hfill
 150 \begin{minipage}{0.55\linewidth}
 151 \footnotesize
 152 %%\begin{boxedverbatim}
 153 \begin{verbatim}
 154 \psSolid[object=grille,base=-4 4 -4 4]%
 155 \psSolid[object=grille,base=-4 4 0 8](0,4,0)
 156 \psSolid[object=grille,base=-4 4 -4 4](-4,0,4)
 157 \defFunction[algebraic]{helice}%
 158    (t){1.3*(1-cos(2.5*t))*cos(6*t)}
 159    {1.3*(1-cos(2.5*t))*sin(6*t)}{t}
 160 \defFunction[algebraic]{helice_xy}%
 161    (t){1.3*(1-cos(2.5*t))*cos(6*t)}
 162    {1.3*(1-cos(2.5*t))*sin(6*t)}{0}
 163 \defFunction[algebraic]{helice_xz}%
 164    (t){1.3*(1-cos(2.5*t))*cos(6*t)}{4}{t}
 165 \defFunction[algebraic]{helice_yz}%
 166    (t){-4}{1.3*(1-cos(2.5*t))*sin(6*t)}{t}
 167 \psset{range=0 8}
 168 \psSolid[object=courbe,
 169         r=0,
 170         linecolor=blue,
 171         linewidth=0.05,
 172         resolution=360,
 173         normal=0 0 1,
 174         function=helice_xy]%
 175 \psSolid[object=courbe,
 176         r=0,
 177         linecolor=green,
 178         linewidth=0.05,
 179         resolution=360,
 180         normal=0 0 1,
 181         function=helice_xz]%
 182 \psSolid[object=courbe,
 183         r=0,
 184         linewidth=0.05,
 185         resolution=360,
 186         normal=0 0 1,
 187         function=helice_yz]%
 188 \psSolid[object=courbe,
 189         r=0,
 190         linecolor=red,
 191         linewidth=0.1,
 192         resolution=360,
 193         function=helice]%
 194 \end{verbatim}
 195 %%\end{boxedverbatim}
 196 \end{minipage}
 197
 198 Ces dernières courbes se trouvent sous forme d'animations sur la page :
 199
 200 \centerline{\url{http://melusine.eu.org/syracuse/pstricks/pst-solides3d/animations/}}
 201
 202