doc-en/par-cylindres-cones-en.tex

   1 \section{Generalization of the notion of a cylinder and a cone}
   2
   3 \subsection{Cylinder or \Index{cylindric area}}
   4
   5 This paragraph generalizes the  notion of a cylinder, or a cylindric
   6 area\footnote{This was written by
   7 Maxime \textsc{Chupin}, as a result of a question on the list
   8 \url{http://melusine.eu.org/cgi-bin/mailman/listinfo/syracuse}}.
   9 A \textit{routing} curve has to be defined by a function and the
  10 direction of the \textit{cylinder} axis needs to be arranged. In
  11 the example below the routing curve is sinusoidal, situated in the plane $z=-2$:
  12 \begin{verbatim}
  13 \defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
  14 \end{verbatim}
  15 The direction of the cylinder is defined by the components of a vector
  16 \texttt{\Lkeyword{axe}=0 1 1}. The drawing calls  \Lkeyword{object}=\Lkeyval{cylindre} which
  17 in addition to the usual parameters---which is the height \texttt{\Lkeyword{h}=4}---
  18 is about the \textbf{length of the generator} and not of the distance
  19 between the two base planes, and needs to define the routing curve
  20 \texttt{\Lkeyword{function}=G1} and the interval of the variable $t$ \texttt{\Lkeyword{range}=-3 3}.
  21
  22 \begin{verbatim}
  23 \psSolid[object=cylindre,
  24    h=4,function=G1,
  25    range=-3 3,
  26    ngrid=3 16,
  27    axe=0 1 1,
  28    incolor=green!50,
  29    fillcolor=yellow!50]
  30 \end{verbatim}
  31
  32
  33 \begin{center}
  34 \psset{unit=0.75}
  35 \begin{pspicture}(-5,-4)(5,4)
  36 \psset{lightsrc=viewpoint,viewpoint=100 10 20 rtp2xyz,Decran=100}
  37 \psSolid[object=grille,base=-4 4 -6 6,linecolor={[rgb]{0.72 0.72 0.5}},action=draw](0,0,-2)
  38 \defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
  39 \defFunction[algebraic]{G2}(t){t}{2*sin(t)+4}{2}
  40 \psSolid[object=courbe,function=G1,
  41    range=-3 3,r=0,
  42    linecolor=blue,
  43    linewidth=2pt]
  44 \psSolid[object=cylindre,
  45    h=5.65685,function=G1,
  46    range=-3 3,
  47    ngrid=3 16,
  48    axe=0 1 1,
  49    incolor=green!50,
  50    fillcolor=yellow!50]
  51 \psSolid[object=courbe,function=G2,
  52    range=-3 3,r=0,
  53    linecolor=blue,
  54    linewidth=2pt]
  55 \psSolid[object=parallelepiped,
  56    a=8,b=12,c=4,action=draw](0,0,0)
  57 \psSolid[object=plan,action=draw,
  58    definition=equation,
  59    args={[0 0 1 -2] 90},
  60    base=-6 6 -4 4,planmarks,showBase]
  61 \psSolid[object=plan,action=draw,
  62    definition=equation,
  63    args={[0 1 0 -6] 180},
  64    base=-4 4 -2 2,planmarks,showBase]
  65 \psSolid[object=plan,action=draw,
  66    definition=equation,
  67    args={[1 0 0 -4] 90},
  68    base=-6 6 -2 2,planmarks,showBase]
  69 \psSolid[object=vecteur,
  70          linecolor=red,
  71          args=0 3 3]
  72 \end{pspicture}
  73 \end{center}
  74
  75 In the following example, before drawing the horizontal planes, we calculate the
  76 distance between these two planes.
  77
  78  \begin{verbatim}
  79 \pstVerb{/ladistance 2 sqrt 2 mul def}
  80  \end{verbatim}
  81
  82 {\psset{unit=0.75,lightsrc=viewpoint,viewpoint=100 -10 20 rtp2xyz,Decran=100}
  83 \begin{LTXexample}[pos=t]
  84 \begin{pspicture}(-1.5,-3)(6.5,6)
  85 \psSolid[object=grille,base=-3 3 -1 8,action=draw]
  86 \pstVerb{/ladistance 2 sqrt 2 mul def}
  87 \defFunction[algebraic]{G3}(t){6*(cos(t))^3*sin(t)}{4*(cos(t))^2}{0}
  88 \defFunction[algebraic]{G4}(t){6*(cos(t))^3*sin(t)}{4*(cos(t))^2+ladistance}{ladistance}
  89 \psSolid[object=courbe,function=G3,range=0 6.28,r=0,linecolor=blue,linewidth=2pt]
  90 \psSolid[object=cylindre,range=0 -6.28,h=4,function=G3,axe=0 1 1,ngrid=3 36,
  91    fillcolor=green!50,incolor=yellow!50]
  92 \psSolid[object=courbe,function=G4,range=0 6.28,r=0,linecolor=blue,linewidth=2pt]
  93 \psSolid[object=vecteur,linecolor=red,args=0 ladistance dup]
  94 \psSolid[object=plan,action=draw,definition=equation,args={[0 0 1 ladistance neg] 90},
  95    base=-1 8 -3 3,planmarks,showBase]
  96 \axesIIID(0,4.5,0)(4,8,5)
  97 \rput(0,-3){\texttt{axe=0 1 1}}
  98 \end{pspicture}
  99 \end{LTXexample}}
 100
 101
 102 \begin{LTXexample}[width=8cm]
 103 \psset{unit=0.75,lightsrc=viewpoint,
 104   viewpoint=100 -10 20 rtp2xyz,Decran=100}
 105 \begin{pspicture}(-1.5,-3)(6.5,6)
 106 \psSolid[object=grille,base=-3 3 -1 6,action=draw]
 107 \defFunction[algebraic]{G5}(t)
 108   {t}{0.5*t^2}{0}
 109 \defFunction[algebraic]{G6}(t)
 110   {t}{0.5*t^2}{4}
 111 \psSolid[object=courbe,function=G5,
 112   range=-3 2,r=0,linecolor=blue,
 113   linewidth=2pt]
 114 \psSolid[object=cylindre,
 115   range=-3 2,h=4,
 116   function=G5,
 117   axe=0 0 1, %% valeur par d\'{e}faut
 118   incolor=green!50,
 119   fillcolor=yellow!50,
 120   ngrid=3 8]
 121 \psSolid[object=courbe,function=G6,
 122   range=-3 2,r=0,linecolor=blue,
 123   linewidth=2pt]
 124 \axesIIID(0,4.5,0)(4,6,5)
 125 \psSolid[object=vecteur,
 126   linecolor=red,args=0 0 4]
 127 \psSolid[object=plan,action=draw,
 128   definition=equation,
 129   args={[0 0 1 -4] 90},
 130   base=-1 6 -3 3,planmarks,showBase]
 131 \end{pspicture}
 132 \end{LTXexample}
 133
 134 \begin{LTXexample}[width=8cm]
 135 \psset{unit=0.75,lightsrc=viewpoint,
 136   viewpoint=100 -10 20 rtp2xyz,Decran=100}
 137 \begin{pspicture}(-3.5,-3)(6.5,6)
 138 \psset{lightsrc=viewpoint,viewpoint=100 45 45,Decran=100}
 139 \psSolid[object=grille,base=-3 3 -2 7,fillcolor=gray!30]
 140 \defFunction[algebraic]{G7}(t)
 141    {2*cos(t)}{2*sin(t)}{0}
 142 \defFunction[algebraic]{G8}(t)
 143     {2*cos(t)}{2*sin(t)+4}{4}
 144 \psSolid[object=courbe,function=G7,
 145   range=0 6.28,r=0,
 146   linecolor=blue,linewidth=2pt]
 147 \psSolid[object=cylindre,
 148   range=0 6.28,h=5.65685,
 149   function=G7,axe=0 1 1,
 150   incolor=green!20,
 151   fillcolor=yellow!50,
 152   ngrid=3 36]
 153 \psSolid[object=courbe,function=G8,
 154   range=0 6.28,r=0,linecolor=blue,
 155   linewidth=2pt]
 156 \axesIIID(2,4.5,2)(4,8,5)
 157 \psSolid[object=vecteur,
 158   linecolor=red,args=0 1 1](0,4,4)
 159 \psSolid[object=plan,action=draw,
 160   definition=equation,
 161   args={[0 0 1 -4] 90},
 162   base=-2 7 -3 3,planmarks,showBase]
 163 \end{pspicture}
 164 \end{LTXexample}
 165
 166
 167 \encadre{The routing curve can be any curve and need not necessarily be a plane horizontal.}
 168
 169 \begin{LTXexample}[width=8cm]
 170 \begin{pspicture}(-3.5,-2)(4,5)
 171 \psset{unit=0.75,lightsrc=viewpoint,viewpoint=100 -5 10 rtp2xyz,Decran=100}
 172 \psSolid[object=grille,base=-4 4 -4 4,ngrid=8. 8.](0,0,-1)
 173 \defFunction[algebraic]{G9}(t)
 174   {3*cos(t)}{3*sin(t)}{1*cos(5*t)}
 175 \psSolid[object=cylindre,
 176   range=0 6.28,h=5,function=G9,
 177   axe=0 0 1,incolor=green!50,
 178   fillcolor=yellow!50,
 179   ngrid=4 72,grid]
 180 \end{pspicture}
 181 \end{LTXexample}
 182
 183 \subsection{Cone or \Index{conic area}}
 184 This paragraph generalizes the  notion of a cone, or a conic
 185 area\footnote{This was written by
 186 Maxime \textsc{Chupin}, as the result of a question on the list
 187 \url{http://melusine.eu.org/cgi-bin/mailman/listinfo/syracuse}}.
 188 A \textit{routing} curve needs to be defined by a function which
 189 defines the base of the cone, and the vertex of the \textit{cone}
 190 which is by default \texttt{\Lkeyword{origine}=0 0 0}.  The parts above and
 191 below the cone are symmetric concerning the vertice.  In the example
 192 below, the routing curve is a parabolic arc, situated in the plane $z=-2$.
 193
 194 \begin{LTXexample}[width=7.5cm]
 195 \begin{pspicture}(-3,-4)(4.5,6)
 196 \psset{unit=0.75,lightsrc=viewpoint,viewpoint=100 10 10 rtp2xyz,Decran=100}
 197 \psSolid[object=grille,base=-4 4 -3 3,action=draw](0,0,-2)
 198 \defFunction[algebraic]{G1}(t){t}{0.25*t^2}{-2}
 199 \defFunction[algebraic]{G2}(t){-t}{-0.25*t^2}{2}
 200 \psSolid[object=courbe,function=G1,
 201   range=-3.46 3,r=0,
 202   linecolor=blue,linewidth=2pt]
 203 \psSolid[object=cone,function=G1,
 204   range=-3.46 3,ngrid=3 16,
 205   incolor=green!50,
 206   fillcolor=yellow!50,
 207   origine=0 0 0]
 208 \psSolid[object=courbe,
 209   function=G2,range=-3.46 3,
 210    r=0,linecolor=blue,
 211    linewidth=2pt]
 212 \psPoint(0,0,0){I}
 213 \uput[l](I){\red$(0,0,0)$}
 214 \psdot[linecolor=red](I)
 215 \gridIIID[Zmin=-2,Zmax=2,spotX=r](-4,4)(-3,3)
 216 \end{pspicture}
 217 \end{LTXexample}
 218
 219 \begin{LTXexample}[width=7.5cm]
 220 \begin{pspicture}(-3,-4)(4.5,6)
 221 \psset{unit=0.7,lightsrc=viewpoint,viewpoint=100 -10 20 rtp2xyz,Decran=100}
 222 \psSolid[object=grille,base=-4 4 -3 3,
 223   linecolor={[rgb]{0.72 0.72 0.5}},action=draw](0,0,-2)
 224 \defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
 225 \defFunction[algebraic]{G2}(t){-t}{-2*sin(t)}{2}
 226 \psSolid[object=courbe,function=G1,
 227    range=-3.14 3.14,r=0,
 228    linecolor=blue,
 229    linewidth=2pt]
 230 \psSolid[object=cone,function=G1,
 231    range=-3.14 3.14,ngrid=3 16,
 232    incolor=green!50,
 233    fillcolor=yellow!50,
 234    origine=0 0 0]
 235 \psSolid[object=courbe,
 236    function=G2,range=-3.14 3.14,
 237    r=0,linecolor=blue,
 238    linewidth=2pt]
 239 \psPoint(0,0,0){I} \uput[l](I){\red$(0,0,0)$}
 240 \psdot[linecolor=red](I)
 241 \gridIIID[Zmin=-2,Zmax=2,spotX=r](-4,4)(-3,3)
 242 \end{pspicture}
 243 \end{LTXexample}
 244
 245 \begin{LTXexample}[width=7.5cm]
 246 \begin{pspicture}(-3,-4)(4.5,6)
 247 \psset{unit=0.7,lightsrc=viewpoint,viewpoint=100 -10 20 rtp2xyz,Decran=100}
 248 \psSolid[object=grille,base=-4 4 -4 4,linecolor={[rgb]{0.72 0.72 0.5}},action=draw](0,0,-2)
 249 \defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
 250 \defFunction[algebraic]{G2}(t){-t}{-2*sin(t)-2}{2}
 251 \psSolid[object=courbe,function=G1,
 252    range=-3.14 3.14,r=0,
 253    linecolor=blue,
 254    linewidth=2pt]
 255 \psSolid[object=cone,
 256    function=G1,range=-3.14 3.14,
 257    ngrid=3 16,incolor=green!50,
 258    fillcolor=yellow!50,
 259    origine=0 -1 0]
 260 \psSolid[object=courbe,
 261    function=G2,range=-3.14 3.14,
 262    r=0,linecolor=blue,
 263    linewidth=2pt]
 264 \psPoint(0,-1,0){I}\uput[l](I){\red$(0,-1,0)$}
 265 \psdot[linecolor=red](I)
 266 \gridIIID[Zmin=-2,Zmax=2,spotX=r](-4,4)(-4,4)
 267 \end{pspicture}
 268 \end{LTXexample}
 269
 270 \encadre{For the cones as well, the routing curve can be any curve and need not necessarily
 271 be a plane horizontal curve, as the following example, written by Maxime
 272 \textsc{Chupin}, will show.}
 273
 274 \centerline{\url{http://melusine.eu.org/lab/bpst/pst-solides3d/cone/cone-dir_02.pst}}
 275
 276 \endinput