\section{Generalization of the notion of a cylinder and a cone}

\subsection{Cylinder or \Index{cylindric area}}

This paragraph generalizes the  notion of a cylinder, or a cylindric
area\footnote{This was written by
Maxime \textsc{Chupin}, as a result of a question on the list
\url{http://melusine.eu.org/cgi-bin/mailman/listinfo/syracuse}}.
A \textit{routing} curve has to be defined by a function and the
direction of the \textit{cylinder} axis needs to be arranged. In
the example below the routing curve is sinusoidal, situated in the plane $z=-2$:
\begin{verbatim}
\defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
\end{verbatim}
The direction of the cylinder is defined by the components of a vector
\texttt{\Lkeyword{axe}=0 1 1}. The drawing calls  \Lkeyword{object}=\Lkeyval{cylindre} which
in addition to the usual parameters---which is the height \texttt{\Lkeyword{h}=4}---
is about the \textbf{length of the generator} and not of the distance
between the two base planes, and needs to define the routing curve
\texttt{\Lkeyword{function}=G1} and the interval of the variable $t$ \texttt{\Lkeyword{range}=-3 3}.

\begin{verbatim}
\psSolid[object=cylindre,
   h=4,function=G1,
   range=-3 3,
   ngrid=3 16,
   axe=0 1 1,
   incolor=green!50,
   fillcolor=yellow!50]
\end{verbatim}


\begin{center}
\psset{unit=0.75}
\begin{pspicture}(-5,-4)(5,4)
\psset{lightsrc=viewpoint,viewpoint=100 10 20 rtp2xyz,Decran=100}
\psSolid[object=grille,base=-4 4 -6 6,linecolor={[rgb]{0.72 0.72 0.5}},action=draw](0,0,-2)
\defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
\defFunction[algebraic]{G2}(t){t}{2*sin(t)+4}{2}
\psSolid[object=courbe,function=G1,
   range=-3 3,r=0,
   linecolor=blue,
   linewidth=2pt]
\psSolid[object=cylindre,
   h=5.65685,function=G1,
   range=-3 3,
   ngrid=3 16,
   axe=0 1 1,
   incolor=green!50,
   fillcolor=yellow!50]
\psSolid[object=courbe,function=G2,
   range=-3 3,r=0,
   linecolor=blue,
   linewidth=2pt]
\psSolid[object=parallelepiped,
   a=8,b=12,c=4,action=draw](0,0,0)
\psSolid[object=plan,action=draw,
   definition=equation,
   args={[0 0 1 -2] 90},
   base=-6 6 -4 4,planmarks,showBase]
\psSolid[object=plan,action=draw,
   definition=equation,
   args={[0 1 0 -6] 180},
   base=-4 4 -2 2,planmarks,showBase]
\psSolid[object=plan,action=draw,
   definition=equation,
   args={[1 0 0 -4] 90},
   base=-6 6 -2 2,planmarks,showBase]
\psSolid[object=vecteur,
         linecolor=red,
         args=0 3 3]
\end{pspicture}
\end{center}

In the following example, before drawing the horizontal planes, we calculate the
distance between these two planes.

 \begin{verbatim}
\pstVerb{/ladistance 2 sqrt 2 mul def}
 \end{verbatim}

{\psset{unit=0.75,lightsrc=viewpoint,viewpoint=100 -10 20 rtp2xyz,Decran=100}
\begin{LTXexample}[pos=t]
\begin{pspicture}(-1.5,-3)(6.5,6)
\psSolid[object=grille,base=-3 3 -1 8,action=draw]
\pstVerb{/ladistance 2 sqrt 2 mul def}
\defFunction[algebraic]{G3}(t){6*(cos(t))^3*sin(t)}{4*(cos(t))^2}{0}
\defFunction[algebraic]{G4}(t){6*(cos(t))^3*sin(t)}{4*(cos(t))^2+ladistance}{ladistance}
\psSolid[object=courbe,function=G3,range=0 6.28,r=0,linecolor=blue,linewidth=2pt]
\psSolid[object=cylindre,range=0 -6.28,h=4,function=G3,axe=0 1 1,ngrid=3 36,
   fillcolor=green!50,incolor=yellow!50]
\psSolid[object=courbe,function=G4,range=0 6.28,r=0,linecolor=blue,linewidth=2pt]
\psSolid[object=vecteur,linecolor=red,args=0 ladistance dup]
\psSolid[object=plan,action=draw,definition=equation,args={[0 0 1 ladistance neg] 90},
   base=-1 8 -3 3,planmarks,showBase]
\axesIIID(0,4.5,0)(4,8,5)
\rput(0,-3){\texttt{axe=0 1 1}}
\end{pspicture}
\end{LTXexample}}


\begin{LTXexample}[width=8cm]
\psset{unit=0.75,lightsrc=viewpoint,
  viewpoint=100 -10 20 rtp2xyz,Decran=100}
\begin{pspicture}(-1.5,-3)(6.5,6)
\psSolid[object=grille,base=-3 3 -1 6,action=draw]
\defFunction[algebraic]{G5}(t)
  {t}{0.5*t^2}{0}
\defFunction[algebraic]{G6}(t)
  {t}{0.5*t^2}{4}
\psSolid[object=courbe,function=G5,
  range=-3 2,r=0,linecolor=blue,
  linewidth=2pt]
\psSolid[object=cylindre,
  range=-3 2,h=4,
  function=G5,
  axe=0 0 1, %% valeur par d\'{e}faut
  incolor=green!50,
  fillcolor=yellow!50,
  ngrid=3 8]
\psSolid[object=courbe,function=G6,
  range=-3 2,r=0,linecolor=blue,
  linewidth=2pt]
\axesIIID(0,4.5,0)(4,6,5)
\psSolid[object=vecteur,
  linecolor=red,args=0 0 4]
\psSolid[object=plan,action=draw,
  definition=equation,
  args={[0 0 1 -4] 90},
  base=-1 6 -3 3,planmarks,showBase]
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=8cm]
\psset{unit=0.75,lightsrc=viewpoint,
  viewpoint=100 -10 20 rtp2xyz,Decran=100}
\begin{pspicture}(-3.5,-3)(6.5,6)
\psset{lightsrc=viewpoint,viewpoint=100 45 45,Decran=100}
\psSolid[object=grille,base=-3 3 -2 7,fillcolor=gray!30]
\defFunction[algebraic]{G7}(t)
   {2*cos(t)}{2*sin(t)}{0}
\defFunction[algebraic]{G8}(t)
    {2*cos(t)}{2*sin(t)+4}{4}
\psSolid[object=courbe,function=G7,
  range=0 6.28,r=0,
  linecolor=blue,linewidth=2pt]
\psSolid[object=cylindre,
  range=0 6.28,h=5.65685,
  function=G7,axe=0 1 1,
  incolor=green!20,
  fillcolor=yellow!50,
  ngrid=3 36]
\psSolid[object=courbe,function=G8,
  range=0 6.28,r=0,linecolor=blue,
  linewidth=2pt]
\axesIIID(2,4.5,2)(4,8,5)
\psSolid[object=vecteur,
  linecolor=red,args=0 1 1](0,4,4)
\psSolid[object=plan,action=draw,
  definition=equation,
  args={[0 0 1 -4] 90},
  base=-2 7 -3 3,planmarks,showBase]
\end{pspicture}
\end{LTXexample}


\encadre{The routing curve can be any curve and need not necessarily be a plane horizontal.}

\begin{LTXexample}[width=8cm]
\begin{pspicture}(-3.5,-2)(4,5)
\psset{unit=0.75,lightsrc=viewpoint,viewpoint=100 -5 10 rtp2xyz,Decran=100}
\psSolid[object=grille,base=-4 4 -4 4,ngrid=8. 8.](0,0,-1)
\defFunction[algebraic]{G9}(t)
  {3*cos(t)}{3*sin(t)}{1*cos(5*t)}
\psSolid[object=cylindre,
  range=0 6.28,h=5,function=G9,
  axe=0 0 1,incolor=green!50,
  fillcolor=yellow!50,
  ngrid=4 72,grid]
\end{pspicture}
\end{LTXexample}

\subsection{Cone or conic area}
This paragraph generalizes the  notion of a cone, or a conic
area\footnote{This was written by
Maxime \textsc{Chupin}, as the result of a question on the list
\url{http://melusine.eu.org/cgi-bin/mailman/listinfo/syracuse}}.
A \textit{routing} curve needs to be defined by a function which
defines the base of the cone, and the vertex of the \textit{cone}
which is by default \texttt{\Lkeyword{origine}=0 0 0}.  The parts above and
below the cone are symmetric concerning the vertice.  In the example
below, the routing curve is a parabolic arc, situated in the plane $z=-2$.

\begin{LTXexample}[width=7.5cm]
\begin{pspicture}(-3,-4)(4.5,6)
\psset{unit=0.75,lightsrc=viewpoint,viewpoint=100 10 10 rtp2xyz,Decran=100}
\psSolid[object=grille,base=-4 4 -3 3,action=draw](0,0,-2)
\defFunction[algebraic]{G1}(t){t}{0.25*t^2}{-2}
\defFunction[algebraic]{G2}(t){-t}{-0.25*t^2}{2}
\psSolid[object=courbe,function=G1,
  range=-3.46 3,r=0,
  linecolor=blue,linewidth=2pt]
\psSolid[object=cone,function=G1,
  range=-3.46 3,ngrid=3 16,
  incolor=green!50,
  fillcolor=yellow!50,
  origine=0 0 0]
\psSolid[object=courbe,
  function=G2,range=-3.46 3,
   r=0,linecolor=blue,
   linewidth=2pt]
\psPoint(0,0,0){I}
\uput[l](I){\red$(0,0,0)$}
\psdot[linecolor=red](I)
\gridIIID[Zmin=-2,Zmax=2,spotX=r](-4,4)(-3,3)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=7.5cm]
\begin{pspicture}(-3,-4)(4.5,6)
\psset{unit=0.7,lightsrc=viewpoint,viewpoint=100 -10 20 rtp2xyz,Decran=100}
\psSolid[object=grille,base=-4 4 -3 3,
  linecolor={[rgb]{0.72 0.72 0.5}},action=draw](0,0,-2)
\defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
\defFunction[algebraic]{G2}(t){-t}{-2*sin(t)}{2}
\psSolid[object=courbe,function=G1,
   range=-3.14 3.14,r=0,
   linecolor=blue,
   linewidth=2pt]
\psSolid[object=cone,function=G1,
   range=-3.14 3.14,ngrid=3 16,
   incolor=green!50,
   fillcolor=yellow!50,
   origine=0 0 0]
\psSolid[object=courbe,
   function=G2,range=-3.14 3.14,
   r=0,linecolor=blue,
   linewidth=2pt]
\psPoint(0,0,0){I} \uput[l](I){\red$(0,0,0)$}
\psdot[linecolor=red](I)
\gridIIID[Zmin=-2,Zmax=2,spotX=r](-4,4)(-3,3)
\end{pspicture}
\end{LTXexample}

\begin{LTXexample}[width=7.5cm]
\begin{pspicture}(-3,-4)(4.5,6)
\psset{unit=0.7,lightsrc=viewpoint,viewpoint=100 -10 20 rtp2xyz,Decran=100}
\psSolid[object=grille,base=-4 4 -4 4,linecolor={[rgb]{0.72 0.72 0.5}},action=draw](0,0,-2)
\defFunction[algebraic]{G1}(t){t}{2*sin(t)}{-2}
\defFunction[algebraic]{G2}(t){-t}{-2*sin(t)-2}{2}
\psSolid[object=courbe,function=G1,
   range=-3.14 3.14,r=0,
   linecolor=blue,
   linewidth=2pt]
\psSolid[object=cone,
   function=G1,range=-3.14 3.14,
   ngrid=3 16,incolor=green!50,
   fillcolor=yellow!50,
   origine=0 -1 0]
\psSolid[object=courbe,
   function=G2,range=-3.14 3.14,
   r=0,linecolor=blue,
   linewidth=2pt]
\psPoint(0,-1,0){I}\uput[l](I){\red$(0,-1,0)$}
\psdot[linecolor=red](I)
\gridIIID[Zmin=-2,Zmax=2,spotX=r](-4,4)(-4,4)
\end{pspicture}
\end{LTXexample}

\encadre{For the cones as well, the routing curve can be any curve and need not necessarily
be a plane horizontal curve, as the following example, written by Maxime
\textsc{Chupin}, will show.}

\url{http://melusine.eu.org/lab/bpst/pst-solides3d/cone/cone-dir_02.pst}

\endinput