\section {Courbes de fonctions de $R$ vers $R^3$}

%% \section{Fonctions R --> R\textsuperscript{3}}

Le tracé de la fonction ainsi définie fait appel à l'objet
\Cadre{courbe} et à l'option \Cadre{function}. 

On pourra réaliser le tracé de l'hélice :

%% \[
%% \left\{
%% \begin{array}{lcl}
%% x&=&3\cos(4t)\\
%% y&=&3\sin(4t)\\
%% z&=&t
%% \end{array}
%% \right.
%% \]
en notation algébrique avec la fonction :

%%\begin{boxedverbatim}
\begin{verbatim}
\defFunction[algebraic]{helice}(t){3*cos(4*t)}{3*sin(4*t)}{t}
\end{verbatim}
%%\end{boxedverbatim}

%\defFunction[algebraic]{f}(t){t}{(t^2)/10-1}{}
\begin{minipage}{0.4\linewidth}
\psset{unit=0.5}
\psset{lightsrc=10 -20 50,viewpoint=50 -20 20 rtp2xyz,Decran=50}
\begin{pspicture}(-6,-3)(6,8)
\psframe*[linecolor=blue!50](-6,-3)(6,8)
\psSolid[object=grille,base=-4 4 -4 4,linecolor=red,linewidth=0.5\pslinewidth]%
\axesIIID(0,0,0)(4,4,7)
\defFunction[algebraic]{helice}(t){3*cos(4*t)}{3*sin(4*t)}{t}
\psSolid[object=courbe,
        r=0,
        range=0 6,
        linecolor=blue,linewidth=0.1,
        resolution=360,
        function=helice]%
\end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.55\linewidth}
%%\begin{boxedverbatim}
\begin{verbatim}
\defFunction[algebraic]%
   {helice}(t){cos(3*t)}{sin(3*t)}{t}
\psSolid[object=courbe,
        range=0 6,
        r=0,
        linecolor=blue,
        resolution=360,
        function=helice]%
\end{verbatim}
%%\end{boxedverbatim}
\end{minipage}

\begin{minipage}{0.4\linewidth}
\psset{unit=0.5}
\begin{pspicture}(-6,-3)(6,8)
\psframe*[linecolor=blue!50](-6,-3)(6,8)
\psset{lightsrc=10 -20 50,viewpoint=50 -20 30 rtp2xyz,Decran=50}
\psSolid[object=grille,base=-4 4 -4 4,linecolor=red,linewidth=0.5\pslinewidth]%
\axesIIID(0,0,0)(4,4,7)
\psset{range=-4 4}
\defFunction{CosRad}(t){t 2 mul Cos 4 mul}{t}{0}
\psSolid[object=courbe,linewidth=0.1,
        r=0,
        linecolor=red,
        resolution=360,
        function=CosRad]%
\psSolid[object=grille,base=-4 4 -4 4,linecolor=blue,linewidth=0.5\pslinewidth](0,0,3)
\psPoint(0,0,3){O1}\psPoint(0,0,7){Z1}\psline(O1)(Z1)\psline[linestyle=dashed](O1)(O)
\pstVerb{/tmin -4 def /tmax 4 def}%
\defFunction{sinRad}(t){t}{t Sin 3 mul}{3}
\psSolid[object=courbe,linewidth=0.1,
        r=0,
        linecolor=blue,
        resolution=30,
        function=sinRad]%
\end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.55\linewidth}
%%\begin{boxedverbatim}
\begin{verbatim}
\psset{range=-4 4}
\psSolid[object=grille,base=-4 4 -4 4]%
\defFunction{CosRad}(t){t 2 mul Cos 4 mul}{t}{0}
\psSolid[object=courbe,linewidth=0.1,
        r=0,
        linecolor=red,
        resolution=360,
        function=CosRad]%
\defFunction{sinRad}(t){t}{t Sin 3 mul}{3}
\psSolid[object=grille,base=-4 4 -4 4](0,0,3)
\psSolid[object=courbe,
        r=0,
        linecolor=blue,
        resolution=360,
        function=sinRad]
\end{verbatim}
%%\end{boxedverbatim}
\end{minipage}

\begin{minipage}{0.4\linewidth}
\psset{unit=0.5}
\begin{pspicture}(-5,0)(7,9)
\psset{lightsrc=10 -20 50,viewpoint=50 -20 20 rtp2xyz,Decran=50}
\psSolid[object=grille,base=-4 4 -4 4,linecolor=lightgray,linewidth=0.5\pslinewidth]%
\psSolid[object=grille,base=-4 4 0 8,linecolor=lightgray,RotX=90,linewidth=0.5\pslinewidth](0,4,0)
\psSolid[object=grille,base=-4 4 -4 4,linecolor=lightgray,RotY=90,linewidth=0.5\pslinewidth](-4,0,4)
\defFunction[algebraic]{helice}%
(t){1.3*(1-cos(2.5*t))*cos(6*t)}{1.3*(1-cos(2.5*t))*sin(6*t)}{t}
\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}
\defFunction[algebraic]{helice_xz}(t){1.3*(1-cos(2.5*t))*cos(6*t)}{4}{t}
\defFunction[algebraic]{helice_yz}(t){-4}{1.3*(1-cos(2.5*t))*sin(6*t)}{t}
\psset{range=0 8}
\psSolid[object=courbe,
        r=0,
        linecolor=blue,
        linewidth=0.05,
        resolution=360,
        normal=0 0 1,
        function=helice_xy]%
\psSolid[object=courbe,
        r=0,
        linecolor=green,
        linewidth=0.05,
        resolution=360,
        normal=0 0 1,
        function=helice_xz]%
\psSolid[object=courbe,
        r=0,
        linewidth=0.05,
        resolution=360,
        normal=0 0 1,
        function=helice_yz]%
\psSolid[object=courbe,
        r=0,
        linecolor=red,
        linewidth=0.1,
        resolution=360,
        function=helice]%
 \end{pspicture}
\end{minipage}
\hfill
\begin{minipage}{0.55\linewidth}
\footnotesize
%%\begin{boxedverbatim}
\begin{verbatim}
\psSolid[object=grille,base=-4 4 -4 4]%
\psSolid[object=grille,base=-4 4 0 8](0,4,0)
\psSolid[object=grille,base=-4 4 -4 4](-4,0,4)
\defFunction[algebraic]{helice}%
   (t){1.3*(1-cos(2.5*t))*cos(6*t)}
   {1.3*(1-cos(2.5*t))*sin(6*t)}{t}
\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}
\defFunction[algebraic]{helice_xz}%
   (t){1.3*(1-cos(2.5*t))*cos(6*t)}{4}{t}
\defFunction[algebraic]{helice_yz}%
   (t){-4}{1.3*(1-cos(2.5*t))*sin(6*t)}{t}
\psset{range=0 8}
\psSolid[object=courbe,
        r=0,
        linecolor=blue,
        linewidth=0.05,
        resolution=360,
        normal=0 0 1,
        function=helice_xy]%
\psSolid[object=courbe,
        r=0,
        linecolor=green,
        linewidth=0.05,
        resolution=360,
        normal=0 0 1,
        function=helice_xz]%
\psSolid[object=courbe,
        r=0,
        linewidth=0.05,
        resolution=360,
        normal=0 0 1,
        function=helice_yz]%
\psSolid[object=courbe,
        r=0,
        linecolor=red,
        linewidth=0.1,
        resolution=360,
        function=helice]%
\end{verbatim}
%%\end{boxedverbatim}
\end{minipage}

Ces dernières courbes se trouvent sous forme d'animations sur la page :

\centerline{\url{http://melusine.eu.org/syracuse/pstricks/pst-solides3d/animations/}}


