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-
-\section{Solid rings}
-
-This paragraph discusses the cylindric \Index{rings}. Within the macro
-\Lcs{psSolid}, this object is passed with the option:
-\texttt{\Lkeyword{object}=\Lkeyval{anneau}}, that comes with 3 parameters:
-\begin{itemize}
- \item the inner radius \texttt{\Lkeyword{r}=1.5} (value by default);
- \item the outer radius \texttt{\Lkeyword{R}=4} (value by default);
- \item the height \texttt{\Lkeyword{h}=6} (value by default).
-\end{itemize}
-
-The argument \Lkeyword{ngrid} defines the number of sections used to make a complete
-rotation of $360$~degrees. Its default value is $24$.
-
-The section of the ring, whose shape is \textit{rectangular} was chosen as default,
-and can be redesigned by the user.
-We will discuss different examples of sections for rings.
-
-\subsection{Predefined command: the ring with a rectangular section}
-
-This section is defined in the plane $Oyz$, it is parameterized with the
-triple $(r, R, h)$. The values of the outer radius $R$, inner radius $r$ and the
-height $h$ are passed in the macro \Lcs{psSolid}. By default, one has a ring with
-a variable rectangular section, and the definition takes place at the time
-of the transmission of the values $(r, R, h)$ into the options of \Lcs{psSolid}.
-
-If the user redefines the \TeX {} macro \verb+\Section+ with some numeric values
-instead of the parameters $r$, $R$ and $h$, then
-the ring won't be variable anymore and it is not necessary to transmit the
-values $r$, $R$, and $h$ into the options of \Lcs{psSolid}.
-
-\begin{minipage}{0.45\linewidth}
-\begin{verbatim}
-\newcommand\Section{%
-% y z
- R h 2 div neg % sommet 1
- % S1 (R,-h/2)
- R h 2 div % sommet 2
- % S2 (r,h/2)
- r h 2 div % sommet 3
- % S3 (r,h/2)
- r h 2 div neg % sommet 4
- % S4 (r,-h/2)
- }
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{0.45\linewidth}
-\psset{unit=0.5}
-\begin{pspicture}(-5,-3)(5,3)
-\pstVerb{/R 4 def /r 2 def /h 2 def}%
-\newcommand\RectangularSection{%
- \pspolygon[fillstyle=vlines,linewidth=2\pslinewidth]%
- (!R h 2 div neg)(!R h 2 div)(!r h 2 div)(!r h 2 div neg)}
-\rput(0,0){\RectangularSection}\rput(-6,0){\RectangularSection}
-\psline(-2,1)(2,1)\psline(-2,-1)(2,-1)
-\psline[linestyle=dashed,dash=1 0.2 0.05 0.2]{->}(0,-3)(0,3)
-\psline[linestyle=dashed]{->}(-4,0)(4.2,0)
-\uput[u](0,3){$z$}\uput[r](4.1,0){$y$}\uput[dl](0,0){$O$}
-\psline{->}(0,1.5)(2,1.5)\uput[u](1,1.5){$r$}
-\psline{->}(0,2.5)(4,2.5)\uput[u](2,2.5){$R$}
-\psline{<->}(5,-1)(5,1)\uput[r](5,0){$h$}
-\uput[dr](!R h 2 div neg){1}
-\uput[ur](!R h 2 div ){2}
-\uput[ul](!r h 2 div ){3}
-\uput[dl](!r h 2 div neg){4}
-\end{pspicture}
-\end{minipage}
-\newpage
-\begin{LTXexample}[width=6cm]
-\psset{unit=0.5}
-\begin{pspicture}(-5,-4)(5,4)
-\psset[pst-solides3d]{viewpoint=50 20 40 rtp2xyz,
- Decran=25,lightsrc=10 20 20}
-\psSolid[object=anneau,fillcolor=cyan,
- h=3,R=8,r=6,ngrid=4,RotX=10](0,0,0)
-\end{pspicture}\\
-\begin{pspicture}(-5,-4)(5,4)
-\psset[pst-solides3d]{viewpoint=50 -20 -40 rtp2xyz,
- Decran=25,lightsrc=-10 -20 -20}
-\psSolid[object=anneau,
-fillcolor=yellow,h=3,R=8,r=6,
-RotX=90,RotZ=10](0,0,0)
-\end{pspicture}
-\end{LTXexample}
-
-
-
-\subsection{Example 1: a simple ring with a triangular section}
-
-Below is a very simple ring with a fixed triangular section.
- The section is defined by $3$~points $(6, -2)$, $(10, 0)$
-and $(6, 2)$ within the option \Lkeyword{section} of \Lcs{psSolid}.
-
-\begin{LTXexample}[width=6cm]
-\psset{unit=0.5}
-\begin{pspicture}(-5,-6)(5,6)
-\psset[pst-solides3d]{viewpoint=50 20 40 rtp2xyz,Decran=25,
- lightsrc=10 20 20}
-\psSolid[object=anneau,
- section=6 -2 10 0 6 2,
- fillcolor=cyan,RotX=10]%
-\end{pspicture}
-\end{LTXexample}
-
-%\newpage
-
-\subsection{Example 2: a ring with a variable triangular section}
-
-\newcommand\SectionTriangulaire{
- R h 2 div neg % sommet 1
- R r add 2 div h 2 div % sommet 2
- r h 2 div neg % sommet 3
-}
-
-\begin{minipage}{0.45\linewidth}
-\begin{verbatim}
-\newcommand\SectionTriangulaire{
-% y <----z---->
- R h 2 div neg
- % S1 (R,-h/2)
- R r add 2 div h 2 div
- % S2 ((R+r)/2,h/2)
- r h 2 div neg
- % S3 (r,-h/2)
-}
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{0.45\linewidth}
-\psset{unit=0.5}
-\begin{pspicture}(-5,-3)(5,3)
-%\psgrid
-\pstVerb{/R 4 def /r 2 def /h 2 def}%
-\newcommand\TriangularSection{%
- \pspolygon[fillstyle=vlines,linewidth=2\pslinewidth]%
- (!R h 2 div neg)(!R r add 2 div h 2 div)(!r h 2 div neg)}
-\rput(0,0){\TriangularSection}\rput(-6,0){\TriangularSection}
-\psline(-3,1)(3,1)\psline(-4,-1)(4,-1)
-\psline[linestyle=dashed,dash=1 0.2 0.05 0.2]{->}(0,-3)(0,3)
-\uput[u](0,3){$z$}\uput[r](4.1,0){$y$}\uput[dl](0,0){$O$}
-\psline[linestyle=dashed](2,-1)(2,1.5)
-\psline[linestyle=dashed](4,-1)(4,2.5)
-\psline[linestyle=dashed]{->}(-4,0)(4.2,0)
-\psline{->}(0,1.5)(2,1.5)\uput[u](1,1.5){$r$}
-\psline{->}(0,2.5)(4,2.5)\uput[u](2,2.5){$R$}
-\psline{<->}(5,-1)(5,1)\uput[r](5,0){$h$}
-\uput[dr](!R h 2 div neg){1}
-\uput[u](!R r add 2 div h 2 div){2}
-\uput[dl](!r h 2 div neg){3}
-\end{pspicture}
-\end{minipage}
-\begin{center}
-%%
-\psset{unit=0.5}
-\begin{pspicture}(-5,-6)(5,6)
-\psframe(-5,-4)(5,4)
-\psset[pst-solides3d]{viewpoint=50 20 40 rtp2xyz,Decran=25,lightsrc=10 20 20}
-\psSolid[object=anneau,section=\SectionTriangulaire,fillcolor=cyan,h=3,R=8,r=4,RotX=10]%
-\end{pspicture}
-%%
-\begin{pspicture}(-5,-6)(5,5)
-\psframe(-5,-4)(5,4)
-\psset[pst-solides3d]{viewpoint=50 -20 -40 rtp2xyz,Decran=25,lightsrc=-10 -20 -20}
-\psSolid[object=anneau,section=\SectionTriangulaire,fillcolor=yellow,h=3,R=8,r=4,RotX=-90,RotZ=10]%
-\end{pspicture}
-\end{center}
-
-\begin{verbatim}
-\psSolid[object=anneau,section=\SectionTriangulaire,%
- fillcolor=cyan,h=3,R=8,r=4,RotX=10](0,0,0)
-\psSolid[object=anneau,section=\SectionTriangulaire,%
- fillcolor=yellow,h=3,R=8,r=4,RotX=-90,RotZ=10](0,0,0)
-\end{verbatim}
-
-%%\newpage
-
-\newcommand\SectionPneu{
- /m {90 4 div} bind def
- /Scos {m cos 2 m mul cos add 3 m mul cos add} bind def
- /Z0 {h 4 div} bind def
- /c {Z0 Scos div} bind def
- /Z1 {Z0 c m cos mul add} bind def
- /Z2 {Z1 c m 2 mul cos mul add} bind def
- /R1 {R c m sin mul sub} bind def
- /R2 {R1 c m 2 mul sin mul sub} bind def
- /R3 {R2 c m 3 mul sin mul sub} bind def
- R h 4 div neg % 1
- R h 4 div % 2
- R1 Z1 % 3
- R2 Z2 % 4
- R3 h 2 div % 5
- r h 2 div % 6
- r h 2 div neg % 7
- R3 h 2 div neg % 8
- R2 Z2 neg % 9
- R1 Z1 neg % 10
- }
-
-\subsection{Example 3: a ring with a``tyre''-like section: cylindric ring with chamfered edges}
-
-\begin{minipage}{0.45\linewidth}
-{\small
-\begin{verbatim}
-\renewcommand\SectionPneu{
- /m {90 4 div} bind def
- /Scos {m cos 2 m mul cos add 3 m mul cos add} bind def
- /Z0 {h 4 div} bind def
- /c {Z0 Scos div} bind def
- /Z1 {Z0 c m cos mul add} bind def
- /Z2 {Z1 c m 2 mul cos mul add} bind def
- /R1 {R c m sin mul sub} bind def
- /R2 {R1 c m 2 mul sin mul sub} bind def
- /R3 {R2 c m 3 mul sin mul sub} bind def
- R h 4 div neg % 1
- R h 4 div % 2
- R1 Z1 % 3
- R2 Z2 % 4
- R3 h 2 div % 5
- r h 2 div % 6
- r h 2 div neg % 7
- R3 h 2 div neg % 8
- R2 Z2 neg % 9
- R1 Z1 neg % 10
- }
-\end{verbatim}}
-\end{minipage}
-\hfill
-\begin{minipage}{0.45\linewidth}
-%\psset{unit=0.65}
-\begin{pspicture}(-2,-3)(5,6)
-%\psgrid
-\pstVerb{/R 4 def /r 2 def /h 2 def}%
-\pstVerb{/m {90 4 div} bind def
- /Scos {m cos 2 m mul cos add 3 m mul cos add} bind def
- /Z0 {h 4 div} bind def
- /c {Z0 Scos div} bind def
- /Z1 {Z0 c m cos mul add} bind def
- /Z2 {Z1 c m 2 mul cos mul add} bind def
- /R1 {R c m sin mul sub} bind def
- /R2 {R1 c m 2 mul sin mul sub} bind def
- /R3 {R2 c m 3 mul sin mul sub} bind def}%
-\pnode(!R h 4 div neg){S1}
-\pnode(!R h 4 div){S2}
-\pnode(!R1 Z1){S3}
-\pnode(!R2 Z2){S4}
-\pnode(!R3 h 2 div){S5}
-\pnode(!r h 2 div){S6}
-\pnode(!r h 2 div neg){S7}
-\pnode(!R3 h 2 div neg){S8}
-\pnode(!R2 Z2 neg){S9}
-\pnode(!R1 Z1 neg){S10}
-\newcommand\pneuSection{\pspolygon[fillstyle=vlines,linewidth=2\pslinewidth]%
- (S1)(S2)(S3)(S4)(S5)(S6)(S7)(S8)(S9)(S10)}
-\rput(0,0){\pneuSection}\rput{180}{\pspolygon[fillstyle=vlines,linewidth=2\pslinewidth]%
- (!R h 4 div neg)(!R h 4 div)(!R1 Z1)(!R2 Z2)(!R3 h 2 div)(!r h 2 div)(!r h 2 div neg)(!R3 h 2 div neg)(!R2 Z2 neg)(!R1 Z1 neg)}
-\psline(-3,1)(3,1)\psline(-3,-1)(3,-1)
-\psline[linestyle=dashed,dash=1 0.2 0.05 0.2]{->}(0,-3)(0,3)
-\uput[u](0,3){$z$}\uput[r](4.1,0){$y$}\uput[dl](0,0){$O$}
-\psline[linestyle=dashed](2,-1)(2,1.5)
-\psline[linestyle=dashed](4,-1)(4,2.5)
-\psline[linestyle=dashed]{->}(-4,0)(4.2,0)
-\psline{->}(0,1.5)(2,1.5)\uput[u](1,1.5){$r$}
-\psline{->}(0,2.5)(4,2.5)\uput[u](2,2.5){$R$}
-\psline{<->}(5,-1)(5,1)\uput[r](5,0){$h$}
-\uput[r](S1){1}
-\uput[r](S2){2}
-\uput[u](S3){3}
-\uput[u](S4){4}
-\uput[u](S5){5}
-\uput[ul](S6){6}
-\uput[dl](S7){7}
-\uput[dl](S8){8}
-\uput[dr](S9){9}
-\uput[r](S10){10}
-\psdots[linecolor=red](S1)(S2)(S3)(S4)(S5)(S6)(S7)(S8)(S9)(S10)
-\end{pspicture}
-\end{minipage}
-
-%\iffalse
-\begin{center}
-\psset{unit=0.7}
-\begin{pspicture}(-5,-5)(5,4)
-\psframe(-5,-4)(5,4)
-\psset[pst-solides3d]{viewpoint=50 20 40 rtp2xyz,Decran=25,lightsrc=10 20 20}
-\psSolid[object=anneau,section=\SectionPneu,fillcolor=cyan,h=3,R=8,r=4,RotX=10]%
-\end{pspicture}
-%%
-\begin{pspicture}(-5,-5)(5,4)
-\psframe(-5,-4)(5,4)
-\psset[pst-solides3d]{viewpoint=50 -20 -40 rtp2xyz,Decran=25,lightsrc=-10 -20 -20}
-\psSolid[object=anneau,section=\SectionPneu,fillcolor=yellow,h=3,R=8,r=4,RotX=-90,RotZ=10]%
-\end{pspicture}
-\end{center}
-%%
-
-\begin{verbatim}
-\psSolid[object=anneau,section=\SectionPneu,%
- fillcolor=cyan,h=3,R=8,r=4,RotX=10](0,0,0)
-\psSolid[object=anneau,section=\SectionPneu,%
- fillcolor=yellow,h=3,R=8,r=4,RotX=-90,RotZ=10]%
-\end{verbatim}
-
-%\fi
-\newpage
-
-\subsection{Example 4: an empty bobbin}
-
-\newcommand\SectionBobine{
- r h 2 div % 1
- r h 2 div neg % 2
- R h 2 div neg % 3
- R h 3 div neg % 4
- R h 4 div sub h 3 div neg % 5
- R h 4 div sub h 3 div % 6
- R h 3 div % 7
- R h 2 div % 8
- }
-
-\begin{minipage}{0.45\linewidth}
-\begin{verbatim}
-\newcommand\SectionBobine{
- r h 2 div % 1
- r h 2 div neg % 2
- R h 2 div neg % 3
- R h 3 div neg % 4
- R h 4 div sub h 3 div neg % 5
- R h 4 div sub h 3 div % 6
- R h 3 div % 7
- R h 2 div % 8
- }
-\end{verbatim}
-\end{minipage}
-\hfill
-\begin{minipage}{0.45\linewidth}
-\begin{pspicture}(-2,-2)(5,2)
-\pstVerb{/RB 4 def /rB 2 def /hB 3 def}%
-\pnode(!rB hB 2 div){S1}
-\pnode(!rB hB 2 div neg){S2}
-\pnode(!RB hB 2 div neg){S3}
-\pnode(!RB hB 3 div neg){S4}
-\pnode(!RB hB 4 div sub hB 3 div neg){S5}
-\pnode(!RB hB 4 div sub hB 3 div){S6}
-\pnode(!RB hB 3 div){S7}
-\pnode(!RB hB 2 div){S8}
-\newcommand\pneuSection{\pspolygon[fillstyle=vlines,linewidth=2\pslinewidth]%
- (S1)(S2)(S3)(S4)(S5)(S6)(S7)(S8)}
-\rput(0,0){\pneuSection}\rput{180}{\pspolygon[fillstyle=vlines,linewidth=2\pslinewidth]%
- (!rB hB 2 div)(!rB hB 2 div neg)(!RB hB 2 div neg)(!RB hB 3 div neg)(!RB hB 4 div sub hB 3 div neg)(!RB hB 4 div sub hB 3 div)(!RB hB 3 div)(!RB hB 2 div)}
-\psline(-3,1.5)(3,1.5)\psline(-3,-1.5)(3,-1.5)
-\psline[linestyle=dashed,dash=1 0.2 0.05 0.2]{->}(0,-3)(0,3)
-\uput[u](0,3){$z$}\uput[r](4.1,0){$y$}\uput[dl](0,0){$O$}
-\psline[linestyle=dashed](2,-1)(2,1.5)
-\psline[linestyle=dashed](4,-1)(4,2.5)
-\psline[linestyle=dashed](-4,-1)(4,-1)
-\psline[linestyle=dashed](-4,1)(4,1)
-\psline[linestyle=dashed]{->}(-4,0)(4.2,0)
-\psline{->}(0,1.8)(2,1.8)\uput[u](1,1.8){$r$}
-\psline{->}(0,2.5)(4,2.5)\uput[u](2,2.5){$R$}
-\psline{<->}(5,-1.5)(5,1.5)\uput[r](5,0){$h$}
-\uput[u](S1){1}
-\uput[d](S2){2}
-\uput[d](S3){3}
-\uput[r](S4){4}
-\uput[ur](S5){5}
-\uput[dr](S6){6}
-\uput[r](S7){7}
-\uput[r](S8){8}
-\psdots[linecolor=red](S1)(S2)(S3)(S4)(S5)(S6)(S7)(S8)
-\end{pspicture}
-\end{minipage}
-\begin{center}
-\begin{pspicture}(-5,-6)(5,5)
-\psframe*[linecolor=blue!50](-5,-5)(5,4)
-\psset[pst-solides3d]{viewpoint=70 40 10 rtp2xyz,Decran=25,lightsrc=0 30 100}
-\psSolid[object=grille,base=-15 15 -15 15,fillcolor=yellow!30!black!10](0,0,-8)
-%\psSolid[object=prisme,h=2,base=-15 1 -15 -1 15 -1 15 1](0,0,-8)
-\psSolid[object=anneau,section=\SectionBobine,fillcolor=gray!50,h=6,R=8,r=4,RotX=90,linecolor=gray]%
-\end{pspicture}
-\end{center}
-\begin{verbatim}
-\psSolid[object=grille,base=-15 15 -15 15,fillcolor=yellow!30](0,0,-8)
-\psSolid[object=anneau,section=\SectionBobine,%
- fillcolor=gray!50,h=6,R=8,r=4,RotX=90,linecolor=gray]%
-\end{verbatim}
-
-
-
-\subsection{Some other rings}
-
-Three other examples are available on the website:
-
-\centerline{\url{http://syracuse.eu.org/lab/bpst/pst-solides3d/anneaux}}
-
-
-\endinput