\section{Sectioning a solid with a plane}
\subsection{Drawing the \Index{intersection} between a plane and a solid}
\subsubsection{The parameters}
The option \texttt{intersectionplan=\{[a b c d]\}} allows the user
to draw the intersection between a plane and a solid. The numbers
between the braces are the coefficients of the affine plane with
equation: $ax+by+cz+d=0$. It is possible to draw the intersection
between a solid and more than one plane by placing the appropriate
parameters in order, as in the following example.
The drawing is activated with \texttt{\texttt{\Lkeyword{intersectiontype}=0}} or any
value $\geq0$.
The colour of the intersection line is chosen with the option
\texttt{\Lkeyword{intersectioncolor}=(bleu) (rouge) etc.}. In the same order,
the thickness of the appropriate line
\texttt{\Lkeyword{intersectionlinewidth}=1 2 etc.} (dimensions in picas) is
set up.
The hidden parts, drawn with dashed lines, will be shown with
\Lkeyword{action}=\Lkeyval{draw}.
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3,-2)(3,7.5)
\psset{viewpoint=50 20 20 rtp2xyz,Decran=50}
\psset{lightsrc=viewpoint}
\psSolid[object=cylindre,
ngrid=1 24,
r=2,
fillcolor=yellow!25,
intersectiontype=0,
intersectionplan={
[0 0 1 -1]
[0 0 1 -2]
[0 0 1 -3]
[0.894 0 0.447 -1.8]},
intersectioncolor=(bleu) (rouge) (vert) (rose),
intersectionlinewidth=1 1.5 1.8 2.2]
\axesIIID(2,2,6)(3,3,7)
\end{pspicture}
\end{LTXexample}
\subsection{Slicing a solid}
\subsubsection{Slicing a filled solid}
The object under consideration is a cylinder. The plane that
slices the object is defined by:
\begin{verbatim}
plansepare={[a b c d]}
\end{verbatim}
The two parts are not drawn, but memorised with the name
\texttt{\Lkeyword{name}=partiescylindre}:
\begin{verbatim}
\psset{solidmemory}
\psSolid[object=cylindre,
r=2,h=6
ngrid=6 24,
plansepare={[0.707 0 0.707 0]},
name=partiescylindre,
action=none](0,0,-3)
\end{verbatim}
Then they are displayed separately using their respective index
numbers. The numbering of the two parts is determined by the
direction of the normal to the \Index{slicing} plane: 0 if above the
normal, 1 if below. For both parts, the sliced face carries the
number 0. If there are several sliced faces, as may happen in the
case of a torus, they are numbered 0, 1 etc.
\begin{verbatim}
\psSolid[object=load,
load=partiescylindre1,
fillcolor={[rgb]{0.7 1 0.7 }},
fcol=0 (1 1 0.7 setrgbcolor)]
\psSolid[object=load,
load=partiescylindre0,RotZ=60,
fillcolor={[rgb]{0.7 1 0.7 }},
fcol=0 (1 1 0.7 setrgbcolor)](0,4,0)
\end{verbatim}
\begin{center}
\begin{pspicture}(-4,-5)(7,4)
\psframe(-4,-5)(7,4)
\psset{viewpoint=50 -40 10 rtp2xyz,Decran=50,linecolor=darkgray}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,action=draw,
base=-3 5 -3 5,
linecolor=red](0,0,-3)
\psset{solidmemory}
\psSolid[object=cylindre,
r=2,h=6,
ngrid=6 24,
plansepare={[0.707 0 0.707 0]},
name=partiescylindre,
action=none](0,0,-3)
\psSolid[object=load,
load=partiescylindre1,
fillcolor={[rgb]{0.7 1 0.7 }},
fcol=0 (1 1 0.7 setrgbcolor)]
\psSolid[object=load,
load=partiescylindre0,RotZ=90,
fillcolor={[rgb]{0.7 1 0.7 }},
fcol=0 (1 1 0.7 setrgbcolor)](0,4,0)
\psSolid[object=plan,action=draw,
definition=equation,
args={[0.707 0 0.707 0] 90},
base=-2 2 -3 3,planmarks,showBase]
\axesIIID(0,0,0)(2.5,2.5,3.5)
\end{pspicture}
\end{center}
\subsubsection{Slicing a \Index{hollow solid}}
The options \verb+rm=0,hollow+ allow us to not only remove a
face \verb+rm=0+ but also to see inside it \Lkeyword{hollow}.
\begin{center}
\begin{pspicture}(-4,-5)(7,4)
\psframe(-4,-5)(7,4)
\psset{viewpoint=50 -40 10 rtp2xyz,Decran=50,linecolor=darkgray}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,action=draw,
base=-3 5 -3 5,
linecolor=red](0,0,-3)
\psset{solidmemory}
\psSolid[object=cylindre,
r=2,h=6,
ngrid=6 24,
plansepare={[0.707 0 0.707 0.5]},
name=partiescylindre,
action=none](0,0,-3)
\psSolid[object=load,
load=partiescylindre1,
fillcolor={[rgb]{0.7 1 0.7 }},
rm=0,hollow,
incolor={[rgb]{1 1 0.7}}]
\psSolid[object=load,
load=partiescylindre0,RotZ=90,
fillcolor={[rgb]{0.7 1 0.7 }},
rm=0,hollow,
incolor={[rgb]{1 1 0.7}}](0,4,0)
\psSolid[object=plan,action=draw,
definition=equation,
args={[0.707 0 0.707 0.5] 90},
base=-2 2 -3 3,planmarks,showBase]
\composeSolid
\end{pspicture}
\end{center}
\subsection{\Index{Slice} of a \Index{pyramid}}
\subsubsection{Highlighting the \Index{contour lines} and first slice}
This pyramid is generated as \texttt{\Lkeyword{object}=\Lkeyval{new}} by giving a list
of the coordinates of the vertices, and the vertices of each face.
\begin{verbatim}
sommets=
0 -2 0 %% 0
-2 0 0 %% 1
0 4 0 %% 2
4 0 0 %% 3
0 0 5, %% 4
faces={
[3 2 1 0]
[4 0 3]
[4 3 2]
[4 2 1]
[4 1 0]
}
\end{verbatim}
In the first diagram, the slicing lines are highlighted.
\begin{verbatim}
intersectiontype=0,
intersectionplan={[0 0 1 -1] [0 0 1 -2]},
intersectionlinewidth=1 2,
intersectioncolor=(bleu) (rouge)
\end{verbatim}
Then we cut off the upper part, and draw the slicing plane as
well.
\begin{verbatim}
\psSolid[object=new,
sommets=
0 -2 0 %% 0
-2 0 0 %% 1
0 4 0 %% 2
4 0 0 %% 3
0 0 5, %% 4
faces={
[3 2 1 0]
[4 0 3]
[4 3 2]
[4 2 1]
[4 1 0]},
plansepare={[0 0 1 -2]},
name=firstSlice,
action=none]
\psSolid[object=load,action=draw*,
load=firstSlice1]
\psSolid[object=plan,
definition=equation,
args={[0 0 1 -2]},
base=-3 5 -3 5,action=draw]
\end{verbatim}
To avoid having to repeatedly type the vertices and faces of the
pyramid, we save these data to the files:
\begin{itemize}
\item \texttt{Pyramid-couleurs.dat}
\item \texttt{Pyramid-faces.dat}
\item \texttt{Pyramid-sommets.dat}
\item \texttt{Pyramid-io.dat}
\end{itemize}
thanks to the command \Lkeyword{action}=\Lkeyval{writesolid}:
\begin{verbatim}
\psSolid[object=new,
sommets=
0 -2 0 %% 0
-2 0 0 %% 1
0 4 0 %% 2
4 0 0 %% 3
0 0 5, %% 4
faces={
[3 2 1 0]
[4 0 3]
[4 3 2]
[4 2 1]
[4 1 0]
},filename=data/Pyramid,fillcolor=yellow!50,
action=writesolid]
\end{verbatim}
All these lines of code could then be removed and, thereafter, we
would recall the data with the command:
\begin{verbatim}
\psSolid[object=datfile,
filename=data/Pyramid]
\end{verbatim}
\begin{center}
\psset{unit=0.75}
\begin{pspicture}(-5,-2)(5,7)
%\psframe(-5,-2)(5,7)
\psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-3 5 -3 5,
linecolor=gray]
%% create the pyramid with base area in xy-plane
\psSolid[object=new,
sommets=
0 -2 0 %% 0
-2 0 0 %% 1
0 4 0 %% 2
4 0 0 %% 3
0 0 5, %% 4
faces={
[3 2 1 0]
[4 0 3]
[4 3 2]
[4 2 1]
[4 1 0]
}, action=draw*,
intersectiontype=0,
intersectionplan={[0 0 1 -1]
[0 0 1 -2]},
intersectionlinewidth=1 2,
intersectioncolor=(bleu) (rouge)]
\axesIIID[linecolor=blue](4,4,5)(5,5,6)
\end{pspicture}
\hfill
\begin{pspicture}(-5,-2)(5,7)
%\psframe(-5,-2)(5,7)
\psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-3 5 -3 5,
linecolor=gray]
\psset{solidmemory}
%% create the pyramid with base area in xy-plane
\psSolid[object=new,
sommets=
0 -2 0 %% 0
-2 0 0 %% 1
0 4 0 %% 2
4 0 0 %% 3
0 0 5, %% 4
faces={
[3 2 1 0]
[4 0 3]
[4 3 2]
[4 2 1]
[4 1 0]
},
plansepare={[0 0 1 -2]},
name=firstSlice,
action=none]
\psSolid[object=load,action=draw*,
load=firstSlice1]
\psSolid[object=plan,
definition=equation,
args={[0 0 1 -2]},
base=-3 5 -3 5,action=draw]
\axesIIID[linecolor=blue](4,4,2)(5,5,6)
\end{pspicture}
\end{center}
\subsubsection{The second \Index{slice} and its insertion within the \Index{pyramid}}
Having removed the upper part \texttt{firstSlice0} (which no
longer appears), we slice the frustum of the pyramid
\texttt{firstSlice1}, and keep the upper part of this as
\texttt{secondSlice0}, then we record it and insert it into a wire
frame model of the pyramid:
\begin{verbatim}
\psset{solidmemory}
\psSolid[object=datfile,
filename=data/Pyramid,
plansepare={[0 0 1 -2]},
name=firstSlice,
action=none]
\psSolid[object=load,
load=firstSlice1,
action=none,
plansepare={[0 0 1 -1]},
name=secondSlice]
\psSolid[object=load,action=draw*,
load=secondSlice0]
\psSolid[object=load,
load=secondSlice0,
filename=data/slicePyramid,
action=writesolid]
\psSolid[object=datfile,fillcolor=yellow!50,
filename=data/slicePyramid]
\end{verbatim}
\begin{center}
\psset{unit=0.75}
\begin{pspicture}(-5,-2)(4,7)
%\psframe(-4,-2)(4,7)
\psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-3 5 -3 5,
linecolor=gray]
\psset{solidmemory}
%% create the pyramid with base area in xy-plane
%\psSolid[object=new,
% sommets=
% 0 -2 0 %% 0
% -2 0 0 %% 1
% 0 4 0 %% 2
% 4 0 0 %% 3
% 0 0 5, %% 4
% faces={
% [3 2 1 0]
% [4 0 3]
% [4 3 2]
% [4 2 1]
% [4 1 0]
%},filename=data/Pyramid,fillcolor=yellow!50,
% action=writesolid]
\psSolid[object=datfile,
filename=data/Pyramid,
plansepare={[0 0 1 -2]},
name=firstSlice,
action=none]
\psSolid[object=plan,
definition=equation,
args={[0 0 1 -1]},action=draw,
base=-3 5 -3 5]
\psSolid[object=load,
load=firstSlice1,
action=none,
plansepare={[0 0 1 -1]},
name=secondSlice]
%\psSolid[object=load,action=draw*,
% load=secondSlice0]
%\psSolid[object=load,
% load=secondSlice0,
% filename=data/slicePyramid,
% action=writesolid]
\psSolid[object=datfile,fillcolor=yellow!50,
filename=data/slicePyramid]
\psSolid[object=plan,
definition=equation,
args={[0 0 1 -2]},
base=-3 5 -3 5,action=draw]
\axesIIID[linecolor=blue](0,0,2)(5,5,6)
\end{pspicture}
\hfill
\begin{pspicture}(-4,-2)(6,7)
\psset{viewpoint=50 20 10 rtp2xyz,Decran=50}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-3 5 -3 5,
linecolor=gray]
\psSolid[object=datfile,fillcolor=yellow!50,
filename=data/slicePyramid]
\psSolid[object=datfile,
filename=data/Pyramid,action=draw]
\axesIIID[linecolor=blue](4,4,2)(5,5,6)
\end{pspicture}
\end{center}
\subsection{Slicing an \Index{octahedron} with a plane parallel to one of its faces}
\subsubsection{The view inside}
Recall that there are options \verb+rm=0,hollow+ that allow us,
on the one hand, to remove a face \verb+rm=0+ and, on the other,
to look inside \Lkeyword{hollow}.
In the following example, we shall start by generating the
required objects without drawing them (\texttt{\Lkeyword{action}=\Lkeyval{none}}).
We construct the octahedron, giving the center of the face with
index $1$ the name $G$, then define the point $H$ which satisfies
$\overrightarrow{OH} = 0.8\,\overrightarrow{OG}$. After that we
define $P$ to be the plane through $H$ parallel to the face of the
octahedron with index $1$. Finally, we slice the octahedron using
the plane $P$.
\begin{LTXexample}[width=7.5cm]
\begin{pspicture}(-3.5,-3)(4.5,5)
\psset{viewpoint=100 5 10 rtp2xyz,Decran=80,
lightsrc=viewpoint,solidmemory,action=none}
\psSolid[object=octahedron,
a=4,name=my_octahedron,]
\psSolid[object=point,
definition=solidcentreface,
args=my_octahedron 1,
name=G,]
\psSolid[object=point,
definition=mulv3d,
args=G .8,
name=H,]
\psSolid[object=plan,
definition=solidface,
args=my_octahedron 1,
base=-4 4 -4 4,
name=P,](H,,)
\psSolid[object=load,
load=my_octahedron,
plansepare=P,
name=part]
\psSolid[object=load,load=part1,
rm=0,hollow,action=draw**,
fillcolor={[rgb]{0.7 1 0.7}},
incolor={[rgb]{1 1 0.7}},]
\psSolid[object=plan,args=P,
action=draw,showBase]
\psSolid[object=line,
args=0 0 0 H,
linestyle=dashed,]
\psProjection[object=point,plan=P,args=0 0,
fontsize=20,pos=cl,text=H,phi=90,]
\axesIIID[linecolor=blue,linewidth=0.4pt](0,0,0)(4,4,4)
\end{pspicture}
\end{LTXexample}
\subsubsection{Regarding the solid as filled}
The option \verb+fcol=0 (YellowOrange)+ allows us to colour the
face with index 0.
\begin{LTXexample}[width=7.5cm]
\begin{pspicture}(-3.5,-3)(4.5,5)
\psset{viewpoint=100 5 10 rtp2xyz,Decran=80,
lightsrc=viewpoint,solidmemory,action=none}
\psSolid[object=octahedron,
a=4,name=my_octahedron,]
\psSolid[object=point,
definition=solidcentreface,
args=my_octahedron 1,
name=G,]
\psSolid[object=point,
definition=mulv3d,
args=G .8,
name=H,]
\psSolid[object=plan,
definition=solidface,
args=my_octahedron 1,
base=-4 4 -4 4,
name=P,](H,,)
\psSolid[object=load,
load=my_octahedron,
plansepare=P,
name=part]
\psSolid[object=load,
load=part1,
fcol=0 (YellowOrange),
action=draw**,
fillcolor={[rgb]{0.7 1 0.7}},]
\psSolid[object=plan,args=P,
action=draw,showBase]
\psSolid[object=line,
args=0 0 0 H,
linestyle=dashed,]
\psProjection[object=point,plan=P,args=0 0,
fontsize=20,pos=cl,text=H,phi=90,]
\axesIIID[linecolor=blue,linewidth=0.4pt](0,0,0)(4,4,4)
\end{pspicture}
\end{LTXexample}
\subsubsection{The two parts of a sliced solid}
You will recall that the direction of the normal of the slicing
plane determines the numbering of the two parts: 0 if above the
normal, 1 if below. For both parts, the sliced face carries the
number 0. If there are several sliced faces, as in the case of the
torus, they are numbered 0, 1 etc.
Using two steps, we memorise both parts of the sliced solid:
\begin{verbatim}
\psSolid[object=load,
load=my_octahedron,
plansepare=P,
name=part]
\end{verbatim}
Then we position and render each part:
\begin{verbatim}
\psSolid[object=load,
fcol=0 (YellowOrange),
fillcolor={[rgb]{0.7 1 0.7}},
load=part1]
\psSolid[object=load,
fillcolor={[rgb]{0.7 1 0.7}},
load=part0](H 2 mulv3d,,)
\composeSolid
\end{verbatim}
\begin{LTXexample}[width=7.5cm]
\begin{pspicture}(-3.5,-3)(4.5,5)
\psset{viewpoint=100 5 20 rtp2xyz,Decran=150,
lightsrc=viewpoint,solidmemory,action=none}
\psSolid[object=octahedron,
a=2,name=my_octahedron,]
\psSolid[object=point,
definition=solidcentreface,
args=my_octahedron 1,
name=G,]
\psSolid[object=point,
definition=mulv3d,
args=G .7,
name=H,]
\psSolid[object=plan,
definition=solidface,
args=my_octahedron 1,
base=-4 4 -4 4,
name=P,](H,,)
\psSolid[object=load,
load=my_octahedron,
plansepare=P,
name=part]
\psset{action=draw**}
\psSolid[object=load,
load=part1,
fcol=0 (YellowOrange),
fillcolor={[rgb]{0.7 1 0.7}},]
\psSolid[object=load,
fillcolor={[rgb]{0.7 1 0.7}},
load=part0](H 2 mulv3d,,)
\composeSolid
\end{pspicture}
\end{LTXexample}
\subsection{Slices of a cube}
\subsubsection{Highlighting the edges of the cut}
\begin{LTXexample}[width=8cm]
\psset{viewpoint=100 30 20 rtp2xyz,Decran=150}
\begin{pspicture}(-4,-3)(4,5)
\psset{solidmemory}
\psSolid[object=plan,definition=normalpoint,
args={1 1 1 [1 1 1]},action=none,name=P]
\psSolid[object=cube,a=2,action=draw,
intersectiontype=0,
intersectionplan=P,
intersectionlinewidth=2,
intersectioncolor=(rouge),
](1,1,1)
\psProjection[object=point,
args=0 0,fontsize=10,pos=dc,
text=H,phi=-30,plan=P,
]
\psSolid[object=line,
linestyle=dashed,
args=0 0 0 1 1 1]
\psSolid[object=vecteur,
linecolor=red,
args=1 1 1 .7 mulv3d](1,1,1)
\axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
\end{pspicture}
\end{LTXexample}
\subsubsection{Showing the sliced cube with its hexagonal cut face}
\begin{LTXexample}[width=8cm]
\psset{viewpoint=100 30 20 rtp2xyz,Decran=150}
\begin{pspicture}(-4,-3)(4,5)
\psset{solidmemory}
\psSolid[object=plan,action=none,definition=normalpoint,
args={1 1 1 [1 1 1]},name=P]
\psSolid[object=cube,a=2,
plansepare=P,
action=none,
name=parts_cube,
](1,1,1)
\psSolid[object=load,
load=parts_cube1,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
]
\psProjection[object=point,
args=0 0,fontsize=10,pos=dc,
text=H,phi=-30,plan=P,
]
\psSolid[object=vecteur,
linecolor=red,
args=1 1 1 .7 mulv3d](1,1,1)
\axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
\end{pspicture}
\end{LTXexample}
\subsubsection{The sliced cube in various positions}
Where we use the option that allows us to memorise a solid, in
order to put the truncated cube, after undergoing various
transformations, down on its cut face.
\begin{verbatim}
\psset{solidmemory}
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
name=C1,
action=none,
filename=data/cubeHexagone]
\end{verbatim}
\begin{center}
\begin{pspicture}(-3,-3)(3,3)
\psframe(-3,-2)(3,3)
\psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-2 3 -2 3,
linecolor=gray]
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
filename=data/cubeHexagone]
\axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
\end{pspicture}
\hfil
\begin{pspicture}(-2,-3)(4,3)
\psframe(-2,-2)(4,3)
\psset{viewpoint=100 -30 20 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-2 3 -2 3,
linecolor=gray]
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
filename=data/cubeHexagone]
\axesIIID[linecolor=blue](2,2,2)(2.5,2.5,2.5)
\end{pspicture}
\begin{pspicture}(-3,-2)(3,3)
\psframe(-3,-2)(3,3)
\psset{viewpoint=100 225 20 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-2 3 -2 3,
linecolor=gray]
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
filename=data/cubeHexagone]
\axesIIID[linecolor=blue](0,0,0)(2.5,2.5,2.5)
\end{pspicture}
\hfil
\begin{pspicture}(-3,-2)(3,3)
\psframe(-3,-2)(3,3)
\psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-2 3 -2 3,
linecolor=gray]
\psset{solidmemory}
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
name=C1,
action=none,
filename=data/cubeHexagone]
\codejps{C1 {-1.5 -1.5 0 translatepoint3d} solidtransform
drawsolid**
}
\axesIIID[linecolor=blue](0,0,0)(2.5,2.5,2.5)
\end{pspicture}
\begin{pspicture}(-3,-2)(3,4)
\psframe(-3,-2)(3,3)
\psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-2 3 -2 3,
linecolor=gray]
\psset{solidmemory}
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
name=C1,
action=none,
filename=data/cubeHexagone]
\codejps{C1 {-1.5 -1.5 0 translatepoint3d} solidtransform
{0 0 45 rotateOpoint3d} solidtransform
drawsolid**
}
%\composeSolid
\axesIIID[linecolor=blue](0,0,0)(2.5,2.5,2.5)
\end{pspicture}
\hfil
\begin{pspicture}(-3,-2)(3,4)
\psframe(-3,-2)(3,3)
\psset{viewpoint=100 30 20 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\psSolid[object=grille,
base=-2 3 -2 3,
linecolor=gray]
\psset{solidmemory}
\psSolid[object=datfile,
fcol=0 (Dandelion),
fillcolor={[rgb]{0.7 1 0.7}},
name=C1,
action=none,
filename=data/cubeHexagone]
\codejps{C1 {-1.5 -1.5 0 translatepoint3d} solidtransform
{0 0 45 rotateOpoint3d} solidtransform
{-35.2644 -90 add 0 0 rotateOpoint3d} solidtransform
drawsolid*
}
\axesIIID[linecolor=blue](1,2.5,0.5)(2.5,3,2.5)
\end{pspicture}
\end{center}
\subsection{Multiple sections}
\subsubsection{Slicing a sphere with PStricks}
\begin{LTXexample}[width=8cm]
\begin{pspicture}(-4,-4)(4,4)
\psset{viewpoint=100 20 20 rtp2xyz,Decran=75}
\psset{solidmemory,lightsrc=viewpoint}
\codejps{
/coeff 0.75 def /rO 4 def /OH coeff rO mul neg def}%
\psSolid[object=sphere,
r=rO,ngrid=9 18,
plansepare={[1 0 0 OH]},
name=part,
action=none]
\psSolid[object=load,
load=part1,plansepare={[-1 0 0 OH]},action=none,name=part]
\psSolid[object=load,
load=part1,plansepare={[0 1 0 OH]},action=none,name=part]
\psSolid[object=load,
load=part1,plansepare={[0 -1 0 OH]},action=none,name=part]
\psSolid[object=load,
load=part1,plansepare={[0 0 1 OH]},action=none,name=part]
\psSolid[object=load,
load=part1,plansepare={[0 0 -1 OH]},action=none,name=part]
\psSolid[object=load,hue=.1 .8 0.5 1,
load=part1](0,0,0)
\composeSolid
\end{pspicture}
\end{LTXexample}
\subsubsection{Multiple sections of a \Index{parallelepiped}}
Multiple sections are better carried out inside a PostScript loop,
within \verb+\codejps+; it's easier and quicker!
In this example, the original solid is a parallelepiped.
Truncations of the vertices and chamfering of the edges are
effected by means of slicing planes, starting off with the
vertices and finishing with the edges.
\begin{LTXexample}[width=7cm]
\begin{pspicture}(-3.5,-4)(3.5,4)
\psset{viewpoint=100 -20 10 rtp2xyz,Decran=100}
%\lightsource
\psset{lightsrc=viewpoint}
\codejps{
4 4 6 newparallelepiped
45 90 360 {
/iAngle exch def
/n_x iAngle cos 35.2644 cos mul def
/n_y iAngle sin 35.2644 cos mul def
/n_z 35.2644 sin def
/distance 2 3 add 3 sqrt div neg def
[ n_x n_y n_z distance]
solidplansepare
} for
45 90 360 {
/iAngle exch def
/n_x iAngle cos 35.2644 cos mul def
/n_y iAngle sin 35.2644 cos mul def
/n_z 35.2644 sin neg def
/distance 2 3 add 3 sqrt div neg def
[ n_x n_y n_z distance]
solidplansepare
} for
45 90 360 {
/iAngle exch def
% plan : ax+by+cz-d=0
[ iAngle cos % a
iAngle sin % b
0 % c
-2.5 % -d
] solidplansepare
} for
dup [.5 .2] solidputhuecolors
solidlightOn
drawsolid*}
\end{pspicture}
\end{LTXexample}
\subsection{Sections of a torus}
%\begin{pspicture}(-6,-4)(6,4)
%\psSolid[r1=3,r0=1.5,
% object=tore,
%% ngrid=18 60,
% filename=data/tore1860,action=writesolid]
%\end{pspicture}
\begin{center}
\begin{pspicture}(-6,-4)(6,4)
\pstVerb{/Ampl 3 2 sqrt mul def}%
\psset{viewpoint=50 -10 30 rtp2xyz,Decran=50}
%\lightsource
\psset{lightsrc=viewpoint}
\psframe[fillstyle=solid,fillcolor=gray!50](-6,-4)(6,4)
%\psset{solidmemory}
%\psSolid[object=datfile,filename=data/tore1860,
% plansepare={[1 0 0 -1.5]},
% name=lemniscate,
% action=none](0,0,0)
%\psSolid[object=load,
% load=lemniscate1,
% filename=data/tore1860lemniscate,action=writesolid](0,0,0)
\defFunction[algebraic]{lemniscate}(t){1.5}{Ampl*sin(t)/(1+cos(t)*cos(t))}{Ampl*sin(t)*cos(t)/(1+cos(t)*cos(t))}
\psSolid[object=datfile,filename=data/tore1860lemniscate,
fcol=0 (0.5 0.72 0.5 setrgbcolor)
1 (0.5 0.72 0.5 setrgbcolor),
fillcolor=green!30,
intersectiontype=0,
intersectionplan={
[1 0 0 -1.5]},
intersectioncolor=(rouge),
intersectionlinewidth=2.2]
\psSolid[object=courbe,r=0,linewidth=2pt,
range=0 6.28,
linecolor=red,
function=lemniscate]%
\psSolid[object=plan,action=draw,
definition=equation,
args={[1 0 0 -1.5] 90},
base=-5 5 -2 2,planmarks,showBase]
%\composeSolid
\end{pspicture}
\end{center}
\begin{center}
\begin{pspicture}(-6,-4)(6,4)
\pstVerb{/Ampl 3 2 sqrt mul def}%
\psset{viewpoint=50 -10 30 rtp2xyz,Decran=50}
%\lightsource
\psset{lightsrc=viewpoint}
\psframe[fillstyle=solid,fillcolor=gray!50](-6,-4)(6,4)
\defFunction[algebraic]{lemniscate}(t){1.5}{Ampl*sin(t)/(1+cos(t)*cos(t))}{Ampl*sin(t)*cos(t)/(1+cos(t)*cos(t))}
\psSolid[object=datfile,filename=data/tore1860lemniscate,
hollow,
rm=0 1,
fillcolor=green!30,incolor=yellow!50]
\psSolid[object=courbe,r=0,linewidth=2pt,
range=0 6.28,
linecolor=red,
function=lemniscate]%
\psSolid[object=plan,action=draw,
definition=equation,
args={[1 0 0 -1.5] 90},
base=-5 5 -2 2,planmarks,showBase]
\end{pspicture}
\end{center}
\subsection{Some more examples}
\begin{enumerate}
\item
You will find a \textit{jps} coded version of this document
within the \verb+\codejps+ command in the following document:
\centerline{\url{
http://melusine.eu.org/syracuse/mluque/solides3d2007/sections}}
\item A lesson about conic sections on:
\centerline{\url{
http://melusine.eu.org/syracuse/mluque/solides3d2007/sections/sections-cone}}
\item A lesson about cylindrical sections on:
\centerline{\url{
http://melusine.eu.org/syracuse/mluque/solides3d2007/sections/section-cylindre}}
\item A lesson about sections of a torus on:
\centerline{\url{
http://melusine.eu.org/syracuse/mluque/solides3d2007/sections/section-tore}}
\end{enumerate}
\endinput
