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\section{The object \texttt{plan}}

\subsection{Presentation: type \texttt{plan\/} and type \texttt{solid} }

The  object
\Lkeyval{plan} is special in
\texttt{pst-solides3d}. However, all the objects presented until now have had a common structure:
 they are of type \verb+solid+: in other words, they are defined by a list of vertices, faces and colours.

For many applications, it is necessary to have some additional information for a \Index{plane}: an origin, an
orientation, a reference base etc.

To fulfill all these requirements, another
data structure of type \Lkeyval{plan} was created, which allows one to save all this necessary information. These manipulations of the plane will be controlled
by such an object.
Only when rendering takes place will an object of type \Lkeyval{plan} be converted to an object of type \verb+solid+ which conforms to the macro \Lcs{psSolid}.

An object of type \Lkeyval{plan} is used to describe an oriented affine plane.
For a complete definition of such an object,
 an origin
$I$, a basis $(\vec u, \vec v)$ for that plane, a scaling of the axis $(I, \vec u)$ and a scaling of the axis
$(I, \vec v)$ are needed.
In addition, we can specify the fineness of the grid---in other words, the number of faces---used to represent that portion of the affine plane
while transforming in an object of the type \verb+solid+.%I'm confused by this last phrase.

This type of object can be used to define planes of section; it is then necessary to define a plane for projection.%check if this keeps your sense

Its usage is quite easy to understand for users of PSTricks.
The only thing that you need to know is that, if we manipulate a
\texttt{\Lkeyword{object}=\Lkeyval{plan}} with the macro \Lcs{psSolid}, we manipulate two objects at the same time: one of type \Lkeyval{plan} and
the other of type \verb+solid+. When we select a backup
of that object (see chapter ``\textit{Advanced usage}'') with the name $monplan$ for example with the option \texttt{\Lkeyword{name}=monplan}, there are
in fact 2 backups that are effected.
The first, with the name \texttt{monplan}, is an object of  type \Lkeyval{plan}, and the second, with the name \texttt{monplan\_s}, is an object of  type \verb+solid+.


\subsection{Defining an oriented plane}

To generate such an object, one uses \texttt{\Lkeyword{object}=\Lkeyval{plan}} which comes with a few arguments:

\begin{itemize}

\item \Lkeyword{definition} which specifies the method to defining the plane.

\item \Lkeyword{args} which specifies the necessary arguments for the method chosen.

\item \texttt{\Lkeyword{base}=$xmin$ $xmax$ $ymin$ $ymax$} which specifies the dimensions of each axis.

\item \verb+[phi]+ (value $0$ by default) which specifies the angle of rotation (in degrees) of the plane around its normal.



\end{itemize}

\subsection{Special options}

The object \verb+plan+ comes with some special options for  viewing:
\begin{itemize}
\item \Lkeyword{planmarks} which shows axes and scaling (with ticks),
\item \Lkeyword{plangrid} which shows the grid,
\item \Lkeyword{showbase} which shows the basis vectors for the plane, and
\item \Lkeyword{showBase} (note the capital letters) which shows the basis vectors of the plane
and draws the associated normal vector.
\end{itemize}
These options apply regardless of the method of definition of the plane.

\begin{center}
\psset{unit=0.4}
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,fontsize=10}
\begin{pspicture*}(-5,-4)(6,4)
\psframe(-5,-4)(5,3)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2]
\end{pspicture*}
%%
\psset{unit=1}
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,fontsize=10}
\begin{pspicture*}(-5,-4)(6,4)
\psframe(-5,-4)(5,3)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2,
   planmarks]
\end{pspicture*}
%%
\psset{unit=1}
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,fontsize=10}
\begin{pspicture*}(-5,-4)(6,4)
\psframe(-5,-4)(5,3)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2,
   plangrid]
\end{pspicture*}
%%
\psset{unit=1}
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,fontsize=10}
\begin{pspicture*}(-5,-4)(6,4)
\psframe(-5,-4)(5,3)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2,
   showbase]
\end{pspicture*}
%%
\psset{unit=1}
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,fontsize=10}
\begin{pspicture*}(-5,-4)(6,4)
\psframe(-5,-4)(5,3)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2,
   showBase]
\end{pspicture*}
%%
\psset{unit=1}
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,fontsize=10}
\begin{pspicture*}(-5,-4)(6,4)
\psframe(-5,-4)(5,3)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2,
   plangrid,
   showBase,
   action=none
]
\end{pspicture*}
\end{center}

These options can be used, even if the plane is not drawn.

\subsection{Defining a plane with a cartesian equation}

The \textit{cartesian equation} of a plane is of the form
\[
   ax+by+cz+d=0
\]
The coefficients $a$, $b$, $c$ and $d$ determine an affine plane.

\subsubsection{Usage with default orientation and origin}

To define an affine plane, we can use
\texttt{\Lkeyword{definition}=\Lkeyval{equation}}, and \texttt{\Lkeyword{args}=\{[$a$ $b$ $c$
$d$]\}}. The orientation and origin of the affine plane must be given.

For example, the quadruple $(a, b, c, d) = (0, 0, 1, 0)$ determines
the plane with the equation $z=0$:

\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,
  fontsize=10,unit=0.65}
\begin{pspicture*}(-5,-4)(5,4)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   planmarks,
   base=-2.2 2.2 -3.2 3.2,
   showbase]
\axesIIID(0,0,0)(2.2,3.2,4)
\end{pspicture*}
\end{LTXexample}

The parameter \texttt{\Lkeyword{base}=$xmin$ $xmax$ $ymin$ $ymax$} specifies the extent along each axis.

\subsubsection{Specifying the origin}

The parameter \texttt{\Lkeyword{origine}=$x_0$ $y_0$ $z_0$} specifies
the origin of the affine plane.
If the chosen point $(x_0, y_0, z_0)$ doesn't fit the equation of the plane, it will be ignored.% The meaning of this is unclear to me.

For example, a plane with the equation $z=0$ for which $(1, 2, 0)$ has been chosen as a possible origin:%(finish the sentence---it does what?)


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,Decran=10,
  fontsize=10,unit=0.65cm}
\begin{pspicture*}(-4,-5.5)(6,4)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0]},
   fillcolor=Aquamarine,
   origine=1 2 0,
   base=-2.2 2.2 -3.2 3.2,
   planmarks]
\axesIIID(0,0,0)(2.2,3.2,4)
\end{pspicture*}
\end{LTXexample}


\subsubsection{Specifying the orientation}

If the chosen orientation is unsatisfactory,
we can specify an angle of rotation $\alpha $ (in degrees) around the normal of the plane with the syntax
\texttt{\Lkeyword{args}=\{[a b c d] $\alpha $\}}.


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,
  Decran=10,fontsize=10,unit=0.65cm}
\begin{pspicture*}(-5,-4)(5,4)
\psSolid[object=plan,
   definition=equation,
   args={[0 0 1 0] 90},
   fillcolor=Aquamarine,
   base=-2.2 2.2 -3.2 3.2,
   planmarks]
\axesIIID(0,0,0)(3.2,2.2,4)
\end{pspicture*}
\end{LTXexample}


\subsection{Defining a plane using a normal vector and a point}

It is also possible to define a plane by giving a point and a normal vector.
 In this case one uses the parameter \texttt{\Lkeyword{definition}=\Lkeyval{normalpoint}}.

If wanted, we can specify the orientation, but it can be omitted.

\subsubsection{First Method: orientation Unspecified}

We use \texttt{\Lkeyword{args}=\{$x_0$ $y_0$ $z_0$ [$a$ $b$ $c$]\}} where $(x_0,
y_0, z_0)$ is the origin of the affine plane, and $(a, b, c)$ is a vector normal to that plane.


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,
  Decran=10,fontsize=10,unit=0.65cm}
\begin{pspicture*}(-5,-4)(5,4)
\psSolid[object=plan,
   definition=normalpoint,
   args={0 0 0 [0 0 1]},
   fillcolor=Aquamarine,
   planmarks,
   base=-2.2 2.2 -3.2 3.2,
   showbase]
\axesIIID(0,0,0)(2.2,3.2,4)
\end{pspicture*}
\end{LTXexample}


\subsubsection{Second Method: Specifying an angle of rotation}

We use \texttt{\Lkeyword{args}=\{$x_0$ $y_0$ $z_0$ [$a$ $b$ $c$ $\alpha
$]\}} where $(x_0, y_0, z_0)$ is the origin of the affine plane, $(a, b,
c)$ a normal vector of that plane, and $\alpha $ the angle of rotation (in
degrees) around the normal vector of that plane.



\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,
  Decran=10,fontsize=10,unit=0.65}
\begin{pspicture*}(-5,-4)(5,4)
\psSolid[object=plan,
   definition=normalpoint,
   args={0 0 0 [0 0 1 45]},
   fillcolor=Aquamarine,
   planmarks,
   base=-2.2 2.2 -3.2 3.2,
   showbase]
\axesIIID(0,0,0)(2.2,3.2,4)
\end{pspicture*}
\end{LTXexample}


\subsubsection{Third Method: Specifying the first basis vector}

We use \texttt{\Lkeyword{args}=\{$x_0$ $y_0$ $z_0$ [$u_x$ $u_y$ $u_z$ $a$ $b$
$c$ ]\}} where $(x_0, y_0, z_0)$ is the origin of the affine plane,
$(a, b, c)$ a normal vector of that plane, and $(u_x, u_y, u_z)$ the first basis vector for that plane.


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,
  Decran=10,fontsize=10,unit=0.65cm}
\begin{pspicture*}(-5,-4)(5,4)
\psSolid[object=plan,
   definition=normalpoint,
   args={0 0 0 [1 1 0 0 0 1]},
   fillcolor=Aquamarine,
   planmarks,
   base=-2.2 2.2 -3.2 3.2,
   showbase,
]
\axesIIID(0,0,0)(2.2,3.2,4)
\end{pspicture*}
\end{LTXexample}


\subsubsection{Fourth Method: Specifying the first basis vector and an angle of rotation}

We use \texttt{\Lkeyword{args}=\{$x_0$ $y_0$ $z_0$ [$u_x$ $u_y$ $u_z$ $a$ $b$
$c$ $\alpha $]\}} where $(x_0, y_0, z_0)$ is the origin of the affine plane,
$(a, b, c)$ is a normal vector of that plane, $(u_x, u_y, u_z)$ is the first basis vector for that plane and $\alpha $ (in degrees) is a rotation around the axis of the normal vector.


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 60 rtp2xyz,
  Decran=10,fontsize=10,unit=0.65cm}
\begin{pspicture*}(-5,-4)(5,4)
\psSolid[object=plan,
   definition=normalpoint,
   args={0 0 0 [1 1 0 0 0 1 45]},
   fillcolor=Aquamarine,
   planmarks,
   base=-2.2 2.2 -3.2 3.2,
   showbase]
\axesIIID(0,0,0)(2.2,3.2,4)
\end{pspicture*}
\end{LTXexample}


\subsection{Defining a plane from a face of a solid}

We use \texttt{\texttt{\Lkeyword{definition}=\Lkeyval{solidface}}} with the arguments
\texttt{\texttt{\Lkeyword{args}=$name$ $i$}} where $name$ is the name of the designated solid and
$i$ is the index of the face. The origin is taken as the centre of the chosen face.

In the example below, the plane is defined through the face with the index 0 from the cube named $A$.


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 20 rtp2xyz,Decran=8}
\begin{pspicture}(-3.5,-2)(3,2.5)
\psset{solidmemory}
\psSolid[object=cube,a=2,fontsize=20,numfaces=all,name=A]
\psSolid[object=plan,
   definition=solidface,
   args=A 0,
   showBase]
\end{pspicture}
\end{LTXexample}

If the user specifies the coordinates $(x, y, z)$ within the macro
\verb+\psSolid[...](+$x,y,z$\verb+)+, a plane is generated parallel to the face with  index $i$ of the solid $name$, and translated to the point $(x, y, z)$ which now is taken as the origin.


\begin{LTXexample}[width=6.5cm]
\psset{viewpoint=10 18 20 rtp2xyz,Decran=8}
\begin{pspicture}(-3.5,-1.5)(3,3)
\psset{solidmemory}
\psSolid[object=cube,a=2,fontsize=20,numfaces=all,name=A]
\psSolid[object=plan,
   definition=solidface,
   args=A 0,
   showBase](0,0,2)
\end{pspicture}
\end{LTXexample}

\endinput