% Copyright 2006 by Till Tantau
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Free Documentation License.
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% See the file doc/generic/pgf/licenses/LICENSE for more details.


\section{Specifying Coordinates}

\label{section-points}

\subsection{Overview}

Most \pgfname\ commands expect you to provide the coordinates of a
\emph{point} (also called \emph{coordinate}) inside your
picture. Points are always ``local'' to your picture, that is, they
never refer to an absolute position on the page, but to a position
inside the current |{pgfpicture}| environment. To specify a coordinate
you can use commands that start with |\pgfpoint|.


\subsection{Basic Coordinate Commands}

The following commands are the most basic  for specifying a
coordinate.

\begin{command}{\pgfpoint\marg{x coordinate}\marg{y coordinate}}
  Yields a point location. The coordinates are given as \TeX\
  dimensions.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathcircle{\pgfpoint{1cm}{1cm}} {2pt}
  \pgfpathcircle{\pgfpoint{2cm}{5pt}} {2pt}
  \pgfpathcircle{\pgfpoint{0pt}{.5in}}{2pt}
  \pgfusepath{fill}
\end{tikzpicture}   
\end{codeexample}
\end{command}

\begin{command}{\pgfpointorigin}
  Yields the origin. Same as |\pgfpoint{0pt}{0pt}|.
\end{command}

\begin{command}{\pgfpointpolar\marg{degree}{\ttfamily\char`\{}\meta{radius}\opt{|/|\meta{y-radius}}{\ttfamily\char`\}}}
  Yields a point location given in polar coordinates. You can specify
  the angle only in degrees, radians are not supported, currently.

  If the optional \meta{y-radius} is given, the polar coordinate is
  actually a coordinate on an ellipse whose $x$-radius is given by
  \meta{radius} and whose $y$-radius is given by \meta{y-radius}.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);

  \foreach \angle in {0,10,...,90}
    {\pgfpathcircle{\pgfpointpolar{\angle}{1cm}}{2pt}}
  \pgfusepath{fill}
\end{tikzpicture}
\end{codeexample}
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);

  \foreach \angle in {0,10,...,90}
    {\pgfpathcircle{\pgfpointpolar{\angle}{1cm/2cm}}{2pt}}
  \pgfusepath{fill}
\end{tikzpicture}   
\end{codeexample}
\end{command}



\subsection{Coordinates in the XY-Coordinate System}

Coordinates can also be specified as multiples of an $x$-vector and a
$y$-vector. Normally, the $x$-vector points one centimeter in the
$x$-direction and the $y$-vector points one centimeter in the
$y$-direction, but using the commands |\pgfsetxvec| and
|\pgfsetyvec| they can be changed. Note that the $x$- and
$y$-vector do not necessarily point ``horizontally'' and
``vertically.''

\begin{command}{\pgfpointxy\marg{$s_x$}\marg{$s_y$}}
  Yields a point that is situated at $s_x$ times the
  $x$-vector plus $s_y$ times the $y$-vector.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathmoveto{\pgfpointxy{1}{0}}
  \pgfpathlineto{\pgfpointxy{2}{2}}
  \pgfusepath{stroke}
\end{tikzpicture}   
\end{codeexample}
\end{command}


\begin{command}{\pgfsetxvec\marg{point}}
  Sets that current $x$-vector for usage in the $xyz$-coordinate
  system. 
  \example
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  
  \pgfpathmoveto{\pgfpointxy{1}{0}}
  \pgfpathlineto{\pgfpointxy{2}{2}}
  \pgfusepath{stroke}

  \color{red}
  \pgfsetxvec{\pgfpoint{0.75cm}{0cm}}
  \pgfpathmoveto{\pgfpointxy{1}{0}}
  \pgfpathlineto{\pgfpointxy{2}{2}}
  \pgfusepath{stroke}
\end{tikzpicture}   
\end{codeexample}
\end{command}

\begin{command}{\pgfsetyvec\marg{point}}
  Works like |\pgfsetyvec|.
\end{command}



\begin{command}{\pgfpointpolarxy\marg{degree}{\ttfamily\char`\{}\meta{radius}\opt{|/|\meta{y-radius}}{\ttfamily\char`\}}}
  This command is similar to the |\pgfpointpolar| command, but the
  \meta{radius} is now a factor to be interpreted in the
  $xy$-coordinate system. This means that a degree of |0| is the same
  as the $x$-vector of the $xy$-coordinate  system times \meta{radius}
  and a degree of |90| is the $y$-vecotr times \meta{radius}. As for
  |\pgfpointpolar|, a \meta{radius} can also be a pair separated by a
  slash. In this case, the $x$- and $y$-vectors are multiplied by
  different factors.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);

  \begin{scope}[x={(1cm,-5mm)},y=1.5cm]
    \foreach \angle in {0,10,...,90}
      {\pgfpathcircle{\pgfpointpolarxy{\angle}{1}}{2pt}}
    \pgfusepath{fill}
  \end{scope}
\end{tikzpicture}
\end{codeexample}
\end{command}



\subsection{Three Dimensional Coordinates}

It is also possible to specify a point as a multiple of three vectors,
the $x$-, $y$-, and $z$-vector. This is useful for creating simple
three dimensional graphics.

\begin{command}{\pgfpointxyz\marg{$s_x$}\marg{$s_y$}\marg{$s_z$}}
  Yields a point that is situated at $s_x$ times the
  $x$-vector plus $s_y$ times the $y$-vector plus  $s_z$ times the
  $z$-vector.
\begin{codeexample}[]
\begin{pgfpicture}
  \pgfsetarrowsend{to}
  
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathlineto{\pgfpointxyz{0}{0}{1}}
  \pgfusepath{stroke}
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathlineto{\pgfpointxyz{0}{1}{0}}
  \pgfusepath{stroke}
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathlineto{\pgfpointxyz{1}{0}{0}}
  \pgfusepath{stroke}
\end{pgfpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgfsetzvec\marg{point}}
  Works like |\pgfsetzvec|.
\end{command}

Inside the $xyz$-coordinate system, you can also specify points
using spherical and cylindrical coordinates.


\begin{command}{\pgfpointcylindrical\marg{degree}\marg{radius}\marg{height}}
  This command yields the same as
\begin{verbatim}
\pgfpointadd{\pgfpointpolarxy{degree}{radius}}{\pgfpointxyz{0}{0}{height}}
\end{verbatim}
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [->] (0,0) -- (1,0,0) node [right] {$x$};
  \draw [->] (0,0) -- (0,1,0) node [above] {$y$};
  \draw [->] (0,0) -- (0,0,1) node [below left] {$z$};

  \pgfpathcircle{\pgfpointcylindrical{80}{1}{.5}}{2pt}
  \pgfusepath{fill}

  \draw[red] (0,0) -- (0,0,.5) -- +(80:1);
\end{tikzpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgfpointspherical\marg{longitude}\marg{latitude}\marg{radius}}
  This command yields a point ``on the surface of the earth''
  specified by the \meta{longitude} and the \marg{latitude}. The
  radius of the earth is given by \meta{radius}. The equator lies in
  the $xy$-plane.
\begin{codeexample}[]
\begin{tikzpicture}
  \pgfsetfillcolor{lightgray}

  \foreach \latitude in {-90,-75,...,30}
  {  
    \foreach \longitude in {0,20,...,360}
    {
      \pgfpathmoveto{\pgfpointspherical{\longitude}{\latitude}{1}}
      \pgfpathlineto{\pgfpointspherical{\longitude+20}{\latitude}{1}}
      \pgfpathlineto{\pgfpointspherical{\longitude+20}{\latitude+15}{1}}
      \pgfpathlineto{\pgfpointspherical{\longitude}{\latitude+15}{1}}
      \pgfpathclose
    }
    \pgfusepath{fill,stroke}
  }
\end{tikzpicture}
\end{codeexample}
\end{command}



\subsection{Building Coordinates From Other Coordinates}

Many commands allow you to construct a coordinate in terms of other
coordinates.


\subsubsection{Basic Manipulations of Coordinates}

\begin{command}{\pgfpointadd\marg{$v_1$}\marg{$v_2$}}
  Returns the sum vector $\meta{$v_1$} + \meta{$v_2$}$.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathcircle{\pgfpointadd{\pgfpoint{1cm}{0cm}}{\pgfpoint{1cm}{1cm}}}{2pt}
  \pgfusepath{fill} 
\end{tikzpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgfpointscale\marg{factor}\marg{coordinate}}
  Returns the vector $\meta{factor}\meta{coordinate}$.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathcircle{\pgfpointscale{1.5}{\pgfpoint{1cm}{0cm}}}{2pt}
  \pgfusepath{fill} 
\end{tikzpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgfpointdiff\marg{start}\marg{end}}
  Returns the difference vector $\meta{end} - \meta{start}$.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathcircle{\pgfpointdiff{\pgfpoint{1cm}{0cm}}{\pgfpoint{1cm}{1cm}}}{2pt}
  \pgfusepath{fill} 
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgfpointnormalised\marg{point}}
  This command returns a normalized version of \meta{point}, that is,
  a vector of length 1pt pointing in the direction of \meta{point}. If
  \meta{point} is the $0$-vector or extremely short, a vector of
  length 1pt pointing upwards is returned.

  This command is \emph{not} implemented by calculating the length of
  the vector, but rather by calculating the angle of the vector and
  then using (something equivalent to) the |\pgfpointpolar|
  command. This ensures that the point will really have length 1pt,
  but it is not guaranteed that the vector will \emph{precisely} point
  in the direction of \meta{point} due to the fact that the polar
  tables are accurate only up to one degree. Normally, this is not a
  problem.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathcircle{\pgfpoint{2cm}{1cm}}{2pt}
  \pgfpathcircle{\pgfpointscale{20}
    {\pgfpointnormalised{\pgfpoint{2cm}{1cm}}}}{2pt}
  \pgfusepath{fill} 
\end{tikzpicture}
\end{codeexample}  
\end{command}


\subsubsection{Points Traveling along Lines and Curves}

\label{section-pointsattime}

The commands in this section allow you to specify points on a line or
a curve. Imaging a point ``traveling'' along a curve from some point
$p$ to another point $q$. At time $t=0$ the point is at $p$ and at
time $t=1$ it is at $q$ and at time, say, $t=1/2$ it is ``somewhere in
the middle.'' The exact location at time $t=1/2$ will not necessarily
be the ``halfway point,'' that is, the point whose distance on the
curve from $p$ and $q$ is equal. Rather, the exact location will
depend on the ``speed'' at which the point is traveling, which in
turn depends on the lengths of the support vectors in a complicated
manner. If you are interested in the details, please see a good book
on Bézier curves.



\begin{command}{\pgfpointlineattime\marg{time $t$}\marg{point $p$}\marg{point $q$}}
  Yields a point that is the $t$th fraction between $p$
  and~$q$, that is, $p + t(q-p)$. For $t=1/2$ this is the middle of
  $p$ and $q$.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathlineto{\pgfpoint{2cm}{2cm}}
  \pgfusepath{stroke}
  \foreach \t in {0,0.25,...,1.25}
    {\pgftext[at=
      \pgfpointlineattime{\t}{\pgfpointorigin}{\pgfpoint{2cm}{2cm}}]{\t}}
\end{tikzpicture}    
\end{codeexample}
\end{command}

\begin{command}{\pgfpointlineatdistance\marg{distance}\marg{start point}\marg{end point}}
  Yields a point that is located \meta{distance} many units removed
  from the start point in the direction of the end point. In other
  words, this is the point that results if we travel \meta{distance}
  steps from \meta{start point} towards \meta{end point}.
  \example
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathlineto{\pgfpoint{3cm}{2cm}}
  \pgfusepath{stroke}
  \foreach \d in {0pt,20pt,40pt,70pt}
    {\pgftext[at=
      \pgfpointlineatdistance{\d}{\pgfpointorigin}{\pgfpoint{3cm}{2cm}}]{\d}}
\end{tikzpicture}    
\end{codeexample}
\end{command}

\begin{command}{\pgfpointcurveattime\marg{time $t$}\marg{point
      $p$}\marg{point $s_1$}\marg{point $s_2$}\marg{point $q$}} 
  Yields a point that is on the Bézier curve from $p$ to $q$ with the
  support points $s_1$ and $s_2$. The time $t$ is used to determine
  the location, where $t=0$ yields $p$ and $t=1$ yields $q$.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathmoveto{\pgfpointorigin}
  \pgfpathcurveto
    {\pgfpoint{0cm}{2cm}}{\pgfpoint{0cm}{2cm}}{\pgfpoint{3cm}{2cm}}
  \pgfusepath{stroke}
  \foreach \t in {0,0.25,0.5,0.75,1}
    {\pgftext[at=\pgfpointcurveattime{\t}{\pgfpointorigin}
                                         {\pgfpoint{0cm}{2cm}}
                                         {\pgfpoint{0cm}{2cm}}
                                         {\pgfpoint{3cm}{2cm}}]{\t}}
\end{tikzpicture}    
\end{codeexample}
\end{command}

\subsubsection{Points on Borders of Objects}

The following commands are useful for specifying a point that lies on
the border of special shapes. They are used, for example, by the shape
mechanism to determine border points of shapes.

\begin{command}{\pgfpointborderrectangle\marg{direction point}\marg{corner}}
  This command returns a point that lies on the intersection of a line
  starting at the origin and going towards the point \meta{direction
    point} and a rectangle whose center is in the origin and whose
  upper right corner is at \meta{corner}.

  The \meta{direction point} should have length ``about 1pt,'' but it
  will be normalized automatically. Nevertheless, the ``nearer'' the
  length is to 1pt, the less rounding errors.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (2,1.5);
  \pgfpathrectanglecorners{\pgfpoint{-1cm}{-1.25cm}}{\pgfpoint{1cm}{1.25cm}}
  \pgfusepath{stroke}

  \pgfpathcircle{\pgfpoint{5pt}{5pt}}{2pt}
  \pgfpathcircle{\pgfpoint{-10pt}{5pt}}{2pt}
  \pgfusepath{fill}
  \color{red}
  \pgfpathcircle{\pgfpointborderrectangle
    {\pgfpoint{5pt}{5pt}}{\pgfpoint{1cm}{1.25cm}}}{2pt}
  \pgfpathcircle{\pgfpointborderrectangle
    {\pgfpoint{-10pt}{5pt}}{\pgfpoint{1cm}{1.25cm}}}{2pt}
  \pgfusepath{fill}
\end{tikzpicture}    
\end{codeexample}
\end{command}


\begin{command}{\pgfpointborderellipse\marg{direction point}\marg{corner}}
  This command works like the corresponding command for rectangles,
  only this time the \meta{corner} is the corner of the bounding
  rectangle of an ellipse.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (2,1.5);
  \pgfpathellipse{\pgfpointorigin}{\pgfpoint{1cm}{0cm}}{\pgfpoint{0cm}{1.25cm}}
  \pgfusepath{stroke}

  \pgfpathcircle{\pgfpoint{5pt}{5pt}}{2pt}
  \pgfpathcircle{\pgfpoint{-10pt}{5pt}}{2pt}
  \pgfusepath{fill}
  \color{red}
  \pgfpathcircle{\pgfpointborderellipse
    {\pgfpoint{5pt}{5pt}}{\pgfpoint{1cm}{1.25cm}}}{2pt}
  \pgfpathcircle{\pgfpointborderellipse
    {\pgfpoint{-10pt}{5pt}}{\pgfpoint{1cm}{1.25cm}}}{2pt}
  \pgfusepath{fill}
\end{tikzpicture}    
\end{codeexample}
\end{command}


\subsubsection{Points on the Intersection of Lines}


\begin{command}{\pgfpointintersectionoflines\marg{$p$}\marg{$q$}\marg{$s$}\marg{$t$}}
  This command returns the intersection of a line going through $p$
  and $q$ and a line going through $s$ and $t$. If the lines do not
  intersection, an arithmetic overflow will occur.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (2,2);
  \draw (.5,0) -- (2,2);
  \draw (1,2) -- (2,0);
  \pgfpathcircle{%
    \pgfpointintersectionoflines
      {\pgfpointxy{.5}{0}}{\pgfpointxy{2}{2}}
      {\pgfpointxy{1}{2}}{\pgfpointxy{2}{0}}}
    {2pt}
  \pgfusepath{stroke}
\end{tikzpicture}    
\end{codeexample}
\end{command}


\subsubsection{Points on the Intersection of Two Circles}


\begin{command}{\pgfpointintersectionofcircles\marg{$p_1$}\marg{$p_2$}\marg{$r_1$}\marg{$r_2$}\marg{solution}}
  This command returns the intersection of the two circles centered at
  $p_1$ and $p_2$ with radii $r_1$ and $r_2$. If \meta{solution} is
  |1|, the first intersection is returned, otherwise the second one is
  returned.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (2,2);
  \draw (0.5,0) circle (1);
  \draw (1.5,1) circle (.8);
  \pgfpathcircle{%
    \pgfpointintersectionofcircles
      {\pgfpointxy{.5}{0}}{\pgfpointxy{1.5}{1}}
      {1cm}{0.8cm}{1}}
    {2pt}
  \pgfusepath{stroke}
\end{tikzpicture}    
\end{codeexample}
\end{command}

\subsubsection{Points on the Intersection of Two Paths}


\begin{pgflibrary}{intersections}
  This library defines the below command and allows you to calculate
  the intersections of  two arbitrary paths. However, due to the low accuracy of
  \TeX, the paths should not be ``too complicated''.
  In particular, you should not try to intersect paths consisting 
  lots of very small segments such as plots or decorated paths.
\end{pgflibrary}
   
\begin{command}{\pgfintersectionofpaths\marg{path 1}\marg{path 2}}
  This command finds the intersection points on the paths 
  \meta{path 1} and \meta{path 2}. The number of intersection points
  (``solutions'') that are found will be stored, and each point 
  can be accessed afterward. The code for \meta{path 1} and 
  \meta{path 2} is executed within a \TeX{} group and so can contain
  transformations (which will be in addition to any existing
  transformations). The code should not use the path in any way, 
  unless the path is saved first and restored afterward.
  \pgfname{} will regard solutions as ``a bit
  special'', in that the points returned  will be ``absolute'' and 
  unaffected by any further transformations.

\begin{codeexample}[]
\begin{pgfpicture}
\pgfintersectionofpaths
{
  \pgfpathellipse{\pgfpointxy{0}{0}}{\pgfpointxy{1}{0}}{\pgfpointxy{0}{2}}
  \pgfgetpath\temppath
  \pgfusepath{stroke}
  \pgfsetpath\temppath
}
{
  \pgftransformrotate{-30}
  \pgfpathrectangle{\pgfpointorigin}{\pgfpointxy{2}{2}}
  \pgfgetpath\temppath
  \pgfusepath{stroke}
  \pgfsetpath\temppath
}
\foreach \s in {1,...,\pgfintersectionsolutions}
  {\pgfpathcircle{\pgfpointintersectionsolution{\s}}{2pt}}
\pgfusepath{stroke}
\end{pgfpicture}
\end{codeexample}

  \begin{command}{\pgfintersectionsolutions}
    After using the |\pgfintersectionofpaths| command, this \TeX-macro
    will indicate the number of solutions found.
  \end{command}
  
  \begin{command}{\pgfpointintersectionsolution\marg{number}}
    After using the |\pgfintersectionofpaths| command, this command
    will return the point for solution \meta{number} or the origin
    if this solution was not found. 
    By default, the intersections are simply returned in the order that 
		the intersection algorithm finds them. Unfortunately, this is not 
		necessarily a ``helpful'' ordering. However the following two
		commands can be used to order the solutions more helpfully.
  \end{command}

\let\ifpgfintersectionsortbyfirstpath=\relax
  \begin{command}{\pgfintersectionsortbyfirstpath}
    Using this command will mean the solutions will be sorted along
    \meta{path 1}. 
  \end{command}
  
\let\ifpgfintersectionsortbysecondpath=\relax 
  \begin{command}{\pgfintersectionsortbysecondpath}
    Using this command will mean the solutions will be sorted along
    \meta{path 2}.  
  \end{command}

\end{command}
 



\subsection{Extracting Coordinates}

There are two commands that can be used to ``extract'' the $x$- or
$y$-coordinate of a coordinate. 

\begin{command}{\pgfextractx\marg{dimension}\marg{point}}
  Sets the \TeX-\meta{dimension} to the $x$-coordinate of the point.

\begin{codeexample}[code only]
\newdimen\mydim
\pgfextractx{\mydim}{\pgfpoint{2cm}{4pt}}
%% \mydim is now 2cm
\end{codeexample}
\end{command}

\begin{command}{\pgfextracty\marg{dimension}\marg{point}}
  Like |\pgfextractx|, except for the $y$-coordinate.
\end{command}

\begin{command}{\pgfgetlastxy\marg{macro for $x$}\marg{macro for $y$}}
	Stores the most recently used $(x,y)$ coordinates into two macros.
\begin{codeexample}[]
\pgfpoint{2cm}{4cm}
\pgfgetlastxy{\macrox}{\macroy}
Macro $x$ is `\macrox' and macro $y$ is `\macroy'.
\end{codeexample}
	Since $(x,y)$ coordinates are usually assigned globally, it is safe to use this command after path operations.
\end{command}



\subsection{Internals of How Point Commands Work}

\label{section-internal-pointcmds}

As a normal user of \pgfname\ you do not need to read this section. It
is relevant only if you need to understand how the point commands work
internally. 

When a command like |\pgfpoint{1cm}{2pt}| is called, all that happens
is that the two \TeX-dimension variables |\pgf@x| and |\pgf@y| are set
to |1cm| and |2pt|, respectively. These variables belong to the set of
internal \pgfname\ registers, see section~\ref{section-internal-registers}
for details. A command like |\pgfpathmoveto| that
takes a coordinate as parameter will just execute this parameter and
then use the values of |\pgf@x| and |\pgf@y| as the coordinates to
which it will move the pen on the current path.

since commands like |\pgfpointnormalised| modify other variables
besides |\pgf@x| and |\pgf@y| during the computation of the final values of
|\pgf@x| and |\pgf@y|, it is a good idea to enclose a call of a
command like |\pgfpoint| in a \TeX-scope and then make the changes of
|\pgf@x| and |\pgf@y| global as in the following example:
\begin{codeexample}[code only]
...
{ % open scope
  \pgfpointnormalised{\pgfpoint{1cm}{1cm}}
  \global\pgf@x=\pgf@x % make the change of \pgf@x persist past the scope
  \global\pgf@y=\pgf@y % make the change of \pgf@y persist past the scope
}
% \pgf@x and \pgf@y are now set correctly, all other variables are
% unchanged 
\end{codeexample}

\makeatletter
Since this situation arises very often, the macro |\pgf@process| can
be used to perform the above code:
\begin{command}{\pgf@process\marg{code}}
  Executes the \meta{code} in a scope and then makes |\pgf@x| and
  |\pgf@y| global.
\end{command}

Note that this macro is used often internally. For this reason, it is
not a good idea to keep anything important in the variables |\pgf@x|
and |\pgf@y| since they will be overwritten and changed
frequently. Instead, intermediate values can ge stored in the
\TeX-dimensions |\pgf@xa|, |\pgf@xb|, |\pgf@xc| and their
|y|-counterparts |\pgf@ya|, |\pgf@yb|, |pgf@yc|. For example, here is
the code of the command |\pgfpointadd|:
\begin{codeexample}[code only]
\def\pgfpointadd#1#2{%
  \pgf@process{#1}%
  \pgf@xa=\pgf@x%
  \pgf@ya=\pgf@y%
  \pgf@process{#2}%
  \advance\pgf@x by\pgf@xa%
  \advance\pgf@y by\pgf@ya}
\end{codeexample}



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