% Copyright 2019 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.
%
% 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 and 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 |\pgfsetxvec|.
\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
    $y$-vector 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 |\pgfsetxvec|.
\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 \meta{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 normalised 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.
Imagine 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 away 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}{\pgfpointarcaxesattime\marg{time $t$}\marg{center}\marg{0-degree axis}\marg{90-degree axis}\marg{start angle}\\\marg{end angle}}
    Yields a point on the arc between \meta{start angle} and \meta{end angle}
    on an ellipse whose center is at \meta{center} and whose two principal axes
    are \meta{0-degree axis} and \meta{90-degree axis}. For $t=0$ the point at
    the \meta{start angle} is returned and for $t=1$ the point at the \meta{end
    angle}.
    %
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfpathmoveto{\pgfpoint{2cm}{1cm}}
  \pgfpatharcaxes{0}{60}{\pgfpoint{2cm}{0cm}}{\pgfpoint{0cm}{1cm}}
  \pgfusepath{stroke}
  \foreach \t in {0,0.25,0.5,0.75,1}
    {\pgftext[at=\pgfpointarcaxesattime{\t}{\pgfpoint{0cm}{1cm}}
       {\pgfpoint{2cm}{0cm}}{\pgfpoint{0cm}{1cm}}{0}{60}]{\t}}
\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 of 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}[preamble={\usetikzlibrary{intersections}}]
\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 be 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}


%%% Local Variables:
%%% mode: latex
%%% TeX-master: "pgfmanual"
%%% End: