% 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.
%
% See the file doc/generic/pgf/licenses/LICENSE for more details.

\section{Coordinate, Canvas, and Nonlinear Transformations}

\subsection{Overview}

\pgfname\ offers different ways of scaling, shifting, and rotating
(these operations are generally known as \emph{transformations})
graphics: You can apply \emph{coordinate transformations} to all
coordinates, you can apply \emph{canvas transformations} to the
canvas on which you draw, and you can apply \emph{additional nonlinear
  transformations.} (The names ``coordinate'' and ``canvas''
transformations are not standard, I introduce them only for the
purposes of this manual.) 

The differences are the following:

\begin{itemize}
\item
  As the name ``coordinate transformation'' suggests, coordinate
  transformations apply only to coordinates. For example, when you
  specify a coordinate like |\pgfpoint{1cm}{2cm}| and you wish to
  ``use'' this coordinate---for example as an argument to a
  |\pgfpathmoveto| command---then the coordinate transformation matrix
  is applied to the coordinate, resulting in a new
  coordinate. Continuing the example, if the current coordinate
  transformation is ``scale by a factor of two,'' the coordinate
  |\pgfpoint{1cm}{2cm}| actually designates the point
  $(2\mathrm{cm},4\mathrm{cm})$. 

  Note that coordinate transformations apply \emph{only} to
  coordinates. They do not apply to, say, line width or shadings or
  text.
\item
  The effect of a ``canvas transformation'' like ``scale by a factor
  of two'' can be imagined as follows: You first draw your picture on
  a ``rubber canvas'' normally. Then, once you are done, the whole
  canvas is transformed, in this case stretched by a factor of
  two. In the resulting image \emph{everything} will be larger: Text,
  lines, coordinates, and shadings.
\item
  Nonlinear transformations are a special form of coordinate
  transformations that are, as the name suggests, not linear. The
  support for nonlinear transformations is quite different from the
  support for linear coordinate transformations, the main reason being
  speed: While linear coordinate transformations can be applied very
  quickly (\pgfname\ does so almost constantly), nonlinear
  transformations are much harder to apply and also to use. For this
  reason, nonlinear transformations are implemented in a special
  module |nonlineartransformations| that has to be loaded
  explicitly. By default, they are not available.
\end{itemize}

In many cases, it is preferable that you use coordinate
transformations and not canvas transformations. When canvas
transformations are used, \pgfname\ looses track of the coordinates of
nodes and shapes. Also, canvas transformations often cause undesirable
effects like changing text size. For these reasons, \pgfname\ makes it
easy to setup the coordinate transformation, but a bit harder to
change the canvas transformation. Because of the speed penalties
caused by nonlinear transformations, they are even harder to set up.


\subsection{Coordinate Transformations}
\label{section-linear-coordinate-transformations}

\subsubsection{How PGF Keeps Track of the Coordinate Transformation
  Matrix}
\label{section-transform-cm}

\pgfname\ has an internal coordinate transformation matrix. This
matrix is applied to coordinates ``in certain situations.'' This means
that the matrix is not always applied to every coordinate ``no matter
what.'' Rather, \pgfname\ tries to be reasonably smart at when and how
this matrix should be applied. The most prominent examples are the
path construction commands, which apply the coordinate transformation
matrix to their inputs.

The coordinate transformation matrix consists of four numbers $a$,
$b$, $c$, and $d$, and two dimensions $s$ and $t$. When the coordinate
transformation matrix is applied to a coordinate $(x,y)$, the new
coordinate $(ax+by+s,cx+dy+t)$ results. For more details on how
transformation matrices work in general, please see, for example, the
\textsc{pdf} or PostScript reference or a textbook on computer
graphics.

The coordinate transformation matrix is equal to the identity matrix
at the beginning. More precisely, $a=1$, $b=0$, $c=0$, $d=1$,
$s=0\mathrm{pt}$, and $t=0\mathrm{pt}$.

The different coordinate transformation commands will modify the
matrix by concatenating it with another transformation matrix. This
way the effect of applying several transformation commands will
\emph{accumulate}.

The coordinate transformation matrix is local to the current \TeX\
group (unlike the canvas transformation matrix, which is local to the
current |{pgfscope}|). Thus, the effect of adding a coordinate
transformation to the coordinate transformation matrix will last only
till the end of the current \TeX\ group.




\subsubsection{Commands for Relative Coordinate Transformations}

The following commands add a basic coordinate transformation to the
current coordinate transformation matrix. For all commands, the
transformation is applied \emph{in addition} to any previous
coordinate transformations.

\begin{command}{\pgftransformshift\marg{point}}
  Shifts coordinates by \meta{point}.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformshift{\pgfpoint{1cm}{1cm}}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgftransformxshift\marg{dimensions}}
  Shifts coordinates by \meta{dimension} along the $x$-axis.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformxshift{.5cm}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgftransformyshift\marg{dimensions}}
  Like |\pgftransformxshift|, only for the $y$-axis.
\end{command}

\begin{command}{\pgftransformscale\marg{factor}}
  Scales coordinates by \meta{factor}.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformscale{.75}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

\begin{command}{\pgftransformxscale\marg{factor}}
  Scales coordinates by \meta{factor} in the $x$-direction.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformxscale{.75}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgftransformyscale\marg{factor}}
  Like |\pgftransformxscale|, only for the $y$-axis.
\end{command}


\begin{command}{\pgftransformxslant\marg{factor}}
  Slants coordinates by \meta{factor} in the $x$-direction. Here, a
  factor of |1| means $45^\circ$.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformxslant{.5}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgftransformyslant\marg{factor}}
  Slants coordinates by \meta{factor} in the $y$-direction.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformyslant{-1}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

  

\begin{command}{\pgftransformrotate\marg{angles}}
  Rotates coordinates counterclockwise by \meta{angles} given in degrees.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformrotate{30}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

  

\begin{command}{\pgftransformtriangle\marg{a}\marg{b}\marg{c}}
  This command transforms the coordinate system in such a way that the
  triangle given by the points \meta{a}, \meta{b} and \meta{c} lies at
  the coordinates $(0,0)$, $(1\mathrm{pt},0\mathrm{pt})$ and
  $(0\mathrm{pt},1\mathrm{pt})$. 
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgftransformtriangle
  {\pgfpoint{1cm}{0cm}}
  {\pgfpoint{0cm}{2cm}}
  {\pgfpoint{3cm}{1cm}}
  
  \draw (0,0) -- (1pt,0pt) -- (0pt,1pt) -- cycle;
\end{tikzpicture}
\end{codeexample}
\end{command}

  
\begin{command}{\pgftransformcm\marg{a}\marg{b}\marg{c}\marg{d}\marg{point}}
  Applies the transformation matrix given by $a$, $b$, $c$, and $d$
  and the shift \meta{point} to coordinates (in addition to any
  previous transformations already in force).
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformcm{1}{1}{0}{1}{\pgfpoint{.25cm}{.25cm}}
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

  
\begin{command}{\pgftransformarrow\marg{start}\marg{end}}
  Shifts coordinates to the end of the line going from \meta{start} 
  to \meta{end} with the correct rotation. 
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (3,1);
  \pgftransformarrow{\pgfpointorigin}{\pgfpoint{3cm}{1cm}}
  \pgftext{tip}
\end{tikzpicture}
\end{codeexample}
\end{command}

  
\begin{command}{\pgftransformlineattime\marg{time}\marg{start}\marg{end}}
  Shifts coordinates by a specific point on a line at a specific
  time. The point by which the coordinate is shifted is calculated by
  calling |\pgfpointlineattime|, see
  Section~\ref{section-pointsattime}.

  In addition to shifting the coordinate, a rotation \emph{may} also
  be applied. Whether this is the case depends on whether the \TeX\ if
  |\ifpgfslopedattime| is set to true or not.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1);
  \pgftransformlineattime{.25}{\pgfpointorigin}{\pgfpoint{2cm}{1cm}}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) -- (2,1);
  \pgfslopedattimetrue
  \pgftransformlineattime{.25}{\pgfpointorigin}{\pgfpoint{2cm}{1cm}}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
  If |\ifpgfslopedattime| is true, another \TeX\ |\if| is important:
  |\ifpgfallowupsidedowattime|. If this is false, \pgfname\ will
  ensure that the rotation is done in such a way that text is never
  ``upside down.''

  There is another \TeX\ |\if| that influences this command. If you set
  |\ifpgfresetnontranslationattime| to true, then, between
  shifting the coordinate and (possibly) rotating/sloping the
  coordinate, the command |\pgftransformresetnontranslations| is
  called. See the description of this command for details.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgftransformscale{1.5}
  \draw      (0,0) -- (2,1);
  \pgfslopedattimetrue
  \pgfresetnontranslationattimefalse
  \pgftransformlineattime{.25}{\pgfpointorigin}{\pgfpoint{2cm}{1cm}}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgftransformscale{1.5}
  \draw      (0,0) -- (2,1);
  \pgfslopedattimetrue
  \pgfresetnontranslationattimetrue
  \pgftransformlineattime{.25}{\pgfpointorigin}{\pgfpoint{2cm}{1cm}}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgftransformcurveattime\marg{time}\marg{start}\marg{first
      support}\marg{second support}\marg{end}}
  Shifts coordinates by a specific point on a curve at a specific
  time, see  Section~\ref{section-pointsattime} once more.

  As for the line-at-time transformation command, |\ifpgfslopedattime|
  decides whether an additional rotation should be applied. Again, the
  value of |\ifpgfallowupsidedowattime| is also considered. 
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) .. controls (0,2) and (1,2) .. (2,1);
  \pgftransformcurveattime{.25}{\pgfpointorigin}
    {\pgfpoint{0cm}{2cm}}{\pgfpoint{1cm}{2cm}}{\pgfpoint{2cm}{1cm}}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \draw      (0,0) .. controls (0,2) and (1,2) .. (2,1);
  \pgfslopedattimetrue
  \pgftransformcurveattime{.25}{\pgfpointorigin}
    {\pgfpoint{0cm}{2cm}}{\pgfpoint{1cm}{2cm}}{\pgfpoint{2cm}{1cm}}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
  The value of |\ifpgfresetnontranslationsattime| is also taken into account.
\end{command}


\begin{command}{\pgftransformarcaxesattime\marg{time
      $t$}\marg{center}\marg{0-degree axis}\marg{90-degree
      axis}\marg{start angle}\marg{end angle}}
  Shifts coordinates by a specific point on an arc at a specific
  time, see  Section~\ref{section-pointsattime} once more.
  
  As for the previous commands, |\ifpgfslopedattime|
  decides whether an additional rotation should be applied and
  |\ifpgfallowupsidedowattime| is also considered.  
\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}
  \pgfslopedattimetrue
  \pgftransformarcaxesattime{.25}
    {\pgfpoint{0cm}{1cm}}
    {\pgfpoint{2cm}{0cm}}{\pgfpoint{0cm}{1cm}}
    {0}{60}
  \pgftext{Hi!}
\end{tikzpicture}
\end{codeexample}
  The value of |\ifpgfresetnontranslationsattime| is also taken into account.
\end{command}


{
  \let\ifpgfslopedattime=\relax
  \begin{textoken}{\ifpgfslopedattime}
    Decides whether the ``at time'' transformation commands also
    rotate coordinates or not.
  \end{textoken}
}
{
  \let\ifpgfallowupsidedowattime=\relax
  \begin{textoken}{\ifpgfallowupsidedowattime}
    Decides whether the ``at time'' transformation commands should
    allow the rotation be done in such a way that ``upside-down text''
    can result.
  \end{textoken}
}
{
  \let\ifpgfresetnontranslationsattime=\relax
  \begin{textoken}{\ifpgfresetnontranslationsattime}
    Decides whether the ``at time'' transformation commands should
    reset the non-translations between shifting and rotating.
  \end{textoken}
}


\subsubsection{Commands for Absolute Coordinate Transformations}

The coordinate transformation commands introduced up to now are always
applied in addition to any previous transformations. In contrast, the
commands presented in the following can be used to change the
transformation matrix ``in absolute terms.'' Note that this is, in general,
dangerous and will often produce unexpected effects. You should use
these commands only if you really know what you are doing.

\begin{command}{\pgftransformreset}
  Resets the coordinate transformation matrix to the identity
  matrix. Thus, once this command is given no transformations are
  applied till the end of the scope.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgftransformrotate{30}
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransformreset
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgftransformresetnontranslations}
  This command sets the $a$, $b$, $c$, and $d$ part of the coordinate
  transformation matrix to $a=1$, $b=0$, $c=0$, and $d=1$. However,
  the current shifting of the matrix is not modified.

  The effect of this command is that any rotation/scaling/slanting is
  undone in the current \TeX\ group, but the origin is not ``moved
  back.''

  This command is mostly useful directly before a |\pgftext| command
  to ensure that the text is not scaled or rotated.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgftransformscale{2}
  \pgftransformrotate{30}
  \pgftransformxshift{1cm}
  {\color{red}\pgftext{rotated}}
  \pgftransformresetnontranslations
  \pgftext{shifted only}
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgftransforminvert}
  Replaces the coordinate transformation matrix by a coordinate
  transformation matrix that ``exactly undoes the original
  transformation.'' For example, if the original transformation was
  ``scale by 2 and then shift right by 1cm'' the new one is ``shift
  left by 1cm and then scale by $1/2$.''

  This command will produce an error if the determinant of
  the matrix is too small, that is, if the matrix is near-singular.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgftransformrotate{30}
  \draw      (0,0) -- (2,1) -- (1,0);
  \pgftransforminvert
  \draw[red] (0,0) -- (2,1) -- (1,0);
\end{tikzpicture}
\end{codeexample}
\end{command}

\subsubsection{Saving and Restoring the Coordinate Transformation
  Matrix}

There are two commands for saving and restoring coordinate
transformation matrices.

\begin{command}{\pgfgettransform\marg{macro}}
  This command will (locally) define \meta{macro} to a representation
  of the current coordinate transformation matrix. This matrix can
  later on be reinstalled using |\pgfsettransform|.
\end{command}


\begin{command}{\pgfsettransform\marg{macro}}
  Reinstalls a coordinate transformation matrix that was previously
  saved using |\pgfgettransform|.
\end{command}

\begin{command}{\pgfgettransformentries\marg{macro for a}\marg{macro for b}\marg{macro for c}\marg{macro for d}\marg{macro for shift x}\marg{macro for shift y}}
	This command is similar to |\pgfgettransform| except that it stores the current coordinate transformation matrix in a set of six macros.

	The matrix can later on be reinstalled using |\pgfsettransformentries|. Furthermore, all these macros (or just a few of them) can be used as arguments for |\pgftransformcm|.
\end{command}

\begin{command}{\pgfsettransformentries\marg{a}\marg{b}\marg{c}\marg{d}\marg{shiftx}\marg{shifty}}
	Reinstalls a coordinate transformation matrix that was previously saved using the storage command |\pgfgettransformentries|. This command can also be used to replace any previously existing coordinate transformation matrix (it is thus equivalent to |\pgftransformreset| followed by |\pgftransformcm|).
\end{command}


\subsection{Canvas Transformations}

The canvas transformation matrix is not managed by \pgfname, but by
the output format like \pdf\ or PostScript. All that \pgfname\ does is
to call appropriate low-level |\pgfsys@| commands to change the canvas
transformation matrix.

Unlike coordinate transformations, canvas transformations apply to
``everything,'' including images, text, shadings, line thickness, and
so on. The idea is that a canvas transformation really stretches and
deforms the canvas after the graphic is finished.

Unlike coordinate transformations, canvas transformations are local to
the current |{pgfscope}|, not to the current \TeX\ group. This is due
to the fact that they are managed by the backend driver, not by \TeX\
or \pgfname.

Unlike the coordinate transformation matrix, it is not possible to
``reset'' the canvas transformation matrix. The only way to change it
is to concatenate it with another canvas transformation matrix or to
end the current |{pgfscope}|.

Unlike coordinate transformations, \pgfname\ does not ``keep track''
of canvas transformations. In particular, it will not be able to
correctly save the coordinates of shapes or nodes when a canvas
transformation is used.

\pgfname\ does not offer a whole set of special commands for modifying
the canvas transformation matrix. Instead, different commands allow
you to concatenate the canvas transformation matrix with a coordinate
transformation matrix (and there are numerous commands for specifying
a coordinate transformation, see the previous section).

\begin{command}{\pgflowlevelsynccm}
  This command concatenates the canvas transformation matrix with the
  current coordinate transformation matrix. Afterward, the coordinate
  transformation matrix is reset.

  The effect of this command is to ``synchronize'' the coordinate
  transformation matrix and the canvas transformation matrix. All
  transformations that were previously applied by the coordinate
  transformations matrix are now applied by the canvas transformation
  matrix.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfsetlinewidth{1pt}
  \pgftransformscale{5}
  \draw      (0,0) -- (0.4,.2);
  \pgftransformxshift{0.2cm}
  \pgflowlevelsynccm
  \draw[red] (0,0) -- (0.4,.2);
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgflowlevel\marg{transformation code}}
  This command concatenates the canvas transformation matrix with the
  coordinate transformation specified by \meta{transformation code}.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfsetlinewidth{1pt}
  \pgflowlevel{\pgftransformscale{5}}
  \draw      (0,0) -- (0.4,.2);
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgflowlevelobj\marg{transformation code}\marg{code}}
  This command creates a local |{pgfscope}|. Inside this scope,
  |\pgflowlevel| is first called with the argument
  \meta{transformation code}, then the \meta{code} is inserted. 

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfsetlinewidth{1pt}
  \pgflowlevelobj{\pgftransformscale{5}}    {\draw (0,0) -- (0.4,.2);}
  \pgflowlevelobj{\pgftransformxshift{-1cm}}{\draw (0,0) -- (0.4,.2);}
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{environment}{{pgflowlevelscope}\marg{transformation code}}
  This environment first surrounds the \meta{environment contents} by
  a |{pgfscope}|. Then it calls |\pgflowlevel| with the argument
  \meta{transformation code}.

\begin{codeexample}[]
\begin{tikzpicture}
  \draw[help lines] (0,0) grid (3,2);
  \pgfsetlinewidth{1pt}
  \begin{pgflowlevelscope}{\pgftransformscale{5}}
    \draw (0,0) -- (0.4,.2);
  \end{pgflowlevelscope}
  \begin{pgflowlevelscope}{\pgftransformxshift{-1cm}}
    \draw (0,0) -- (0.4,.2);
  \end{pgflowlevelscope}
\end{tikzpicture}
\end{codeexample}
\end{environment}


\begin{plainenvironment}{{pgflowlevelscope}\marg{transformation code}}
  Plain \TeX\ version of the environment.
\end{plainenvironment}

\begin{contextenvironment}{{pgflowlevelscope}\marg{transformation code}}
  Con\TeX t version of the environment.
\end{contextenvironment}


\subsection{Nonlinear Transformations}
\label{section-nonlinear-transformations}
In order to use nonlinear transformations, you first have to load the
following \pgfname\ module:

\begin{pgfmodule}{nonlineartransformations}
  Loads the necessary functionality for nonlinear transformations.
\end{pgfmodule}


\subsubsection{Introduction}

The difference between the coordinate transformations introduced in
Section~\ref{section-linear-coordinate-transformations} above to
nonlinear transformations is, of course, that the transformations can
be nonlinear. An example of a nonlinear transformation is the
transformation underlying polar coordinates: A polar coordinate
$(r,d)$ gets transformed to the canvas position $(d\cos r,d\sin r)$,
which is clearly not a linear transformation.

Nonlinear transformations work somewhat like the normal linear
coordinate transformations in the sense that they apply to coordinate
and thereby to the construction of paths, but not to things like text
or line width or shadings. (Indeed, it is not possible to apply
nonlinear transformations to, say, text.)

This means that there is a fundamental
difference between, on the one hand, calling a function like
|\pgfpointpolar| or specifying a coordinate as |(45:2)| in \tikzname\
and, on the other hand, installing the nonlinear transformation
``polar coordinates'' using the command |\pgftransformnonlinear|: In a
coordinate like |(45:2)| the user explicitly says ``please evaluate
this one  coordinate in polar coordinate and then continue in the
normal coordinate system with the result.'' Otherwise nothing changes
and a line between two points specified in this way is still a
straight line.

Things are quite different when we install a polar
\emph{transformation} using |\pgftransformnonlinear|. Now, even a
seemingly low-level Cartesian coordinate |\pgfqpoint{1pt}{1pt}| will
get transformed. Even more drastically, what is specified as a
straight line like
\begin{codeexample}[code only]
\draw (0,1) -- (1,1);
\end{codeexample}
can become curved since \emph{everything} gets transformed.


\subsubsection{Installing Nonlinear Transformation}

\makeatletter
\def\polartransformation{
  % \pgf@x will contain the radius
  % \pgf@y will contain the distance
  \pgfmathsincos@{\pgf@x}%
  % pgfmathresultx is now the cosine of radius and
  % pgfmathresulty is the sine of radius
  \pgf@x=\pgfmathresultx\pgf@y%
  \pgf@y=\pgfmathresulty\pgf@y%
}
\makeatother

\begin{command}{\pgftransformnonlinear\marg{transformation code}}
  This command adds the \meta{transformation code} to the list of non-linear
  transformations currently in force. Thus, similar to linear
  coordinate transformations, each additional call to this function
  adds another transformation to the current \TeX\ scope and the
  effect ends at the end of the current scope. In practice, however,
  you typically will not have more than one active nonlinear
  transformation.

  The job of the \meta{transformation code} is to map a point~$p$
  given in the registers |\pgf@x| and |\pgf@y| to a new
  coordinate~$f(p)$, which should be returned in |\pgf@x| and |\pgf@y|
  as well. As an example, suppose we wish to install polar coordinates
  as the nonlinear transformation. For this, we need a bit of code:
\begin{codeexample}[code only]
\def\polartransformation{%
  % \pgf@x will contain the radius
  % \pgf@y will contain the distance
  \pgfmathsincos@{\pgf@x}%
  % pgfmathresultx is now the cosine of radius and
  % pgfmathresulty is the sine of radius
  \pgf@x=\pgfmathresultx\pgf@y%
  \pgf@y=\pgfmathresulty\pgf@y%
}
\end{codeexample}
  (In case you wonder why you cannot just call |\pgfpointpolar| at
  this point: You can, but this function internally uses |\pgf@x|
  and |\pgf@y| in complicated ways, so you would first have to safe
  them so some other registers. Also, the above is faster.)
  
  If we were to call this function again, we would get something funny
  like ``polar-polar coordinates,'' so let's not do this. Let us
  instead have a look at the effect this call has: Once a nonlinear
  transformation is installed, all subsequent path constructions are
  affected by this transformation. In particular, a normal grid now
  becomes the typical ``polar grid.''

\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  % Start nonlinear transformation
  \pgftransformnonlinear{\polartransformation}% see above
  
  % Draw something with this transformation in force
  \draw (0pt,0mm) grid [xstep=10pt, ystep=5mm] (90pt, 20mm);
\end{tikzpicture}
\end{codeexample}
\end{command}


\subsubsection{Applying Nonlinear Transformations to Points}

\begin{command}{\pgfpointtransformednonlinear\marg{point}}
  Works like |\pgfpointtransformed|, but also applies the current
  nonlinear transformation; that is, it first applies the current
  linear transformation and then the current nonlinear
  transformations. Note that, just like |\pgfpointtransformed|, you
  normally do not call this function directly since it is called
  internally by the path drawing commands.
\end{command}


\subsubsection{Applying Nonlinear Transformations to Paths}

When a nonlinear transformation is installed, the normal path
construction commands like |\pgfpathmoveto| get adjusted so that the
``honour'' the nonlinear transformations currently in force. For
|\pgfpathmoveto| this is pretty simple: Instead of just applying the
linear transformation matrix to the point to which the path should
``jump'' next, we also apply the nonlinear transformation. However,
for a command like |\pgfpathlineto|, things are much more difficult:
A straight line will no longer be a straight line!

In order to make straight lines ``bend'', the following changes are in
force while a nonlinear transformation is installed:

\begin{enumerate}
\item Whenever a straight line between two points $p$ and $q$ should
  be added to the path, either through |\pgfpathlineto| or through
  |\pgfpathclose|, we replace this straight line by a ``degenerated
  curve'' from $p$ to $q$ whose control points are at one third and
  two third of the distance between $p$ and $q$ on the line between
  $p$ and $q$. In this way, while nonlinear transformations are in
  force, we only need to transform curves.
\item Next, suppose we wish to transform a curve from $p$ to $q$ with
  supports $s$ and $t$. For this, we simply apply the nonlinear
  transformation $f$ to all four points and draw a line with the
  results. Note that this mapping is actually not quite satisfactory
  for long lines that are strongly curved:
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  {
    \pgftransformnonlinear{\polartransformation}
    % The curve with the controls computed by pgf: a nice quarter arc
    \draw [red] (0,20mm) -- (90pt,20mm);
  }
  % Here is the curve with controls just transformed:
  \draw (0:20mm) .. controls (30pt:20mm) and (60pt:20mm) .. (90pt:20mm);  
\end{tikzpicture}
\end{codeexample}
  As the example shows, the control points now lie on the arc; but in
  reality they should point along the tangents at the start and the
  end. This is exactly when \pgfname\ does through the computation
  described above.
\item
  To overcome the effect of the control points being ``off,'' it is
  necessary to split up longer curves into smaller parts, which are
  drawn individually to increase the accuracy. When such splitting
  occurs, can be configured using the following command: 
  \begin{command}{\pgfsettransformnonlinearflatness\marg{dimension}
      (initially 5pt)}
    Whenever in a to-be-drawn curve the $L^\infty$-distance (maximum
    of the distances in $x$- and $y$-directions) between the start of
    a curve and its first control point or between the first and
    second control points or between the second control point and the
    end is more than \meta{distance}, the curve gets split in the
    middle (more precisely, at time $t= 0.5$) and we draw the two
    parts individually (for them, splitting may occur again, if the
    curve is still too long).
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  \draw[red] (0:20mm) arc [start angle=0, end angle=90, radius=2cm];
  {
    \pgftransformnonlinear{\polartransformation}
    \pgfsettransformnonlinearflatness{2pt} % very precise
    \draw (0,20mm) -- (90pt,20mm);
  }
\end{tikzpicture}
\end{codeexample}
  \end{command}
\end{enumerate}


\subsubsection{Applying Nonlinear Transformations to Text}

Earlier, it was pointed that nonlinear transformations do not apply
to text. Nevertheless, when you use |\pgftext| or |\pgfnode|,
\pgfname\ will do a sort of ``best effort'' to render the text in
the nonlinear coordinate system: The point where the text should be
shown can obviously be computed easily. When then temporarily reset
the nonlinear transformation and, instead, setup a linear
transformation that matches the nonlinear transformation at the
point where the text should be. Then, the text is shown. This means
that if the text is longer, it will not ``follow'' the nonlinear
transformation, but near the origin of the text it will look
``correct.'' As an example, let us add some text at the grid point
of the above example:

\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  \pgftransformnonlinear{\polartransformation}% see above
  
  % Draw something with this transformation in force
  \draw (0pt,0mm) grid [xstep=10pt, ystep=5mm] (90pt, 20mm);
  
  \foreach \angle in {0,30,60,90}
    \foreach \dist in {1,2}
      {
        \pgftransformshift{\pgfpoint{\angle pt}{\dist cm}}
        \pgftext{\angle$^\circ$}
      }
\end{tikzpicture}
\end{codeexample}


\subsubsection{Approximating Nonlinear Transformations Using Linear Transformations}

At any given point, the current nonlinear transformation can be
approximated using a linear transformation. The following two
functions allow you to install such a local approximation:

\begin{command}{\pgfapproximatenonlineartransformation}
  This command will do two things:
  \begin{enumerate}
  \item It clears the nonlinear transformations for the rest of the
    current \TeX\ scope, so only linear transformations apply.
  \item However, before removing the nonlinear transformations, the
    linear transformation matrix is modified so that it mimics the
    effect the nonlinear transformation had at the origin. That is,
    after you call this command, drawing something near the origin
    will look almost the same as if you had not called it.
  \end{enumerate}

\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  \pgftransformnonlinear{\polartransformation}% see above
  \draw (0pt,0mm) grid [xstep=10pt, ystep=5mm] (90pt, 20mm);

  \begin{scope}[shift={(45pt,20mm)}]
    % Draw something near "origin":
    \draw [red] (-10pt,-10pt) -- (10pt,10pt);    
    \draw [red] (10pt,-10pt) -- (-10pt,10pt);

    % Now draw the same, but in the "approximate" coordinate system:
    \pgfapproximatenonlineartransformation
    \draw [] (-10pt,-10pt) -- (10pt,10pt);    
    \draw [] (10pt,-10pt) -- (-10pt,10pt);
    \pgftext{foo};
  \end{scope}
\end{tikzpicture}
\end{codeexample}  
  This command is used by |\pgftext| and |\pgfnode| to transform text
  when a nonlinear transformation is in force.
\end{command}

\begin{command}{\pgfapproximatenonlineartranslation}
  This command works like the normal approximation command, but it
  will only approximate how the origin gets translated, it will not
  approximate the rotation, skewing, or scaling that is involved. This
  is useful for drawing text at the right position, but without
  ``mutilating'' the text.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  \pgftransformnonlinear{\polartransformation}% see above
  \draw (0pt,0mm) grid [xstep=10pt, ystep=5mm] (90pt, 20mm);

  \begin{scope}[shift={(45pt,20mm)}]
    % Draw something near "origin":
    \draw [red] (-10pt,-10pt) -- (10pt,10pt);    
    \draw [red] (10pt,-10pt) -- (-10pt,10pt);

    % Now draw the same, but in the "approximate" coordinate system:
    \pgfapproximatenonlineartranslation
    \draw [] (-10pt,-10pt) -- (10pt,10pt);    
    \draw [] (10pt,-10pt) -- (-10pt,10pt);
    \pgftext{foo};
  \end{scope}
\end{tikzpicture}
\end{codeexample}  
\end{command}


\subsubsection{Nonlinear Transformation Libraries}
\label{section-library-curvilinear}

\begin{pgflibrary}{curvilinear}
  This library defines commands for computing nonlinear
  transformations ``along B\'ezier curves''.
\end{pgflibrary}

Up to now, our running example for a nonlinear transformation was
polar transformation. However, is \pgfname\ nonlinear transformations
are \emph{actually} mainly used for transforming arrow tips; and these
need to be transformed ``along curves.'' The |curvilinear| library
defines a number of commands that offer the necessary computations for
such transformations.

\begin{command}{\pgfsetcurvilinearbeziercurve\marg{start}\marg{first
      support}\marg{second support}\marg{end}}
  Prior to using any other command from this library, you first call
  this function to ``install'' a B\'ezier curve to which the commands
  will refer. This curve will be local to the current \TeX\ scope and
  you can install only one curve at a time.

  The main job of this command is to store the passed points
  internally and to build a lookup table for distance-to-time
  conversions, see the next command.
\begin{codeexample}[code only]
\pgfsetcurvilinearbeziercurve
  {\pgfpointorigin}
  {\pgfpoint{1cm}{1cm}}
  {\pgfpoint{2cm}{1cm}}
  {\pgfpoint{3cm}{0cm}}
\end{codeexample}
\end{command}

\begin{command}{\pgfcurvilineardistancetotime\marg{distance}}
  This command does a ``distance-to-time-conversion'': It tries to
  compute a time $t$, returned in |\pgf@x|, that corresponds to
  travelling \meta{distance} along the curve that has last been
  installed using the command   |\pgfsetcurvilinearbeziercurve|. The distance-to-time-conversion
  uses the precomputations done by that command. Note that several
  compromises had to be made between speed and accuracy:
  \begin{itemize}
  \item The conversion will be best near the start of the curve.
  \item The more ``degenerate'' the curve, the worse the results.
  \end{itemize}
\end{command}


\begin{command}{\pgfpointcurvilinearbezierorthogonal\marg{distance}\marg{offset}}
  This command computes the following point: Consider the curve
  last installed using the command |\pgfsetcurvilinearbeziercurve|. We
  travel along this curve by \meta{distance}, arriving at a point
  $p$. Then, we turn by $90^\circ$ and travel by \meta{offset} units
  ``aways from the curve,'' arriving at a point $q$. This point $q$
  will now be returned in |\pgf@x| and |\pgf@y|; furthermore, the
  transformed local coordinate system at point $q$ will also be
  returned |\pgf@xa| and the other registers, see
  |\pgftransformnonlinear| for details.
  \makeatletter
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  {
    \pgfsetcurvilinearbeziercurve
      {\pgfpoint{0mm}{20mm}}
      {\pgfpoint{11mm}{20mm}}
      {\pgfpoint{20mm}{11mm}}
      {\pgfpoint{20mm}{0mm}}
    \pgftransformnonlinear{\pgfpointcurvilinearbezierorthogonal\pgf@x\pgf@y}%
    \draw (0,-30pt) grid [step=10pt] (80pt,30pt);
  }
  \draw[red, very thick]
    (0mm,20mm) .. controls (11mm,20mm) and (20mm,11mm) .. (20mm,0mm);
\end{tikzpicture}
\end{codeexample}

\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  {
    \pgfsetcurvilinearbeziercurve
      {\pgfpoint{0mm}{20mm}}
      {\pgfpoint{10mm}{20mm}}
      {\pgfpoint{10mm}{10mm}}
      {\pgfpoint{20mm}{10mm}}
    \pgftransformnonlinear{\pgfpointcurvilinearbezierorthogonal\pgf@x\pgf@y}%
    \draw (0,-30pt) grid [step=10pt] (80pt,30pt);
  }
  \draw[red, very thick]
    (0mm,20mm) .. controls (10mm,20mm) and (10mm,10mm) .. (20mm,10mm);
\end{tikzpicture}
\end{codeexample}
\end{command}


\begin{command}{\pgfpointcurvilinearbezierpolar\marg{x}\marg{y}}
  This command is similar to the previous version, but the
  transformation is different: The idea is that a line form $(0,0)$ to
  $(x,0)$ gets transformed to the curve from the start of the curve to
  a point at distance $x$ along the curve. This is identical to what
  the ``orthogonal'' transformation above also does. The difference is
  that a line from $(0,0)$ to $(0,y)$ gets still transformed to an
  initial segment of the curve of a length of $y$, but now rotated by
  $90^\circ$. In general, the point $p = (x,y)$ gets transferred to a
  point that at distance $|p| = \sqrt{x^2+y^2}$ along the curve, but
  rotated by the angle of $p$ relative to the $x$-axis.

  All of these computations mainly have the following effect: Two
  straight lines from the start of the curve as in a |Straight Barb|
  arrow tip get transformed to an initial segment of the curve whose
  length is the length of the two lines, but this segment gets rotated
  by the angle of the two lines. 
  \makeatletter
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  {
    \pgfsetcurvilinearbeziercurve
      {\pgfpoint{0mm}{20mm}}
      {\pgfpoint{11mm}{20mm}}
      {\pgfpoint{20mm}{11mm}}
      {\pgfpoint{20mm}{0mm}}
    \pgftransformnonlinear{\pgfpointcurvilinearbezierpolar\pgf@x\pgf@y}%
    \draw (0,-30pt) grid [step=10pt] (80pt,30pt);
    % Add a "barb":
    \draw [blue, very thick] (20pt,10pt) -- (0,0) -- (20pt,-10pt);
  }
  \draw[red, very thick]
    (0mm,20mm) .. controls (11mm,20mm) and (20mm,11mm) .. (20mm,0mm);
\end{tikzpicture}
\end{codeexample}

\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);
  {
    \pgfsetcurvilinearbeziercurve
      {\pgfpoint{0mm}{20mm}}
      {\pgfpoint{10mm}{20mm}}
      {\pgfpoint{10mm}{10mm}}
      {\pgfpoint{20mm}{10mm}}
    \pgftransformnonlinear{\pgfpointcurvilinearbezierpolar\pgf@x\pgf@y}%
    \draw (0,-30pt) grid [step=10pt] (80pt,30pt);
    % Add a "barb":
    \draw [blue, very thick] (20pt,10pt) -- (0,0) -- (20pt,-10pt);
  }
  \draw[red, very thick]
    (0mm,20mm) .. controls (10mm,20mm) and (10mm,10mm) .. (20mm,10mm);
\end{tikzpicture}
\end{codeexample}
\end{command}


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% LocalWords:  nonlineartransformations PGF cx dy pdf PostScript pgfscope xstep
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