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+% 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{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
+$(ax+cy+s,bx+dy+t)$ 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}
+
+
+\subsubsection{Applying Coordinate Transformation to Points}
+
+\begin{command}{\pgfpointtransformed\marg{point}}
+ Applies current transformation matrix to \marg{point} $(x,y)$ and returns a
+ transformed point $(ax+cy+s,bx+dy+t)$. Normally, this is done automatically
+ by commands like |\pgfpathlineto| or |\pgfpathmoveto|, but sometimes you
+ may wish to access a transformed point yourself.
+\end{command}
+
+
+\subsubsection{Computing Adjustments for Coordinate Transformations}
+\label{section-adjustment-transformations}
+
+\begin{command}{\pgftransformationadjustments}
+ This command computes ``adjustments'' for the current transformation matrix
+ so that even when you install a transformation matrix that scales
+ everything by a certain factor, you can still draw something of ``an
+ absolute size''. Suppose for instance that you install a transformation
+ matrix that scales everything by a factor of 4 and you now wish to draw a
+ horizontal line of length 1cm. Then, if you do not reset the transformation
+ matrix, you can draw a line of logical length 2.5mm, which will then get
+ scaled to a line of 1cm. Things get more difficult in case you scale things
+ only, say, vertically. In this case, the adjustment necessary for
+ horizontal lines is different from the one needed for vertical lines.
+
+ This function computes two scaling factors, one for horizontal lines
+ and one for vertical lines, and stores them in the following macros:
+ %
+ \begin{command}{\pgfhorizontaltransformationadjustment}
+ When you scale the length of a horizontal line by this factor in the
+ current transformation, you compensate for the scaling. Formally, it is
+ $1/\|\mathit{transform}(1,0)\|_2$, where $\mathit{transform}$ applies
+ the current transformations matrix to the given number.
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (2,2);
+ \begin{scope}[xscale=2,thick]
+ \draw [red] (1,1) -- ++(1,0);
+
+ \pgftransformationadjustments
+ \draw [blue] (1,0) -- ++(\pgfhorizontaltransformationadjustment,0);
+ \end{scope}
+\end{tikzpicture}
+\end{codeexample}
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (2,2);
+ \begin{scope}[xscale=2,thick,rotate=90]
+ \draw [red] (1,1) -- ++(1,0);
+
+ \pgftransformationadjustments
+ \draw [blue] (1,0) -- ++(\pgfhorizontaltransformationadjustment,0);
+ \end{scope}
+\end{tikzpicture}
+\end{codeexample}
+ \end{command}
+ %
+ \begin{command}{\pgfverticaltransformationadjustment}
+ $1/\|\mathit{transform}(0,1)\|_2$.
+ \end{command}
+
+ Note that the ``right'' way to draw a line of absolute length 1cm in a
+ transformed coordinate system is to first compute the start point and to
+ then reset the transformation matrix. The transformation adjustments
+ computed here are important only in situations where you cannot do this,
+ for instance when an |outer xsep| must be set.
+\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.
+
+
+\subsubsection{Applying General Canvas Transformations}
+
+\pgfname\ does not offer many 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}
+
+
+\subsubsection{Establishing View Boxes}
+\label{section-base-view}
+
+A \emph{view box} is like a ``window'' through which you see a graphic. To
+establish a view box, you specify a rectangle -- which is the window -- and
+another rectangle surrounding the to-be-viewed graphic. The graphic will then
+be rescaled and shifted in such a way that the to-be-viewed rectangle matches
+the view box's rectangle as well as possible. Note that establishing a view box
+does, indeed, cause a canvas transformation to be installed.
+
+View boxes are only seldom needed in normal graphics. Their main application is
+with animations since you can \emph{animate} the to-be-viewed rectangle. This
+makes it easy to create animations in which you zoom in, zoom out, and pan a
+graphic.
+
+\begin{environment}{{pgfviewboxscope}\marg{$ll_1$}\marg{$ur_1$}\marg{$ll_2$}\marg{$ur_2$}\marg{meet or slice}}
+ Inside the viewbox scope, the source rectangle (with the two \pgfname\
+ points $ll_1$ and $ur_1$ as corners) will be translated and scaled so that
+ it becomes centered on the target rectangle (with the corners $ll_2$ and
+ $ur_2$) and will, for |meet| as last parameter, be as large as possible so
+ that it fits inside the target and, for |slice|, be as small as possible so
+ that it encompasses the target.
+ %
+\begin{codeexample}[]
+\tikz {
+ \draw [red, very thick] (0,0) rectangle (20mm,20mm);
+ \begin{pgfviewboxscope}
+ {\pgfpoint{5mm}{5mm}}{\pgfpoint{25mm}{15mm}} % Source
+ {\pgfpoint{0mm}{0mm}}{\pgfpoint{20mm}{20mm}} % Target
+ {meet}
+ \draw [blue, very thick] (5mm,5mm) rectangle (25mm,15mm);
+ \draw [thick] (1,1) circle [radius=8mm] node {Hi};
+ \end{pgfviewboxscope} }
+\end{codeexample}
+ %
+\begin{codeexample}[]
+\tikz {
+ \draw [red, very thick] (0,0) rectangle (20mm,20mm);
+ \begin{pgfviewboxscope}
+ {\pgfpoint{5mm}{5mm}}{\pgfpoint{25mm}{15mm}} % Source
+ {\pgfpoint{0mm}{0mm}}{\pgfpoint{20mm}{20mm}} % Target
+ {slice}
+ \draw [blue, very thick] (5mm,5mm) rectangle (25mm,15mm);
+ \draw [thick] (1,1) circle [radius=8mm] node {Hi};
+ \end{pgfviewboxscope} }
+\end{codeexample}
+ %
+\end{environment}
+
+\begin{plainenvironment}{{pgfviewboxscope}\marg{$ll_1$}\marg{$ur_1$}\marg{$ll_2$}\marg{$ur_2$}\marg{meet or slice}}
+ Plain \TeX\ version of the environment.
+\end{plainenvironment}
+
+\begin{contextenvironment}{{pgfviewboxscope}\marg{$ll_1$}\marg{$ur_1$}\marg{$ll_2$}\marg{$ur_2$}\marg{meet or slice}}
+ 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}
+
+\begin{codeexample}[setup code,hidden]
+\makeatletter
+\def\polartransformation{
+ % \pgf@x will contain the radius
+ % \pgf@y will contain the distance
+ \pgfmathsincos@{\pgf@sys@tonumber\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
+\end{codeexample}
+
+\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@sys@tonumber\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}[preamble={\usepgfmodule{nonlineartransformations}}]
+\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}[preamble={\usepgfmodule{nonlineartransformations}}]
+\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}[preamble={\usepgfmodule{nonlineartransformations}}]
+\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}[preamble={\usepgfmodule{nonlineartransformations}}]
+\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}[preamble={\usepgfmodule{nonlineartransformations}}]
+\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}[preamble={\usepgfmodule{nonlineartransformations}}]
+\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 Bézier 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 Bézier 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 ``always 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}[
+ preamble={\usepgfmodule{nonlineartransformations}
+\usetikzlibrary{curvilinear}},
+ pre=\makeatletter]
+\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}[
+ preamble={\usepgfmodule{nonlineartransformations}
+\usetikzlibrary{curvilinear}},
+ pre=\makeatletter]
+\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}[
+ preamble={\usepgfmodule{nonlineartransformations}
+\usetikzlibrary{curvilinear}},
+ pre=\makeatletter]
+\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}[
+ preamble={\usepgfmodule{nonlineartransformations}
+\usetikzlibrary{curvilinear}},
+ pre=\makeatletter]
+\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}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "pgfmanual"
+%%% End:
+
+% LocalWords: nonlineartransformations PGF cx dy pdf PostScript pgfscope xstep
+% LocalWords: Reinstalls shiftx backend pgflowlevelscope ystep ezier lookup xa
+% LocalWords: precomputations