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### 26 Coordinate and Canvas Transformations

#### 26.1 Overview

PGF offers two different ways of scaling, shifting, and rotating (these operations are generally known as transformations) graphics: You can apply coordinate transformations to all coordinates and you can apply canvas transformations to the canvas on which you draw. (The names “coordinate” and “canvas” transformations are not standard, I introduce them only for the purposes of this manual.)

The difference is the following:

- 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 (2cm,4cm).

  Note that coordinate transformations apply only to coordinates. They do not apply to, say, line width or shadings or text.

- 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 everything will be larger: Text, lines, coordinates, and shadings.

In many cases, it is preferable that you use coordinate transformations and not canvas transformations. When canvas transformations are used, PGF looses track of the coordinates of nodes and shapes. Also, canvas transformations often cause undesirable effects like changing text size. For these reasons, PGF makes it easy to setup the coordinate transformation, but a bit harder to change the canvas transformation.

#### 26.2 Coordinate Transformations

##### 26.2.1 How PGF Keeps Track of the Coordinate Transformation Matrix

PGF 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, PGF 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 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 = 0pt, and t = 0pt.

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 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.

##### 26.2.2 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 in addition to any previous coordinate transformations.

\pgftransformshift{<point>}

  Shifts coordinates by <point>.

  \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}

\pgftransformxshift{<dimensions>}

  Shifts coordinates by <dimension> along the x-axis.

  \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}

\pgftransformyshift{<dimensions>}

  Like \pgftransformxshift, only for the y-axis.

\pgftransformscale{<factor>}

  Scales coordinates by <factor>.

  \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}

\pgftransformxscale{<factor>}

  Scales coordinates by <factor> in the x-direction.

  \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}

\pgftransformyscale{<factor>}

  Like \pgftransformxscale, only for the y-axis.

\pgftransformxslant{<factor>}

  Slants coordinates by <factor> in the x-direction. Here, a factor of 1 means 45o.

  \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}

\pgftransformyslant{<factor>}

  Slants coordinates by <factor> in the y-direction.

  \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}

\pgftransformrotate{<degrees>}

  Rotates coordinates counterclockwise by <degrees>.

  \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}

\pgftransformcm{<a>}{<b>}{<c>}{<d>}{<point>}

  Applies the transformation matrix given by a, b, c, and d and the shift <point> to coordinates (in addition to any previous transformations already in force).

  \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}

\pgftransformarrow{<start>}{<end>}

  Shift coordinates to the end of the line going from <start> to <end> with the correct rotation.

  \begin{tikzpicture} \draw[help lines] (0,0) grid (3,2); \draw (0,0) -- (3,1); \pgftransformarrow{\pgfpointorigin}{\pgfpoint{3cm}{1cm}} \pgftext{tip} \end{tikzpicture}

\pgftransformlineattime{<time>}{<start>}{<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 20.4.2.

  In addition to shifting the coordinate, a rotation may also be applied. Whether this is the case depends on whether the TEX if \ifpgfslopedattime is set to true or not.

  \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}

  \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}

  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{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}

  \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}

\pgftransformcurveattime{<time>}{<start>}{<first support>}{<second support>}{<end>}

  Shifts coordinates by a specific point on a curve at a specific time, see Section 20.4.2 once more.

  As for the line-at-time transformation command, \ifpgfslopedattime decides whether an additional rotation should be applied.

  \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}

  \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}

  The value of \ifpgfresetnontranslationsattime is also taken into account.

\ifpgfslopedattime

  Decides whether the “at time” transformation commands also rotate coordinates or not.

\ifpgfresetnontranslationsattime

  Decides whether the “at time” transformation commands should reset the non-translations between shifting and rotating.

##### 26.2.3 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 “absolutely.” 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.

\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{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}

\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{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}

\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{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}

##### 26.2.4 Saving and Restoring the Coordinate Transformation Matrix

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

\pgfgettransform{<macro>}

  This command will (locally) define <macro> to a representation of the current coordinate transformation matrix. This matrix can later on be reinstalled using \pgfsettransform.

\pgfsettransform{<macro>}

  Reinstalls a coordinate transformation matrix that was previously saved using \pgfgettransform.

#### 26.3 Canvas Transformations

The canvas transformation matrix is not managed by PGF, but by the output format like PDF or PostScript. All the PGF 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 PGF.

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, PGF 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.

PGF 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).

\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{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}

\pgflowlevel{<transformation code>}

  This command concatenates the canvas transformation matrix with the coordinate transformation specified by <transformation code>.

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

\pgflowlevelobj{<transformation code>}{<code>}

  This command creates a local {pgfscope}. Inside this scope, \pgflowlevel is first called with the argument <transformation code>, then the <code> is inserted.

  \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}

\begin{pgflowlevelscope}{<transformation code>}
<environment contents>
\end{pgflowlevelscope}

  This environment first surrounds the <environment contents> by a {pgfscope}. Then it calls \pgflowlevel with the argument <transformation code>.

  \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}

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