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### 13 Transformations

P G F has a powerful transformation mechanism that is similar to the transformation capabilities of M E T A F O N T. The present section explains how you can access it in TikZ.

#### 13.1 The Different Coordinate Systems

It is a long process from a coordinate like, say, (1,2) or (1cm,5mathrm pt), to the position a point is finally placed on the display or paper. In order to find out where the point should go, it is constantly “transformed,” which means that it is mostly shifted around and possibly rotated, slanted, scaled, and otherwise mutilated.

In detail, (at least) the following transformations are applied to a coordinate like (1,2) before a point on the screen is chosen:

1. P G F interprets a coordinate like (1,2) in its xy-coordinate system as “add the current x-vector once and the current y-vector twice to obtain the new point.”
2. P G F applies its coordinate transformation matrix to the resulting coordinate. This yields the final position of the point inside the picture.
3. The backend driver (like dvips or pdftex) adds transformation commands such the coordinate is shifted to the correct position in TEX’s page coordinate system.
4. P D F (or PostScript) apply the canvas transformation matrix to the point, which can once more change the position on the page.
5. The viewer application or the printer applies the device transformation matrix to transform the coordinate to its final pixel coordinate on the screen or paper.

In reality, the process is even more involved, but the above should give the idea: A point is constantly transformed by changes of the coordinate system.

In TikZ, you only have access to the first two coordinate systems: The xy-coordinate system and the coordinate transformation matrix (these will be explained later). P G F also allows you to change the canvas transformation matrix, but you have to use commands of the core layer directly to do so and you “better know what you are doing” when you do this. The moment you start modifying the canvas matrix, P G F immediately looses track of all coordinates and shapes, anchors, and bounding box computations will no longer work.

#### 13.2 The Xy- and Xyz-Coordinate Systems

The first and easiest coordinate systems are P G F’s xy- and xyz-coordinate systems. The idea is very simple: Whenever you specify a coordinate like (2,3) this means 2vx + 3vy, where vx is the current x-vector and vy is the current y-vector. Similarly, the coordinate (1,2,3) means vx + 2vy + 3vz.

Unlike other packages, P G F does not insist that vx actually has a y-component of 0, that is, that it is a horizontal vector. Instead, the x-vector can point anywhere you want. Naturally, normally you will want the x-vector to point horizontally.

One undesirable effect of this flexibility is that it is not possible to provide mixed coordinates as in (1,2pt). Life is hard.

To change the x-, y-, and z-vectors, you can use the following options:

- x=<dimension> Sets the x-vector of P G F’s xyz-coordinate system to point <dimension> to the right, that is, to (<dimension>,0pt). The default is 1cm.

  \begin{tikzpicture} \draw (0,0) -- +(1,0); \draw[x=2cm,color=red] (0,0.1) -- +(1,0); \end{tikzpicture}

  \tikz \draw[x=1.5cm] (0,0) grid (2,2);

  The last example shows that the size of steppings in grids, just like all other dimensions, are not affected by the x-vector. After all, the x-vector is only used to determine the coordinate of the upper right corner of the grid.

- x=<coordinate> Sets the x-vector of P G F’s xyz-coordinate system to the specified <coordinate>. If <coordinate> contains a comma, it must be put in braces.

  \begin{tikzpicture} \draw (0,0) -- (1,0); \draw[x={(2cm,0.5cm)},color=red] (0,0) -- (1,0); \end{tikzpicture}

  You can use this, for example, to exchange the meaning of the x- and y-coordinate.

  \begin{tikzpicture}[smooth] \draw plot coordinates{(1,0) (2,0.5) (3,0) (3,1)}; \draw[x={(0cm,1cm)},y={(1cm,0cm)},color=red] plot coordinates{(1,0) (2,0.5) (3,0) (3,1)}; \end{tikzpicture}

- y=<value> Works like the x= option, only if <value> is a dimension, the resulting vector points to (0,<value>).
- z=<value> Works like the z= option, but now a dimension is means the point (<value>,<value>).

  \begin{tikzpicture}[z=-1cm,->,thick] \draw[color=red] (0,0,0) -- (1,0,0); \draw[color=blue] (0,0,0) -- (0,1,0); \draw[color=orange] (0,0,0) -- (0,0,1); \end{tikzpicture}

#### 13.3 Coordinate Transformations

P G F and TikZ allow you to specify coordinate transformations. Whenever you specify a coordinate as in (1,0) or (1cm,1pt) or (30:2cm), this coordinate is first “reduced” to a position of the form “x points to the right and y points upwards.” For example, (1in,5pt) is reduced to “72 points to the right and 5 points upwards” and (90:100pt) means “0pt to the right and 100 points upwards.”

The next step is to apply the current coordinate transformation matrix to the coordinate. For example, the coordinate transformation matrix might currently be set such that it adds a certain constant to the x value. Also, it might be setup such that it, say, exchanges the x and y value. In general, any “standard” transformation like translation, rotation, slanting, or scaling or any combination thereof is possible. (Internally, P G F keeps track of a coordinate transformation matrix very much like the concatenation matrix used by P D F or PostScript.)

\begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) rectangle (1,0.5); \begin{scope}[xshift=1cm] \draw [red] (0,0) rectangle (1,0.5); \draw[yshift=1cm] [blue] (0,0) rectangle (1,0.5); \draw[rotate=30] [orange] (0,0) rectangle (1,0.5); \end{scope} \end{tikzpicture}

The most important aspect of the coordinate transformation matrix is that it applies to coordinates only! In particular, the coordinate transformation has no effect on things like the line width or the dash pattern or the shading angle. In certain cases, it is not immediately clear whether the coordinate transformation matrix should apply to a certain dimension. For example, should the coordinate transformation matrix apply to grids? (It does.) And what about the size of arced corners? (It does not.) The general rule is “If there is no ‘coordinate’ involved, even ‘indirectly,’ the matrix is not applied.” However, sometimes, you simply have to try or look it up in the documentation whether the matrix will be applied.

Setting the matrix cannot be done directly. Rather, all you can do is to “add” another transformation to the current matrix. However, all transformations are local to the current TEX-group. All transformations are added using graphic options, which are described below.

Transformations apply immediately when they are encountered “in the middle of a path” and they apply only to the coordinates on the path following the transformation option.

\tikz \draw (0,0) rectangle (1,0.5) [xshift=2cm] (0,0) rectangle (1,0.5);

A final word of warning: You should refrain from using “aggressive” transformations like a scaling of a factor of 10000. The reason is that all transformations are done using TEX, which has a fairly low accuracy. Furthermore, in certain situations it is necessary that TikZ inverts the current transformation matrix and this will fail if the transformation matrix is badly conditioned or even singular (if you do not know what singular matrices are, you are blessed).

- shift={<coordinate>} adds the <coordinate> to all coordinates.

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[shift={(1,1)},blue] (0,0) -- (1,1) -- (1,0); \draw[shift={(30:1cm)},red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- xshift=<dimension> adds <dimension> to the x value of all coordinates.

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[xshift=2cm,blue] (0,0) -- (1,1) -- (1,0); \draw[xshift=-10pt,red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- yshift=<dimension> adds <dimension> to the y value of all coordinates.
- scale=<factor> multiplies all coordinates by the given <factor>. The <factor> should not be excessively large in absolute terms or very near to zero.

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[scale=2,blue] (0,0) -- (1,1) -- (1,0); \draw[scale=-1,red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- xscale=<factor> multiplies only the x-value of all coordinates by the given <factor>.

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[xscale=2,blue] (0,0) -- (1,1) -- (1,0); \draw[xscale=-1,red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- yscale=<factor> multiplies only the y-value of all coordinates by <factor>.
- xslant=<factor> slants the coordinate horizontally by the given <factor>:

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[xslant=2,blue] (0,0) -- (1,1) -- (1,0); \draw[xslant=-1,red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- yslant=<factor> slants the coordinate vertically by the given <factor>:

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[yslant=2,blue] (0,0) -- (1,1) -- (1,0); \draw[yslant=-1,red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- rotate=<degree> rotates the coordinate system by <degree>:

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[rotate=40,blue] (0,0) -- (1,1) -- (1,0); \draw[rotate=-20,red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- rotate around={<degree>:<coordinate>} rotates the coordinate system by <degree> around the point <coordinate>.

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[rotate around={40:(1,1)},blue] (0,0) -- (1,1) -- (1,0); \draw[rotate around={-20:(1,1)},red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- cm={<a>,<b>,<c>,<d>,<coordinate>} applies the following transformation to all coordinates: Let (x,y) be the coordinate to be transformed and let <coordinate> specify the point (tx,ty). Then the new coordinate is given by + . Usually, you do not use this option directly.

  \begin{tikzpicture} \draw[style=help lines] (0,0) grid (3,2); \draw (0,0) -- (1,1) -- (1,0); \draw[cm={1,1,0,1,(0,0)},blue] (0,0) -- (1,1) -- (1,0); \draw[cm={0,1,1,0,(1cm,1cm)},red] (0,0) -- (1,1) -- (1,0); \end{tikzpicture}

- reset cm completely resets the coordinate transformation matrix to the identity matrix. This will destroy not only the transformations applied in the current scope, but also all transformations inherited from surrounding scopes. Do not use this option.

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