From 810d9b451473ebe8e27c3f691bf001e01864e6fb Mon Sep 17 00:00:00 2001
From: Karl Berry <karl@freefriends.org>
Date: Sun, 15 Jan 2023 20:58:27 +0000
Subject: pgf (15jan23)

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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{Tutorial: A Picture for Karl's Students}
+
+This tutorial is intended for new users of \tikzname. It does not give an
+exhaustive account of all the features of \tikzname, just of those that you are
+likely to use right away.
+
+Karl is a math and chemistry high-school teacher. He used to create the
+graphics in his worksheets and exams using \LaTeX's |{picture}| environment.
+While the results were acceptable, creating the graphics often turned out to be
+a lengthy process. Also, there tended to be problems with lines having slightly
+wrong angles and circles also seemed to be hard to get right. Naturally, his
+students could not care less whether the lines had the exact right angles and
+they find Karl's exams too difficult no matter how nicely they were drawn. But
+Karl was never entirely satisfied with the result.
+
+Karl's son, who was even less satisfied with the results (he did not have to
+take the exams, after all), told Karl that he might wish to try out a new
+package for creating graphics. A bit confusingly, this package seems to have
+two names: First, Karl had to download and install a package called \pgfname.
+Then it turns out that inside this package there is another package called
+\tikzname, which is supposed to stand for ``\tikzname\ ist \emph{kein}
+Zeichenprogramm''. Karl finds this all a bit strange and \tikzname\ seems to
+indicate that the package does not do what he needs. However, having used
+\textsc{gnu} software for quite some time and ``\textsc{gnu} not being Unix'',
+there seems to be hope yet. His son assures him that \tikzname's name is
+intended to warn people that \tikzname\ is not a program that you can use to
+draw graphics with your mouse or tablet. Rather, it is more like a ``graphics
+language''.
+
+
+\subsection{Problem Statement}
+
+Karl wants to put a graphic on the next worksheet for his students. He is
+currently teaching his students about sine and cosine. What he would like to
+have is something that looks like this (ideally):
+%
+\noindent
+\begin{tikzpicture}
+  [scale=3,line cap=round,
+   % Styles
+   axes/.style=,
+   important line/.style={very thick},
+   information text/.style={rounded corners,fill=red!10,inner sep=1ex}]
+
+  % Local definitions
+  \def\costhirty{0.8660256}
+
+  % Colors
+  \colorlet{anglecolor}{green!50!black}
+  \colorlet{sincolor}{red}
+  \colorlet{tancolor}{orange!80!black}
+  \colorlet{coscolor}{blue}
+
+  % The graphic
+  \draw[help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4);
+
+  \draw (0,0) circle [radius=1cm];
+
+  \begin{scope}[axes]
+    \draw[->] (-1.5,0) -- (1.5,0) node[right] {$x$};
+    \draw[->] (0,-1.5) -- (0,1.5) node[above] {$y$};
+
+    \foreach \x/\xtext in {-1, -.5/-\frac{1}{2}, 1}
+      \draw[xshift=\x cm] (0pt,1pt) -- (0pt,-1pt) node[below,fill=white] {$\xtext$};
+
+    \foreach \y/\ytext in {-1, -.5/-\frac{1}{2}, .5/\frac{1}{2}, 1}
+      \draw[yshift=\y cm] (1pt,0pt) -- (-1pt,0pt) node[left,fill=white] {$\ytext$};
+  \end{scope}
+
+  \filldraw[fill=green!20,draw=anglecolor] (0,0) -- (3mm,0pt) arc(0:30:3mm);
+  \draw (15:2mm) node[anglecolor] {$\alpha$};
+
+  \draw[important line,sincolor]
+    (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} +(0,-.5);
+
+  \draw[important line,coscolor]
+    (0,0) -- node[below=2pt,fill=white] {$\cos \alpha$} (\costhirty,0);
+
+  \draw[important line,tancolor] (1,0) --
+    node [right=1pt,fill=white]
+    {
+      $\displaystyle \tan \alpha \color{black}=
+      \frac{{\color{sincolor}\sin \alpha}}{\color{coscolor}\cos \alpha}$
+    } (intersection of 0,0--30:1cm and 1,0--1,1) coordinate (t);
+
+  \draw (0,0) -- (t);
+
+  \draw[xshift=1.85cm] node [right,text width=6cm,information text]
+    {
+      The {\color{anglecolor} angle $\alpha$} is $30^\circ$ in the
+      example ($\pi/6$ in radians). The {\color{sincolor}sine of
+        $\alpha$}, which is the height of the red line, is
+      \[
+      {\color{sincolor} \sin \alpha} = 1/2.
+      \]
+      By the Theorem of Pythagoras we have ${\color{coscolor}\cos^2 \alpha} +
+      {\color{sincolor}\sin^2\alpha} =1$. Thus the length of the blue
+      line, which is the {\color{coscolor}cosine of $\alpha$}, must be
+      \[
+      {\color{coscolor}\cos\alpha} = \sqrt{1 - 1/4} = \textstyle
+      \frac{1}{2} \sqrt 3.
+      \]%
+      This shows that {\color{tancolor}$\tan \alpha$}, which is the
+      height of the orange line, is
+      \[
+      {\color{tancolor}\tan\alpha} = \frac{{\color{sincolor}\sin
+          \alpha}}{\color{coscolor}\cos \alpha} = 1/\sqrt 3.
+      \]%
+    };
+\end{tikzpicture}
+
+
+\subsection{Setting up the Environment}
+
+In \tikzname, to draw a picture, at the start of the picture you need to tell
+\TeX\ or \LaTeX\ that you want to start a picture. In \LaTeX\ this is done
+using the environment |{tikzpicture}|, in plain \TeX\ you just use
+|\tikzpicture| to start the picture and |\endtikzpicture| to end it.
+
+
+\subsubsection{Setting up the Environment in \LaTeX}
+
+Karl, being a \LaTeX\ user, thus sets up his file as follows:
+%
+\begin{codeexample}[code only]
+\documentclass{article} % say
+\usepackage{tikz}
+\begin{document}
+We are working on
+\begin{tikzpicture}
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+\end{tikzpicture}.
+\end{document}
+\end{codeexample}
+
+When executed, that is, run via |pdflatex| or via |latex| followed by |dvips|,
+the resulting will contain something that looks like this:
+%
+\begin{codeexample}[width=7cm]
+We are working on
+\begin{tikzpicture}
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+\end{tikzpicture}.
+\end{codeexample}
+
+Admittedly, not quite the whole picture, yet, but we do have the axes
+established. Well, not quite, but we have the lines that make up the axes
+drawn. Karl suddenly has a sinking feeling that the picture is still some way
+off.
+
+Let's have a more detailed look at the code. First, the package |tikz| is
+loaded. This package is a so-called ``frontend'' to the basic \pgfname\ system.
+The basic layer, which is also described in this manual, is somewhat more,
+well, basic and thus harder to use. The frontend makes things easier by
+providing a simpler syntax.
+
+Inside the environment there are two |\draw| commands. They mean: ``The path,
+which is specified following the command up to the semicolon, should be
+drawn.'' The first path is specified as |(-1.5,0) -- (1.5,0)|, which means ``a
+straight line from the point at position $(-1.5,0)$ to the point at position
+$(1.5,0)$''. Here, the positions are specified within a special coordinate
+system in which, initially, one unit is 1cm.
+
+Karl is quite pleased to note that the environment automatically reserves
+enough space to encompass the picture.
+
+
+\subsubsection{Setting up the Environment in Plain \TeX}
+
+Karl's wife Gerda, who also happens to be a math teacher, is not a \LaTeX\
+user, but uses plain \TeX\ since she prefers to do things ``the old way''. She
+can also use \tikzname. Instead of |\usepackage{tikz}| she has to write
+|\input tikz.tex| and instead of |\begin{tikzpicture}| she writes
+|\tikzpicture| and instead of |\end{tikzpicture}| she writes |\endtikzpicture|.
+
+Thus, she would use:
+%
+\begin{codeexample}[code only]
+%% Plain TeX file
+\input tikz.tex
+\baselineskip=12pt
+\hsize=6.3truein
+\vsize=8.7truein
+We are working on
+\tikzpicture
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+\endtikzpicture.
+\bye
+\end{codeexample}
+
+Gerda can typeset this file using either |pdftex| or |tex| together with
+|dvips|. \tikzname\ will automatically discern which driver she is using. If
+she wishes to use |dvipdfm| together with |tex|, she either needs to modify the
+file |pgf.cfg| or can write |\def\pgfsysdriver{pgfsys-dvipdfm.def}| somewhere
+\emph{before} she inputs |tikz.tex| or |pgf.tex|.
+
+
+\subsubsection{Setting up the Environment in Con\TeX t}
+
+Karl's uncle Hans uses Con\TeX t. Like Gerda, Hans can also use \tikzname.
+Instead of |\usepackage{tikz}| he says |\usemodule[tikz]|. Instead of
+|\begin{tikzpicture}| he writes |\starttikzpicture| and  instead of
+|\end{tikzpicture}| he writes |\stoptikzpicture|.
+
+His version of the example looks like this:
+%
+\begin{codeexample}[code only]
+%% ConTeXt file
+\usemodule[tikz]
+
+\starttext
+  We are working on
+  \starttikzpicture
+    \draw (-1.5,0) -- (1.5,0);
+    \draw (0,-1.5) -- (0,1.5);
+  \stoptikzpicture.
+\stoptext
+\end{codeexample}
+
+Hans will now typeset this file in the usual way using |texexec| or |context|.
+
+
+\subsection{Straight Path Construction}
+
+The basic building block of all pictures in \tikzname\ is the path. A
+\emph{path} is a series of straight lines and curves that are connected (that
+is not the whole picture, but let us ignore the complications for the moment).
+You start a path by specifying the coordinates of the start position as a point
+in round brackets, as in |(0,0)|. This is followed by a series of ``path
+extension operations''. The simplest is |--|, which we used already. It must be
+followed by another coordinate and it extends the path in a straight line to
+this new position. For example, if we were to turn the two paths of the axes
+into one path, the following would result:
+%
+\begin{codeexample}[]
+\tikz \draw (-1.5,0) -- (1.5,0) -- (0,-1.5) -- (0,1.5);
+\end{codeexample}
+
+Karl is a bit confused by the fact that there is no |{tikzpicture}|
+environment, here. Instead, the little command |\tikz| is used. This command
+either takes one argument (starting with an opening brace as in
+|\tikz{\draw (0,0) -- (1.5,0)}|, which yields \tikz{\draw (0,0) --(1.5,0);}) or
+collects everything up to the next semicolon and puts it inside a
+|{tikzpicture}| environment. As a rule of thumb, all \tikzname\ graphic drawing
+commands must occur as an argument of |\tikz| or inside a |{tikzpicture}|
+environment. Fortunately, the command |\draw| will only be defined inside this
+environment, so there is little chance that you will accidentally do something
+wrong here.
+
+
+\subsection{Curved Path Construction}
+
+The next thing Karl wants to do is to draw the circle. For this, straight lines
+obviously will not do. Instead, we need some way to draw curves. For this,
+\tikzname\ provides a special syntax. One or two ``control points'' are needed.
+The math behind them is not quite trivial, but here is the basic idea: Suppose
+you are at point $x$ and the first control point is $y$. Then the curve will
+start ``going in the direction of~$y$ at~$x$'', that is, the tangent of the
+curve at $x$ will point toward~$y$. Next, suppose the curve should end at $z$
+and the second support point is $w$. Then the curve will, indeed, end at $z$
+and the tangent of the curve at point $z$ will go through $w$.
+
+Here is an example (the control points have been added for clarity):
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \filldraw [gray] (0,0) circle [radius=2pt]
+                   (1,1) circle [radius=2pt]
+                   (2,1) circle [radius=2pt]
+                   (2,0) circle [radius=2pt];
+  \draw (0,0) .. controls (1,1) and (2,1) .. (2,0);
+\end{tikzpicture}
+\end{codeexample}
+
+The general syntax for extending a path in a ``curved'' way is |.. controls|
+\meta{first control point} |and| \meta{second control point} |..|
+\meta{end point}. You can leave out the |and| \meta{second control point},
+which causes the first one to be used twice.
+
+So, Karl can now add the first half circle to the picture:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (-1,0) .. controls (-1,0.555) and (-0.555,1) .. (0,1)
+               .. controls (0.555,1) and (1,0.555) .. (1,0);
+\end{tikzpicture}
+\end{codeexample}
+
+Karl is happy with the result, but finds specifying circles in this way to be
+extremely awkward. Fortunately, there is a much simpler way.
+
+
+\subsection{Circle Path Construction}
+
+In order to draw a circle, the path construction operation |circle| can be
+used. This operation is followed by a radius in brackets as in the following
+example: (Note that the previous position is used as the \emph{center} of the
+circle.)
+%
+\begin{codeexample}[]
+\tikz \draw (0,0) circle [radius=10pt];
+\end{codeexample}
+
+You can also append an ellipse to the path using the |ellipse| operation.
+Instead of a single radius you can specify two of them:
+%
+\begin{codeexample}[]
+\tikz \draw (0,0) ellipse [x radius=20pt, y radius=10pt];
+\end{codeexample}
+
+To draw an ellipse whose axes are not horizontal and vertical, but point in an
+arbitrary direction (a ``turned ellipse'' like \tikz \draw[rotate=30] (0,0)
+ellipse [x radius=6pt, y radius=3pt];) you can use transformations, which are
+explained later. The code for the little ellipse is
+|\tikz \draw[rotate=30] (0,0) ellipse [x radius=6pt, y radius=3pt];|, by the
+way.
+
+So, returning to Karl's problem, he can write
+|\draw (0,0) circle [radius=1cm];| to draw the circle:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+\end{tikzpicture}
+\end{codeexample}
+
+At this point, Karl is a bit alarmed that the circle is so small when he wants
+the final picture to be much bigger. He is pleased to learn that \tikzname\ has
+powerful transformation options and scaling everything by a factor of three is
+very easy. But let us leave the size as it is for the moment to save some
+space.
+
+
+\subsection{Rectangle Path Construction}
+
+The next things we would like to have is the grid in the background. There are
+several ways to produce it. For example, one might draw lots of rectangles.
+Since rectangles are so common, there is a special syntax for them: To add a
+rectangle to the current path, use the |rectangle| path construction operation.
+This operation should be followed by another coordinate and will append a
+rectangle to the path such that the previous coordinate and the next
+coordinates are corners of the rectangle. So, let us add two rectangles to the
+picture:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \draw (0,0) rectangle (0.5,0.5);
+  \draw (-0.5,-0.5) rectangle (-1,-1);
+\end{tikzpicture}
+\end{codeexample}
+
+While this may be nice in other situations, this is not really leading anywhere
+with Karl's problem: First, we would need an awful lot of these rectangles and
+then there is the border that is not ``closed''.
+
+So, Karl is about to resort to simply drawing four vertical and four horizontal
+lines using the nice |\draw| command, when he learns that there is a |grid|
+path construction operation.
+
+
+\subsection{Grid Path Construction}
+
+The |grid| path operation adds a grid to the current path. It will add lines
+making up a grid that fills the rectangle whose one corner is the current point
+and whose other corner is the point following the |grid| operation. For
+example, the code |\tikz \draw[step=2pt] (0,0) grid (10pt,10pt);| produces
+\tikz \draw[step=2pt] (0,0) grid (10pt,10pt);. Note how the optional argument
+for |\draw| can be used to specify a grid width (there are also |xstep| and
+|ystep| to define the steppings independently). As Karl will learn soon, there
+are \emph{lots} of things that can be influenced using such options.
+
+For Karl, the following code could be used:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \draw[step=.5cm] (-1.4,-1.4) grid (1.4,1.4);
+\end{tikzpicture}
+\end{codeexample}
+
+Having another look at the desired picture, Karl notices that it would be nice
+for the grid to be more subdued. (His son told him that grids tend to be
+distracting if they are not subdued.) To subdue the grid, Karl adds two more
+options to the |\draw| command that draws the grid. First, he uses the color
+|gray| for the grid lines. Second, he reduces the line width to |very thin|.
+Finally, he swaps the ordering of the commands so that the grid is drawn first
+and everything else on top.
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsection{Adding a Touch of  Style}
+
+Instead of the options |gray,very thin| Karl could also have said |help lines|.
+\emph{Styles} are predefined sets of options that can be used to organize how a
+graphic is drawn. By saying |help lines| you say ``use the style that I (or
+someone else) has set for drawing help lines''. If Karl decides, at some later
+point, that grids should be drawn, say, using the color |blue!50| instead of
+|gray|, he could provide the following option somewhere:
+%
+\begin{codeexample}[code only]
+help lines/.style={color=blue!50,very thin}
+\end{codeexample}
+%
+The effect of this ``style setter'' is that in the current scope or environment
+the |help lines| option has the same effect as |color=blue!50,very thin|.
+
+Using styles makes your graphics code more flexible. You can change the way
+things look easily in a consistent manner. Normally, styles are defined at the
+beginning of a picture. However, you may sometimes wish to define a style
+globally, so that all pictures of your document can use this style. Then you
+can easily change the way all graphics look by changing this one style. In this
+situation you can use the |\tikzset| command at the beginning of the document
+as in
+%
+\begin{codeexample}[code only]
+\tikzset{help lines/.style=very thin}
+\end{codeexample}
+
+To build a hierarchy of styles you can have one style use another. So in order
+to define a style |Karl's grid| that is based on the |grid| style Karl could
+say
+%
+\begin{codeexample}[code only]
+\tikzset{Karl's grid/.style={help lines,color=blue!50}}
+...
+\draw[Karl's grid] (0,0) grid (5,5);
+\end{codeexample}
+
+Styles are made even more powerful by parametrization. This means that, like
+other options, styles can also be used with a parameter. For instance, Karl
+could parameterize his grid so that, by default, it is blue, but he could also
+use another color.
+%
+\begin{codeexample}[code only]
+\begin{tikzpicture}
+  [Karl's grid/.style  ={help lines,color=#1!50},
+   Karl's grid/.default=blue]
+
+  \draw[Karl's grid]     (0,0) grid (1.5,2);
+  \draw[Karl's grid=red] (2,0) grid (3.5,2);
+\end{tikzpicture}
+\end{codeexample}
+
+ In this example, the definition of the style |Karl's grid| is given as an
+ optional argument to the |{tikzpicture}| environment. Additional styles for other
+ elements would follow after a comma. With many styles in effect, the optional
+ argument of the environment may easily happen to be longer than the actual
+ contents.
+
+\subsection{Drawing Options}
+
+Karl wonders what other options there are that influence how a path is drawn.
+He saw already that the |color=|\meta{color} option can be used to set the
+line's color. The option |draw=|\meta{color} does nearly the same, only it sets
+the color for the lines only and a different color can be used for filling
+(Karl will need this when he fills the arc for the angle).
+
+He saw that the style |very thin| yields very thin lines. Karl is not really
+surprised by this and neither is he surprised to learn that |thin| yields thin
+lines,  |thick| yields thick lines, |very thick| yields very thick lines,
+|ultra thick| yields really, really thick lines and |ultra thin| yields lines
+that are so thin that low-resolution printers and displays will have trouble
+showing them. He wonders what gives lines of ``normal'' thickness. It turns out
+that |thin| is the correct choice, since it gives the same thickness as \TeX's
+|\hrule| command. Nevertheless, Karl would like to know whether there is
+anything ``in the middle'' between |thin| and |thick|. There is: |semithick|.
+
+Another useful thing one can do with lines is to dash or dot them. For this,
+the two styles |dashed| and |dotted| can be used, yielding \tikz[baseline]
+\draw[dashed] (0,.5ex) -- ++(2em,0pt); and \tikz[baseline] \draw[dotted]
+(0,.5ex) -- ++(2em,0pt);. Both options also exist in a loose and a dense
+version, called |loosely dashed|, |densely dashed|, |loosely dotted|, and
+|densely dotted|. If he really, really  needs to, Karl can also define much
+more complex dashing patterns with the |dash pattern| option, but his son
+insists that dashing is to be used with utmost care and mostly distracts.
+Karl's son claims that complicated dashing patterns are evil. Karl's students
+do not care about dashing patterns.
+
+
+\subsection{Arc Path Construction}
+
+Our next obstacle is to draw the arc for the angle. For this, the |arc| path
+construction operation is useful, which draws part of a circle or ellipse. This
+|arc| operation is followed by options in brackets that specify the arc. An
+example would be \texttt{arc[start angle=10, end angle=80, radius=10pt]}, which
+means exactly what it says. Karl obviously needs an arc from $0^\circ$ to
+$30^\circ$. The radius should be something relatively small, perhaps around one
+third of the circle's radius. When one uses the arc path construction
+operation, the specified arc will be added with its starting point at the
+current position. So, we first have to ``get there''.
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \draw (3mm,0mm) arc [start angle=0, end angle=30, radius=3mm];
+\end{tikzpicture}
+\end{codeexample}
+
+Karl thinks this is really a bit small and he cannot continue unless he learns
+how to do scaling. For this, he can add the |[scale=3]| option. He could add
+this option to each |\draw| command, but that would be awkward. Instead, he
+adds it to the whole environment, which causes this option to apply to
+everything within.
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \draw (3mm,0mm) arc [start angle=0, end angle=30, radius=3mm];
+\end{tikzpicture}
+\end{codeexample}
+
+As for circles, you can specify ``two'' radii in order to get an elliptical
+arc.
+%
+\begin{codeexample}[]
+  \tikz \draw (0,0)
+    arc [start angle=0, end angle=315,
+         x radius=1.75cm, y radius=1cm];
+\end{codeexample}
+
+
+\subsection{Clipping a Path}
+
+In order to save space in this manual, it would be nice to clip Karl's graphics
+a bit so that we can focus on the ``interesting'' parts. Clipping is pretty
+easy in \tikzname. You can use the |\clip| command to clip all subsequent
+drawing. It works like |\draw|, only it does not draw anything, but uses the
+given path to clip everything subsequently.
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,0.75);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \draw (3mm,0mm) arc [start angle=0, end angle=30, radius=3mm];
+\end{tikzpicture}
+\end{codeexample}
+
+You can also do both at the same time: Draw \emph{and} clip a path. For this,
+use the |\draw| command and add the |clip| option. (This is not the whole
+picture: You can also use the |\clip| command and add the |draw| option. Well,
+that is also not the whole picture: In reality, |\draw| is just a shorthand for
+|\path[draw]| and |\clip| is a shorthand for |\path[clip]| and you could also
+say |\path[draw,clip]|.) Here is an example:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip[draw] (0.5,0.5) circle (.6cm);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \draw (3mm,0mm) arc [start angle=0, end angle=30, radius=3mm];
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsection{Parabola and Sine Path Construction}
+
+Although Karl does not need them for his picture, he is pleased to learn that
+there are |parabola| and |sin| and |cos| path operations for adding parabolas
+and sine and cosine curves to the current path. For the |parabola| operation,
+the current point will lie on the parabola as well as the point given after the
+parabola operation. Consider the following example:
+%
+\begin{codeexample}[]
+\tikz \draw (0,0) rectangle (1,1)  (0,0) parabola (1,1);
+\end{codeexample}
+
+It is also possible to place the bend somewhere else:
+%
+\begin{codeexample}[]
+\tikz \draw[x=1pt,y=1pt] (0,0) parabola bend (4,16) (6,12);
+\end{codeexample}
+
+The operations |sin| and |cos| add a sine or cosine curve in the interval
+$[0,\pi/2]$ such that the previous current point is at the start of the curve
+and the curve ends at the given end point. Here are two examples:
+%
+\begin{codeexample}[]
+A sine \tikz \draw[x=1ex,y=1ex] (0,0) sin (1.57,1); curve.
+\end{codeexample}
+
+\begin{codeexample}[]
+\tikz \draw[x=1.57ex,y=1ex] (0,0) sin (1,1) cos (2,0) sin (3,-1) cos (4,0)
+                            (0,1) cos (1,0) sin (2,-1) cos (3,0) sin (4,1);
+\end{codeexample}
+
+
+\subsection{Filling and Drawing}
+
+Returning to the picture, Karl now wants the angle to be ``filled'' with a very
+light green. For this he uses |\fill| instead of |\draw|. Here is what Karl
+does:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,0.75);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \fill[green!20!white] (0,0) -- (3mm,0mm)
+    arc [start angle=0, end angle=30, radius=3mm] -- (0,0);
+\end{tikzpicture}
+\end{codeexample}
+
+The color |green!20!white| means 20\% green and 80\% white mixed together. Such
+color expression are possible since \tikzname\ uses Uwe Kern's |xcolor|
+package, see the documentation of that package for details on color
+expressions.
+
+What would have happened, if Karl had not ``closed'' the path using |--(0,0)|
+at the end? In this case, the path is closed automatically, so this could have
+been omitted. Indeed, it would even have been better to write the following,
+instead:
+%
+\begin{codeexample}[code only]
+  \fill[green!20!white] (0,0) -- (3mm,0mm)
+    arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+\end{codeexample}
+%
+The |--cycle| causes the current path to be closed (actually the current part
+of the current path) by smoothly joining the first and last point. To
+appreciate the difference, consider the following example:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[line width=5pt]
+  \draw (0,0) -- (1,0) -- (1,1) -- (0,0);
+  \draw (2,0) -- (3,0) -- (3,1) -- cycle;
+  \useasboundingbox (0,1.5); % make bounding box higher
+\end{tikzpicture}
+\end{codeexample}
+
+You can also fill and draw a path at the same time using the |\filldraw|
+command. This will first draw the path, then fill it. This may not seem too
+useful, but you can specify different colors to be used for filling and for
+stroking. These are specified as optional arguments like this:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,0.75);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \filldraw[fill=green!20!white, draw=green!50!black] (0,0) -- (3mm,0mm)
+    arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsection{Shading}
+
+Karl briefly considers the possibility of making the angle ``more fancy'' by
+\emph{shading} it. Instead of filling the area with a uniform color, a smooth
+transition between different colors is used. For this, |\shade| and
+|\shadedraw|, for shading and drawing at the same time, can be used:
+%
+\begin{codeexample}[]
+  \tikz \shade (0,0) rectangle (2,1)  (3,0.5) circle (.5cm);
+\end{codeexample}
+%
+The default shading is a smooth transition from gray to white. To specify
+different colors, you can use options:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[rounded corners,ultra thick]
+  \shade[top color=yellow,bottom color=black] (0,0) rectangle +(2,1);
+  \shade[left color=yellow,right color=black] (3,0) rectangle +(2,1);
+  \shadedraw[inner color=yellow,outer color=black,draw=yellow] (6,0) rectangle +(2,1);
+  \shade[ball color=green] (9,.5) circle (.5cm);
+\end{tikzpicture}
+\end{codeexample}
+
+For Karl, the following might be appropriate:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,0.75);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \shadedraw[left color=gray,right color=green, draw=green!50!black]
+    (0,0) -- (3mm,0mm)
+    arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+\end{tikzpicture}
+\end{codeexample}
+
+However, he wisely decides that shadings usually only distract without adding
+anything to the picture.
+
+
+\subsection{Specifying Coordinates}
+
+Karl now wants to add the sine and cosine lines. He knows already that he can
+use the |color=| option to set the lines' colors. So, what is the best way to
+specify the coordinates?
+
+There are different ways of specifying coordinates. The easiest way is to say
+something like |(10pt,2cm)|. This means 10pt in $x$-direction and 2cm in
+$y$-directions. Alternatively, you can also leave out the units as in |(1,2)|,
+which means ``one times the current $x$-vector plus twice the current
+$y$-vector''. These vectors default to 1cm in the $x$-direction and 1cm in the
+$y$-direction, respectively.
+
+In order to specify points in polar coordinates, use the notation |(30:1cm)|,
+which means 1cm in direction 30 degree. This is obviously quite useful to ``get
+to the point $(\cos 30^\circ,\sin 30^\circ)$ on the circle''.
+
+You can add a single |+| sign in front of a coordinate or two of them as in
+|+(0cm,1cm)| or |++(2cm,0cm)|. Such coordinates are interpreted differently:
+The first form means ``1cm upwards from the previous specified position'' and
+the second means ``2cm to the right of the previous specified position, making
+this the new specified position''. For example, we can draw the sine line as
+follows:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,0.75);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+      arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[red,very thick] (30:1cm) -- +(0,-0.5);
+\end{tikzpicture}
+\end{codeexample}
+
+Karl used the fact $\sin 30^\circ = 1/2$. However, he very much doubts that his
+students know this, so it would be nice to have a way of specifying ``the point
+straight down from |(30:1cm)| that lies on the $x$-axis''. This is, indeed,
+possible using a special syntax: Karl can write \verb!(30:1cm |- 0,0)!. In
+general, the meaning of |(|\meta{p}\verb! |- !\meta{q}|)| is ``the intersection
+of a vertical line through $p$ and a horizontal line through $q$''.
+
+Next, let us draw the cosine line. One way would be to say
+\verb!(30:1cm |- 0,0) -- (0,0)!. Another way is the following: we ``continue''
+from where the sine ends:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,0.75);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw (-1.5,0) -- (1.5,0);
+  \draw (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+      arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[red,very thick]  (30:1cm) -- +(0,-0.5);
+  \draw[blue,very thick] (30:1cm) ++(0,-0.5) -- (0,0);
+\end{tikzpicture}
+\end{codeexample}
+
+Note that there is no |--| between |(30:1cm)| and |++(0,-0.5)|. In detail, this
+path is interpreted as follows: ``First, the |(30:1cm)| tells me to move my pen
+to $(\cos 30^\circ,1/2)$. Next, there comes another coordinate specification,
+so I move my pen there without drawing anything. This new point is half a unit
+down from the last position, thus it is at $(\cos 30^\circ,0)$. Finally, I move
+the pen to the origin, but this time drawing something (because of the |--|).''
+
+To appreciate the difference between |+| and |++| consider the following
+example:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \def\rectanglepath{-- ++(1cm,0cm)  -- ++(0cm,1cm)  -- ++(-1cm,0cm) -- cycle}
+  \draw (0,0) \rectanglepath;
+  \draw (1.5,0) \rectanglepath;
+\end{tikzpicture}
+\end{codeexample}
+
+By comparison, when using a single |+|, the coordinates are different:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \def\rectanglepath{-- +(1cm,0cm)  -- +(1cm,1cm)  -- +(0cm,1cm) -- cycle}
+  \draw (0,0) \rectanglepath;
+  \draw (1.5,0) \rectanglepath;
+\end{tikzpicture}
+\end{codeexample}
+
+
+Naturally, all of this could have been written more clearly and more
+economically like this (either with a single or a double |+|):
+%
+\begin{codeexample}[]
+\tikz \draw (0,0) rectangle +(1,1)  (1.5,0) rectangle +(1,1);
+\end{codeexample}
+
+
+\subsection{Intersecting Paths}
+
+Karl is left with the line for $\tan \alpha$, which seems difficult to specify
+using transformations and polar coordinates. The first -- and easiest -- thing
+he can do is so simply use the coordinate |(1,{tan(30)})| since \tikzname's
+math engine knows how to compute things like |tan(30)|. Note the added braces
+since, otherwise, \tikzname's parser would think that the first closing
+parenthesis ends the coordinate (in general, you need to add braces around
+components of coordinates when these components contain parentheses).
+
+Karl can, however, also use a more elaborate, but also more ``geometric'' way
+of computing the length of the orange line: He can specify intersections of
+paths as coordinates. The line for $\tan \alpha$ starts at $(1,0)$ and goes
+upward to a point that is at the intersection of a line going ``up'' and a line
+going from the origin through |(30:1cm)|. Such computations are made available
+by the |intersections| library.
+
+What Karl must do is to create two ``invisible'' paths that intersect at the
+position of interest. Creating paths that are not otherwise seen can be done
+using the |\path| command without any options like |draw| or |fill|. Then, Karl
+can add the |name path| option to the path for later reference. Once the paths
+have been constructed, Karl can use the |name intersections| to assign names to
+the coordinate for later reference.
+%
+\begin{codeexample}[code only]
+\path [name path=upward line] (1,0) -- (1,1);
+\path [name path=sloped line] (0,0) -- (30:1.5cm); % a bit longer, so that there is an intersection
+
+% (add `\usetikzlibrary{intersections}' after loading tikz in the preamble)
+\draw [name intersections={of=upward line and sloped line, by=x}]
+  [very thick,orange] (1,0) -- (x);
+\end{codeexample}
+
+
+\subsection{Adding Arrow Tips}
+
+Karl now wants to add the little arrow tips at the end of the axes. He has
+noticed that in many plots, even in scientific journals, these arrow tips seem
+to be missing, presumably because the generating programs cannot produce them.
+Karl thinks arrow tips belong at the end of axes. His son agrees. His students
+do not care about arrow tips.
+
+It turns out that adding arrow tips is pretty easy: Karl adds the option |->|
+to the drawing commands for the axes:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,1.51);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \draw[->] (-1.5,0) -- (1.5,0);
+  \draw[->] (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+        arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[red,very thick]    (30:1cm) -- +(0,-0.5);
+  \draw[blue,very thick]   (30:1cm) ++(0,-0.5) -- (0,0);
+
+  \path [name path=upward line] (1,0) -- (1,1);
+  \path [name path=sloped line] (0,0) -- (30:1.5cm);
+  \draw [name intersections={of=upward line and sloped line, by=x}]
+        [very thick,orange] (1,0) -- (x);
+\end{tikzpicture}
+\end{codeexample}
+
+If Karl had used the option |<-| instead of |->|, arrow tips would have been
+put at the beginning of the path. The option |<->| puts arrow tips at both ends
+of the path.
+
+There are certain restrictions to the kind of paths to which arrow tips can be
+added. As a rule of thumb, you can add arrow tips only to a single open
+``line''. For example, you cannot add tips to, say, a rectangle or a circle.
+However, you can add arrow tips to curved paths and to paths that have several
+segments, as in the following examples:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw [<->] (0,0) arc [start angle=180, end angle=30, radius=10pt];
+  \draw [<->] (1,0) -- (1.5cm,10pt) -- (2cm,0pt) -- (2.5cm,10pt);
+\end{tikzpicture}
+\end{codeexample}
+
+Karl has a more detailed look at the arrow that \tikzname\ puts at the end. It
+looks like this when he zooms it: \tikz[baseline] \draw[->,line width=1pt]
+(0pt,.5ex) -- ++(10pt,0pt);. The shape seems vaguely familiar and, indeed, this
+is exactly the end of \TeX's standard arrow used in something like $f\colon A
+\to B$.
+
+Karl likes the arrow, especially since it is not ``as thick'' as the arrows
+offered by many other packages. However, he expects that, sometimes, he might
+need to use some other kinds of arrow. To do so, Karl can say |>=|\meta{kind of
+end arrow tip}, where \meta{kind of end arrow tip} is a special arrow tip
+specification. For example, if Karl says |>=Stealth|, then he tells \tikzname\
+that he would like  ``stealth-fighter-like'' arrow tips:
+\todosp{remaining instance of bug \#473}
+%
+\begin{codeexample}[preamble={\usetikzlibrary{arrows.meta}}]
+\begin{tikzpicture}[>=Stealth]
+  \draw [->] (0,0) arc [start angle=180, end angle=30, radius=10pt];
+  \draw [<<-,very thick] (1,0) -- (1.5cm,10pt) -- (2cm,0pt) -- (2.5cm,10pt);
+\end{tikzpicture}
+\end{codeexample}
+
+Karl wonders whether such a military name for the arrow type is really
+necessary. He is not really mollified when his son tells him that Microsoft's
+PowerPoint uses the same name. He decides to have his students discuss this at
+some point.
+
+In addition to |Stealth| there are several other predefined kinds of arrow tips
+Karl can choose from, see Section~\ref{section-arrows}. Furthermore, he can
+define arrows types himself, if he needs new ones.
+
+
+\subsection{Scoping}
+
+Karl saw already that there are numerous graphic options that affect how paths
+are rendered. Often, he would like to apply certain options to a whole set of
+graphic commands. For example, Karl might wish to draw three paths using a
+|thick| pen, but would like everything else to be drawn ``normally''.
+
+If Karl wishes to set a certain graphic option for the whole picture, he can
+simply pass this option to the |\tikz| command or to the |{tikzpicture}|
+environment (Gerda would pass the options to |\tikzpicture| and Hans passes
+them to |\starttikzpicture|). However, if Karl wants to apply graphic options
+to a local group, he put these commands inside a |{scope}| environment (Gerda
+uses |\scope| and |\endscope|, Hans uses |\startscope| and |\stopscope|). This
+environment takes graphic options as an optional argument and these options
+apply to everything inside the scope, but not to anything outside.
+
+Here is an example:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[ultra thick]
+  \draw (0,0) -- (0,1);
+  \begin{scope}[thin]
+    \draw (1,0) -- (1,1);
+    \draw (2,0) -- (2,1);
+  \end{scope}
+  \draw (3,0) -- (3,1);
+\end{tikzpicture}
+\end{codeexample}
+
+Scoping has another interesting effect: Any changes to the clipping area are
+local to the scope. Thus, if you say |\clip| somewhere inside a scope, the
+effect of the |\clip| command ends at the end of the scope. This is useful
+since there is no other way of ``enlarging'' the clipping area.
+
+Karl has also already seen that giving options to commands like |\draw| apply
+only to that command. It turns out that the situation is slightly more complex.
+First, options to a command like |\draw| are not really options to the command,
+but they are ``path options'' and can be given anywhere on the path. So,
+instead of |\draw[thin] (0,0) -- (1,0);| one can also write
+|\draw (0,0) [thin] -- (1,0);| or |\draw (0,0) -- (1,0) [thin];|; all of these
+have the same effect. This might seem strange since in the last case, it would
+appear that the |thin| should take effect only ``after'' the line from $(0,0)$
+to $(1,0)$ has been drawn. However, most graphic options only apply to the
+whole path. Indeed, if you say both |thin| and |thick| on the same path, the
+last option given will ``win''.
+
+When reading the above, Karl notices that only ``most'' graphic options apply
+to the whole path. Indeed, all transformation options do \emph{not} apply to
+the whole path, but only to ``everything following them on the path''. We will
+have a more detailed look at this in a moment. Nevertheless, all options given
+during a path construction apply only to this path.
+
+
+\subsection{Transformations}
+
+When you specify a  coordinate like |(1cm,1cm)|, where is that coordinate
+placed on the page? To determine the position, \tikzname, \TeX, and
+\textsc{pdf} or PostScript all apply certain transformations to the given
+coordinate in order to determine the final position on the page.
+
+\tikzname\ provides numerous options that allow you to transform coordinates in
+\tikzname's private coordinate system. For example, the |xshift| option allows
+you to shift all subsequent points by a certain amount:
+
+\begin{codeexample}[]
+\tikz \draw (0,0) -- (0,0.5) [xshift=2pt] (0,0) -- (0,0.5);
+\end{codeexample}
+
+It is important to note that you can change transformation ``in the middle of a
+path'', a feature that is not supported by \pdf\ or PostScript. The reason is
+that \tikzname\ keeps track of its own transformation matrix.
+
+Here is a more complicated example:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[even odd rule,rounded corners=2pt,x=10pt,y=10pt]
+  \filldraw[fill=yellow!80!black] (0,0)   rectangle (1,1)
+        [xshift=5pt,yshift=5pt]   (0,0)   rectangle (1,1)
+                    [rotate=30]   (-1,-1) rectangle (2,2);
+\end{tikzpicture}
+\end{codeexample}
+
+The most useful transformations are |xshift| and |yshift| for shifting, |shift|
+for shifting to a given point as in |shift={(1,0)}| or |shift={+(0,0)}| (the
+braces are necessary so that \TeX\ does not mistake the comma for separating
+options), |rotate| for rotating by a certain angle (there is also a
+|rotate around| for rotating around a given point), |scale| for scaling by a
+certain factor, |xscale| and |yscale| for scaling only in the $x$- or
+$y$-direction (|xscale=-1| is a flip), and |xslant| and |yslant| for slanting.
+If these transformation and those that I have not mentioned are not sufficient,
+the |cm| option allows you to apply an arbitrary transformation matrix. Karl's
+students, by the way, do not know what a transformation matrix is.
+
+
+\subsection{Repeating Things: For-Loops}
+
+Karl's next aim is to add little ticks on the axes at positions $-1$, $-1/2$,
+$1/2$, and $1$. For this, it would be nice to use some kind of ``loop'',
+especially since he wishes to do the same thing at each of these positions.
+There are different packages for doing this. \LaTeX\ has its own internal
+command for this, |pstricks| comes along with the powerful |\multido| command.
+All of these can be used together with \tikzname, so if you are familiar with
+them, feel free to use them. \tikzname\ introduces yet another command, called
+|\foreach|, which I introduced since I could never remember the syntax of the
+other packages. |\foreach| is defined in the package |pgffor| and can be used
+independently of \tikzname, but \tikzname\ includes it automatically.
+
+In its basic form, the |\foreach| command is easy to use:
+%
+\begin{codeexample}[]
+\foreach \x in {1,2,3} {$x =\x$, }
+\end{codeexample}
+
+The general syntax is
+|\foreach| \meta{variable}| in {|\meta{list of values}|} |\meta{commands}.
+Inside the \meta{commands}, the \meta{variable} will be assigned to the
+different values. If the \meta{commands} do not start with a brace, everything
+up to the next semicolon is used as \meta{commands}.
+
+For Karl and the ticks on the axes, he could use the following code:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.1,-0.2) rectangle (1.1,1.51);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+      arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[->] (-1.5,0) -- (1.5,0);
+  \draw[->] (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+
+  \foreach \x in {-1cm,-0.5cm,1cm}
+    \draw (\x,-1pt) -- (\x,1pt);
+  \foreach \y in {-1cm,-0.5cm,0.5cm,1cm}
+    \draw (-1pt,\y) -- (1pt,\y);
+\end{tikzpicture}
+\end{codeexample}
+
+As a matter of fact, there are many different ways of creating the ticks. For
+example, Karl could have put the |\draw ...;| inside curly braces. He could
+also have used, say,
+%
+\begin{codeexample}[code only]
+\foreach \x in {-1,-0.5,1}
+  \draw[xshift=\x cm] (0pt,-1pt) -- (0pt,1pt);
+\end{codeexample}
+
+Karl is curious what would happen in a more complicated situation where there
+are, say, 20 ticks. It seems bothersome to explicitly mention all these numbers
+in the set for |\foreach|. Indeed, it is possible to use |...| inside the
+|\foreach| statement to iterate over a large number of values (which must,
+however, be dimensionless real numbers) as in the following example:
+%
+\begin{codeexample}[]
+\tikz \foreach \x in {1,...,10}
+        \draw (\x,0) circle (0.4cm);
+\end{codeexample}
+
+If you provide \emph{two} numbers before the |...|, the |\foreach| statement
+will use their difference for the stepping:
+%
+\begin{codeexample}[]
+\tikz \foreach \x in {-1,-0.5,...,1}
+       \draw (\x cm,-1pt) -- (\x cm,1pt);
+\end{codeexample}
+
+We can also nest loops to create interesting effects:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \foreach \x in {1,2,...,5,7,8,...,12}
+    \foreach \y in {1,...,5}
+    {
+      \draw (\x,\y) +(-.5,-.5) rectangle ++(.5,.5);
+      \draw (\x,\y) node{\x,\y};
+    }
+\end{tikzpicture}
+\end{codeexample}
+
+The |\foreach| statement can do even trickier stuff, but the above gives the
+idea.
+
+
+\subsection{Adding Text}
+
+Karl is, by now, quite satisfied with the picture. However, the most important
+parts, namely the labels, are still missing!
+
+\tikzname\ offers an easy-to-use and powerful system for adding text and, more
+generally, complex shapes to a picture at specific positions. The basic idea is
+the following: When \tikzname\ is constructing a path and encounters the
+keyword |node| in the middle of a path, it reads a \emph{node specification}.
+The keyword |node| is typically followed by some options and then some text
+between curly braces. This text is put inside a normal \TeX\ box (if the node
+specification directly follows a coordinate, which is usually the case,
+\tikzname\ is able to perform some magic so that it is even possible to use
+verbatim text inside the boxes) and then placed at the current position, that
+is, at the last specified position (possibly shifted a bit, according to the
+given options). However, all nodes are drawn only after the path has been
+completely drawn/filled/shaded/clipped/whatever.
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw (0,0) rectangle (2,2);
+  \draw (0.5,0.5) node [fill=yellow!80!black]
+                       {Text at \verb!node 1!}
+     -- (1.5,1.5) node {Text at \verb!node 2!};
+\end{tikzpicture}
+\end{codeexample}
+
+Obviously, Karl would not only like to place nodes \emph{on} the last specified
+position, but also to the left or the right of these positions. For this, every
+node object that you put in your picture is equipped with several
+\emph{anchors}. For example, the |north| anchor is in the middle at the upper
+end of the shape, the |south| anchor is at the bottom and the |north east|
+anchor is in the upper right corner. When you give the option |anchor=north|,
+the text will be placed such that this northern anchor will lie on the current
+position and the text is, thus, below the current position. Karl uses this to
+draw the ticks as follows:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.6,-0.2) rectangle (0.6,1.51);
+  \draw[step=.5cm,help lines] (-1.4,-1.4) grid (1.4,1.4);
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+    arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[->] (-1.5,0) -- (1.5,0);   \draw[->] (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+
+  \foreach \x in {-1,-0.5,1}
+    \draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\x$};
+  \foreach \y in {-1,-0.5,0.5,1}
+    \draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\y$};
+\end{tikzpicture}
+\end{codeexample}
+
+This is quite nice, already. Using these anchors, Karl can now add most of the
+other text elements. However, Karl thinks that, though ``correct'', it is quite
+counter-intuitive that in order to place something \emph{below} a given point,
+he has to use the \emph{north} anchor. For this reason, there is an option
+called |below|, which does the same as |anchor=north|. Similarly, |above right|
+does the same as |anchor=south west|. In addition, |below| takes an optional
+dimension argument. If given, the shape will additionally be shifted downwards
+by the given amount. So, |below=1pt| can be used to put a text label below some
+point and, additionally shift it  1pt downwards.
+
+Karl is not quite satisfied with the ticks. He would like to have $1/2$ or
+$\frac{1}{2}$ shown instead of $0.5$, partly to show off the nice capabilities
+of \TeX\ and \tikzname, partly because for positions like $1/3$ or $\pi$ it is
+certainly very much preferable to have the ``mathematical'' tick there instead
+of just the ``numeric'' tick. His students, on the other hand, prefer $0.5$
+over $1/2$ since they are not too fond of fractions in general.
+
+Karl now faces a problem: For the |\foreach| statement, the position |\x|
+should still be given as |0.5| since \tikzname\ will not know where
+|\frac{1}{2}| is supposed to be. On the other hand, the typeset text should
+really be  |\frac{1}{2}|. To solve this problem, |\foreach| offers a special
+syntax: Instead of having one variable |\x|, Karl can specify two (or even
+more) variables separated by a slash as in |\x / \xtext|. Then, the elements in
+the set over which |\foreach| iterates must also be of the form
+\meta{first}|/|\meta{second}. In each iteration, |\x| will be set to
+\meta{first} and |\xtext| will be set to \meta{second}. If no \meta{second} is
+given, the \meta{first} will be used again. So, here is the new code for the
+ticks:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}[scale=3]
+  \clip (-0.6,-0.2) rectangle (0.6,1.51);
+  \draw[step=.5cm,help lines] (-1.4,-1.4) grid (1.4,1.4);
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+      arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[->] (-1.5,0) -- (1.5,0); \draw[->] (0,-1.5) -- (0,1.5);
+  \draw (0,0) circle [radius=1cm];
+
+  \foreach \x/\xtext in {-1, -0.5/-\frac{1}{2}, 1}
+    \draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north] {$\xtext$};
+  \foreach \y/\ytext in {-1, -0.5/-\frac{1}{2}, 0.5/\frac{1}{2}, 1}
+    \draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east] {$\ytext$};
+\end{tikzpicture}
+\end{codeexample}
+
+Karl is quite pleased with the result, but his son points out that this is
+still not perfectly satisfactory: The grid and the circle interfere with the
+numbers and decrease their legibility. Karl is not very concerned by this (his
+students do not even notice), but his son insists that there is an easy
+solution: Karl can add the |[fill=white]| option to fill out the background of
+the text shape with a white color.
+
+The next thing Karl wants to do is to add the labels like $\sin \alpha$. For
+this, he would like to place a label ``in the middle of the line''. To do so,
+instead of specifying the label |node {$\sin\alpha$}|  directly after one of
+the endpoints of the line (which would place the label at that endpoint), Karl
+can give the label directly after the |--|, before the coordinate. By default,
+this places the label in the middle of the line, but the |pos=| options can be
+used to modify this. Also, options like |near start| and |near end| can be used
+to modify this position:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}[scale=3]
+  \clip (-2,-0.2) rectangle (2,0.8);
+  \draw[step=.5cm,gray,very thin] (-1.4,-1.4) grid (1.4,1.4);
+  \filldraw[fill=green!20,draw=green!50!black] (0,0) -- (3mm,0mm)
+    arc [start angle=0, end angle=30, radius=3mm] -- cycle;
+  \draw[->] (-1.5,0) -- (1.5,0) coordinate (x axis);
+  \draw[->] (0,-1.5) -- (0,1.5) coordinate (y axis);
+  \draw (0,0) circle [radius=1cm];
+
+  \draw[very thick,red]
+    (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} (30:1cm |- x axis);
+  \draw[very thick,blue]
+    (30:1cm |- x axis) -- node[below=2pt,fill=white] {$\cos \alpha$} (0,0);
+  \path [name path=upward line] (1,0) -- (1,1);
+  \path [name path=sloped line] (0,0) -- (30:1.5cm);
+  \draw [name intersections={of=upward line and sloped line, by=t}]
+    [very thick,orange] (1,0) -- node [right=1pt,fill=white]
+    {$\displaystyle \tan \alpha \color{black}=
+      \frac{{\color{red}\sin \alpha}}{\color{blue}\cos \alpha}$} (t);
+
+  \draw (0,0) -- (t);
+
+  \foreach \x/\xtext in {-1, -0.5/-\frac{1}{2}, 1}
+    \draw (\x cm,1pt) -- (\x cm,-1pt) node[anchor=north,fill=white] {$\xtext$};
+  \foreach \y/\ytext in {-1, -0.5/-\frac{1}{2}, 0.5/\frac{1}{2}, 1}
+    \draw (1pt,\y cm) -- (-1pt,\y cm) node[anchor=east,fill=white] {$\ytext$};
+\end{tikzpicture}
+\end{codeexample}
+
+You can also position labels on curves and, by adding the |sloped| option, have
+them rotated such that they match the line's slope. Here is an example:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+  \draw (0,0) .. controls (6,1) and (9,1) ..
+    node[near start,sloped,above] {near start}
+    node {midway}
+    node[very near end,sloped,below] {very near end} (12,0);
+\end{tikzpicture}
+\end{codeexample}
+
+It remains to draw the explanatory text at the right of the picture. The main
+difficulty here lies in limiting the width of the text ``label'', which is
+quite long, so that line breaking is used. Fortunately, Karl can use the option
+|text width=6cm| to get the desired effect. So, here is the full code:
+%
+\begin{codeexample}[code only]
+\begin{tikzpicture}
+  [scale=3,line cap=round,
+  % Styles
+  axes/.style=,
+  important line/.style={very thick},
+  information text/.style={rounded corners,fill=red!10,inner sep=1ex}]
+
+  % Colors
+  \colorlet{anglecolor}{green!50!black}
+  \colorlet{sincolor}{red}
+  \colorlet{tancolor}{orange!80!black}
+  \colorlet{coscolor}{blue}
+
+  % The graphic
+  \draw[help lines,step=0.5cm] (-1.4,-1.4) grid (1.4,1.4);
+
+  \draw (0,0) circle [radius=1cm];
+
+  \begin{scope}[axes]
+    \draw[->] (-1.5,0) -- (1.5,0) node[right] {$x$} coordinate(x axis);
+    \draw[->] (0,-1.5) -- (0,1.5) node[above] {$y$} coordinate(y axis);
+
+    \foreach \x/\xtext in {-1, -.5/-\frac{1}{2}, 1}
+      \draw[xshift=\x cm] (0pt,1pt) -- (0pt,-1pt) node[below,fill=white] {$\xtext$};
+
+    \foreach \y/\ytext in {-1, -.5/-\frac{1}{2}, .5/\frac{1}{2}, 1}
+      \draw[yshift=\y cm] (1pt,0pt) -- (-1pt,0pt) node[left,fill=white] {$\ytext$};
+  \end{scope}
+
+  \filldraw[fill=green!20,draw=anglecolor] (0,0) -- (3mm,0pt)
+    arc [start angle=0, end angle=30, radius=3mm];
+  \draw (15:2mm) node[anglecolor] {$\alpha$};
+
+  \draw[important line,sincolor]
+    (30:1cm) -- node[left=1pt,fill=white] {$\sin \alpha$} (30:1cm |- x axis);
+
+  \draw[important line,coscolor]
+    (30:1cm |- x axis) -- node[below=2pt,fill=white] {$\cos \alpha$} (0,0);
+
+  \path [name path=upward line] (1,0) -- (1,1);
+  \path [name path=sloped line] (0,0) -- (30:1.5cm);
+  \draw [name intersections={of=upward line and sloped line, by=t}]
+    [very thick,orange] (1,0) -- node [right=1pt,fill=white]
+    {$\displaystyle \tan \alpha \color{black}=
+      \frac{{\color{red}\sin \alpha}}{\color{blue}\cos \alpha}$} (t);
+
+  \draw (0,0) -- (t);
+
+  \draw[xshift=1.85cm]
+    node[right,text width=6cm,information text]
+    {
+      The {\color{anglecolor} angle $\alpha$} is $30^\circ$ in the
+      example ($\pi/6$ in radians). The {\color{sincolor}sine of
+        $\alpha$}, which is the height of the red line, is
+      \[
+      {\color{sincolor} \sin \alpha} = 1/2.
+      \]
+      By the Theorem of Pythagoras ...
+    };
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsection{Pics: The Angle Revisited}
+
+Karl expects that the code of certain parts of the picture he created might be
+so useful that he might wish to reuse them in the future. A natural thing to do
+is to create \TeX\ macros that store the code he wishes to reuse. However,
+\tikzname\ offers another way that is integrated directly into its parser:
+pics!
+
+A ``pic'' is ``not quite a full picture'', hence the short name. The idea is
+that a pic is simply some code that you can add to a picture at different
+places using the |pic| command whose syntax is almost identical to the |node|
+command. The main difference is that instead of specifying some text in curly
+braces that should be shown, you specify the name of a predefined picture that
+should be shown.
+
+Defining new pics is easy enough, see Section~\ref{section-pics}, but right now
+we just want to use one such predefined pic: the |angle| pic. As the name
+suggests, it is a small drawing of an angle consisting of a little wedge and an
+arc together with some text (Karl needs to load the |angles| library and the
+|quotes| for the following examples). What makes this pic useful is the fact
+that the size of the wedge will be computed automatically.
+
+The |angle| pic draws an angle between the two lines $BA$ and $BC$, where $A$,
+$B$, and $C$ are three coordinates. In our case, $B$ is the origin, $A$ is
+somewhere on the $x$-axis and $C$ is somewhere on a line at $30^\circ$.
+%
+\begin{codeexample}[preamble={\usetikzlibrary{angles,quotes}}]
+\begin{tikzpicture}[scale=3]
+  \coordinate (A) at (1,0);
+  \coordinate (B) at (0,0);
+  \coordinate (C) at (30:1cm);
+
+  \draw (A) -- (B) -- (C)
+        pic [draw=green!50!black, fill=green!20, angle radius=9mm,
+             "$\alpha$"] {angle = A--B--C};
+\end{tikzpicture}
+\end{codeexample}
+
+Let us see, what is happening here. First we have specified three
+\emph{coordinates} using the |\coordinate| command. It allows us to name a
+specific coordinate in the picture. Then comes something that starts as a
+normal |\draw|, but then comes the |pic| command. This command gets lots of
+options and, in curly braces, comes the most important point: We specify that
+we want to add an |angle| pic and this angle should be between the points we
+named |A|, |B|, and |C| (we could use other names). Note that the text that we
+want to be shown in the pic is specified in quotes inside the options of the
+|pic|, not inside the curly braces.
+
+To learn more about pics, please see Section~\ref{section-pics}.
-- 
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