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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{Specifying Coordinates}
+
+\subsection{Overview}
+
+A \emph{coordinate} is a position on the canvas on which your picture is drawn.
+\tikzname\ uses a special syntax for specifying coordinates. Coordinates are
+always put in round brackets. The general syntax is
+\declare{|(|\opt{|[|\meta{options}|]|}\meta{coordinate specification}|)|}.
+
+The \meta{coordinate specification} specifies coordinates using one of many
+different possible \emph{coordinate systems}. Examples are the Cartesian
+coordinate system or polar coordinates or spherical coordinates. No matter
+which coordinate system is used, in the end, a specific point on the canvas is
+represented by the coordinate.
+
+There are two ways of specifying which coordinate system should be used:
+%
+\begin{description}
+ \item[Explicitly] You can specify the coordinate system explicitly. To do
+ so, you give the name of the coordinate system at the beginning,
+ followed by |cs:|, which stands for ``coordinate system'', followed by
+ a specification of the coordinate using the key--value syntax. Thus,
+ the general syntax for \meta{coordinate specification} in the explicit
+ case is |(|\meta{coordinate system}| cs:|\meta{list of key--value pairs
+ specific to the coordinate system}|)|.
+ \item[Implicitly] The explicit specification is often too verbose when
+ numerous coordinates should be given. Because of this, for the
+ coordinate systems that you are likely to use often a special syntax
+ is provided. \tikzname\ will notice when you use a coordinate
+ specified in a special syntax and will choose the correct coordinate
+ system automatically.
+\end{description}
+
+Here is an example in which explicit the coordinate systems are specified
+explicitly:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw[help lines] (0,0) grid (3,2);
+ \draw (canvas cs:x=0cm,y=2mm)
+ -- (canvas polar cs:radius=2cm,angle=30);
+\end{tikzpicture}
+\end{codeexample}
+%
+In the next example, the coordinate systems are implicit:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw[help lines] (0,0) grid (3,2);
+ \draw (0cm,2mm) -- (30:2cm);
+\end{tikzpicture}
+\end{codeexample}
+
+It is possible to give options that apply only to a single coordinate, although
+this makes sense for transformation options only. To give transformation
+options for a single coordinate, give these options at the beginning in
+brackets:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw[help lines] (0,0) grid (3,2);
+ \draw (0,0) -- (1,1);
+ \draw[red] (0,0) -- ([xshift=3pt] 1,1);
+ \draw (1,0) -- +(30:2cm);
+ \draw[red] (1,0) -- +([shift=(135:5pt)] 30:2cm);
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsection{Coordinate Systems}
+
+\subsubsection{Canvas, XYZ, and Polar Coordinate Systems}
+
+Let us start with the basic coordinate systems.
+
+\begin{coordinatesystem}{canvas}
+ The simplest way of specifying a coordinate is to use the |canvas|
+ coordinate system. You provide a dimension $d_x$ using the |x=| option and
+ another dimension $d_y$ using the |y=| option. The position on the canvas
+ is located at the position that is $d_x$ to the right and $d_y$ above the
+ origin.
+
+ \begin{key}{/tikz/cs/x=\meta{dimension} (initially 0pt)}
+ Distance by which the coordinate is to the right of the origin. You can
+ also write things like |1cm+2pt| since the mathematical engine is used
+ to evaluate the \meta{dimension}.
+ \end{key}
+
+ \begin{key}{/tikz/cs/y=\meta{dimension} (initially 0pt)}
+ Distance by which the coordinate is above the origin.
+ \end{key}
+
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw[help lines] (0,0) grid (3,2);
+
+ \fill (canvas cs:x=1cm,y=1.5cm) circle (2pt);
+ \fill (canvas cs:x=2cm,y=-5mm+2pt) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+ To specify a coordinate in the coordinate system implicitly, you use two
+ dimensions that are separated by a comma as in |(0cm,3pt)| or
+ |(2cm,\textheight)|.
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw[help lines] (0,0) grid (3,2);
+
+ \fill (1cm,1.5cm) circle (2pt);
+ \fill (2cm,-5mm+2pt) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+ %
+\end{coordinatesystem}
+
+\begin{coordinatesystem}{xyz}
+ The |xyz| coordinate system allows you to specify a point as a multiple of
+ three vectors called the $x$-, $y$-, and $z$-vectors. By default, the
+ $x$-vector points 1cm to the right, the $y$-vector points 1cm upwards, but
+ this can be changed arbitrarily as explained in Section~\ref{section-xyz}.
+ The default $z$-vector points to
+ $\bigl(-3.85\textrm{mm},-3.85\textrm{mm}\bigr)$.
+
+ To specify the factors by which the vectors should be multiplied before
+ being added, you use the following three options:
+ %
+ \begin{key}{/tikz/cs/x=\meta{factor} (initially 0)}
+ Factor by which the $x$-vector is multiplied.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/y=\meta{factor} (initially 0)}
+ Works like |x|.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/z=\meta{factor} (initially 0)}
+ Works like |x|.
+ \end{key}
+
+\begin{codeexample}[]
+\begin{tikzpicture}[->]
+ \draw (0,0) -- (xyz cs:x=1);
+ \draw (0,0) -- (xyz cs:y=1);
+ \draw (0,0) -- (xyz cs:z=1);
+\end{tikzpicture}
+\end{codeexample}
+
+ This coordinate system can also be selected implicitly. To do so, you just
+ provide two or three comma-separated factors (not dimensions).
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}[->]
+ \draw (0,0) -- (1,0);
+ \draw (0,0) -- (0,1,0);
+ \draw (0,0) -- (0,0,1);
+\end{tikzpicture}
+\end{codeexample}
+ %
+\end{coordinatesystem}
+
+\emph{Note:} It is possible to use coordinates like |(1,2cm)|, which are
+neither |canvas| coordinates nor |xyz| coordinates. The rule is the following:
+If a coordinate is of the implicit form |(|\meta{x}|,|\meta{y}|)|, then
+\meta{x} and \meta{y} are checked, independently, whether they have a dimension
+or whether they are dimensionless. If both have a dimension, the |canvas|
+coordinate system is used. If both lack a dimension, the |xyz| coordinate
+system is used. If \meta{x} has a dimension and \meta{y} has not, then the sum
+of two coordinate |(|\meta{x}|,0pt)| and |(0,|\meta{y}|)| is used. If \meta{y}
+has a dimension and \meta{x} has not, then the sum of two coordinate
+|(|\meta{x}|,0)| and |(0pt,|\meta{y}|)| is used.
+
+\emph{Note furthermore:} An expression like |(2+3cm,0)| does \emph{not} mean
+the same as |(2cm+3cm,0)|. Instead, if \meta{x} or \meta{y} internally uses a
+mixture of dimensions and dimensionless values, then all dimensionless values
+are ``upgraded'' to dimensions by interpreting them as |pt|. So, |2+3cm| is the
+same dimension as |2pt+3cm|.
+
+\begin{coordinatesystem}{canvas polar}
+ The |canvas polar| coordinate system allows you to specify polar
+ coordinates. You provide an angle using the |angle=| option and a radius
+ using the |radius=| option. This yields the point on the canvas that is at
+ the given radius distance from the origin at the given degree. An angle of
+ zero degrees to the right, a degree of 90 upward.
+ %
+ \begin{key}{/tikz/cs/angle=\meta{degrees}}
+ The angle of the coordinate. The angle must always be given in degrees.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/radius=\meta{dimension}}
+ The distance from the origin.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/x radius=\meta{dimension}}
+ A polar coordinate is, after all, just a point on a circle of the given
+ \meta{radius}. When you provide an $x$-radius and also a $y$-radius,
+ you specify an ellipse instead of a circle. The |radius| option has the
+ same effect as specifying identical |x radius| and |y radius| options.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/y radius=\meta{dimension}}
+ Works like |x radius|.
+ \end{key}
+ %
+\begin{codeexample}[]
+\tikz \draw (0,0) -- (canvas polar cs:angle=30,radius=1cm);
+\end{codeexample}
+
+ The implicit form for canvas polar coordinates is the following: you
+ specify the angle and the distance, separated by a colon as in |(30:1cm)|.
+ %
+\begin{codeexample}[]
+\tikz \draw (0cm,0cm) -- (30:1cm) -- (60:1cm) -- (90:1cm)
+ -- (120:1cm) -- (150:1cm) -- (180:1cm);
+\end{codeexample}
+
+ Two different radii are specified by writing |(30:1cm and 2cm)|.
+
+ For the implicit form, instead of an angle given as a number you can also
+ use certain words. For example, |up| is the same as |90|, so that you can
+ write |\tikz \draw (0,0) -- (2ex,0pt) -- +(up:1ex);| and get
+ \tikz \draw (0,0) -- (2ex,0pt) -- +(up:1ex);. Apart from |up| you can use
+ |down|, |left|, |right|, |north|, |south|, |west|, |east|, |north east|,
+ |north west|, |south east|, |south west|, all of which have their natural
+ meaning.
+\end{coordinatesystem}
+
+\begin{coordinatesystem}{xyz polar}
+ This coordinate system work similarly to the |canvas polar| system.
+ However, the radius and the angle are interpreted in the $xy$-coordinate
+ system, not in the canvas system. More detailed, consider the circle or
+ ellipse whose half axes are given by the current $x$-vector and the current
+ $y$-vector. Then, consider the point that lies at a given angle on this
+ ellipse, where an angle of zero is the same as the $x$-vector and an angle
+ of 90 is the $y$-vector. Finally, multiply the resulting vector by the
+ given radius factor. Voil\`a.
+ %
+ \begin{key}{/tikz/cs/angle=\meta{degrees}}
+ The angle of the coordinate interpreted in the ellipse whose axes are
+ the $x$-vector and the $y$-vector.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/radius=\meta{factor}}
+ A factor by which the $x$-vector and $y$-vector are multiplied prior to
+ forming the ellipse.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/x radius=\meta{dimension}}
+ A specific factor by which only the $x$-vector is multiplied.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/y radius=\meta{dimension}}
+ Works like |x radius|.
+ \end{key}
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}[x=1.5cm,y=1cm]
+ \draw[help lines] (0cm,0cm) grid (3cm,2cm);
+
+ \draw (0,0) -- (xyz polar cs:angle=0,radius=1);
+ \draw (0,0) -- (xyz polar cs:angle=30,radius=1);
+ \draw (0,0) -- (xyz polar cs:angle=60,radius=1);
+ \draw (0,0) -- (xyz polar cs:angle=90,radius=1);
+
+ \draw (xyz polar cs:angle=0,radius=2)
+ -- (xyz polar cs:angle=30,radius=2)
+ -- (xyz polar cs:angle=60,radius=2)
+ -- (xyz polar cs:angle=90,radius=2);
+ \end{tikzpicture}
+\end{codeexample}
+
+ The implicit version of this option is the same as the implicit version of
+ |canvas polar|, only you do not provide a unit.
+
+\begin{codeexample}[]
+\tikz[x={(0cm,1cm)},y={(-1cm,0cm)}]
+ \draw (0,0) -- (30:1) -- (60:1) -- (90:1)
+ -- (120:1) -- (150:1) -- (180:1);
+\end{codeexample}
+ %
+\end{coordinatesystem}
+
+\begin{coordinatesystem}{xy polar}
+ This is just an alias for |xyz polar|, which some people might prefer as
+ there is no z-coordinate involved in the |xyz polar| coordinates.
+\end{coordinatesystem}
+
+
+\subsubsection{Barycentric Systems}
+\label{section-barycentric-coordinates}
+
+In the barycentric coordinate system a point is expressed as the linear
+combination of multiple vectors. The idea is that you specify vectors $v_1$,
+$v_2$, \dots, $v_n$ and numbers $\alpha_1$, $\alpha_2$, \dots, $\alpha_n$. Then
+the barycentric coordinate specified by these vectors and numbers is
+%
+\begin{align*}
+ \frac{\alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n}{\alpha_1
+ + \alpha_2 + \cdots + \alpha_n}
+\end{align*}
+
+The |barycentric cs| allows you to specify such coordinates easily.
+
+\begin{coordinatesystem}{barycentric}
+ For this coordinate system, the \meta{coordinate specification} should be a
+ comma-separated list of expressions of the form \meta{node
+ name}|=|\meta{number}. Note that (currently) the list should not contain
+ any spaces before or after the \meta{node name} (unlike normal key--value
+ pairs).
+
+ The specified coordinate is now computed as follows: Each pair provides one
+ vector and a number. The vector is the |center| anchor of the \meta{node
+ name}. The number is the \meta{number}. Note that (currently) you cannot
+ specify a different anchor, so that in order to use, say, the |north|
+ anchor of a node you first have to create a new coordinate at this north
+ anchor. (Using for instance \texttt{\string\coordinate (mynorth) at
+ (mynode.north);}.)
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \coordinate (content) at (90:3cm);
+ \coordinate (structure) at (210:3cm);
+ \coordinate (form) at (-30:3cm);
+
+ \node [above] at (content) {content oriented};
+ \node [below left] at (structure) {structure oriented};
+ \node [below right] at (form) {form oriented};
+
+ \draw [thick,gray] (content.south) -- (structure.north east) -- (form.north west) -- cycle;
+
+ \small
+ \node at (barycentric cs:content=0.5,structure=0.1 ,form=1) {PostScript};
+ \node at (barycentric cs:content=1 ,structure=0 ,form=0.4) {DVI};
+ \node at (barycentric cs:content=0.5,structure=0.5 ,form=1) {PDF};
+ \node at (barycentric cs:content=0 ,structure=0.25,form=1) {CSS};
+ \node at (barycentric cs:content=0.5,structure=1 ,form=0) {XML};
+ \node at (barycentric cs:content=0.5,structure=1 ,form=0.4) {HTML};
+ \node at (barycentric cs:content=1 ,structure=0.2 ,form=0.8) {\TeX};
+ \node at (barycentric cs:content=1 ,structure=0.6 ,form=0.8) {\LaTeX};
+ \node at (barycentric cs:content=0.8,structure=0.8 ,form=1) {Word};
+ \node at (barycentric cs:content=1 ,structure=0.05,form=0.05) {ASCII};
+\end{tikzpicture}
+\end{codeexample}
+ %
+\end{coordinatesystem}
+
+
+\subsubsection{Node Coordinate System}
+\label{section-node-coordinates}
+
+In \pgfname\ and in \tikzname\ it is quite easy to define a node that you wish
+to reference at a later point. Once you have defined a node, there are
+different ways of referencing points of the node. To do so, you use the
+following coordinate system:
+
+\begin{coordinatesystem}{node}
+ This coordinate system is used to reference a specific point inside or on
+ the border of a previously defined node. It can be used in different ways,
+ so let us go over them one by one.
+
+ You can use three options to specify which coordinate you mean:
+ %
+ \begin{key}{/tikz/cs/name=\meta{node name}}
+ Specifies the node that you wish to use to specify a coordinate. The
+ \meta{node name} is the name that was previously used to name the node
+ using the |name=|\meta{node name} option or the special node name
+ syntax.
+ \end{key}
+ %
+ \begin{key}{/tikz/anchor=\meta{anchor}}
+ Specifies an anchor of the node. Here is an example:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{arrows.meta}}]
+\begin{tikzpicture}
+ \node (shape) at (0,2) [draw] {|class Shape|};
+ \node (rect) at (-2,0) [draw] {|class Rectangle|};
+ \node (circle) at (2,0) [draw] {|class Circle|};
+ \node (ellipse) at (6,0) [draw] {|class Ellipse|};
+
+ \draw (node cs:name=circle,anchor=north) |- (0,1);
+ \draw (node cs:name=ellipse,anchor=north) |- (0,1);
+ \draw [arrows = -{Triangle[open, angle=60:3mm]}]
+ (node cs:name=rect,anchor=north)
+ |- (0,1) -| (node cs:name=shape,anchor=south);
+\end{tikzpicture}
+\end{codeexample}
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/angle=\meta{degrees}}
+ It is also possible to provide an angle \emph{instead} of an anchor.
+ This coordinate refers to a point of the node's border where a ray shot
+ from the center in the given angle hits the border. Here is an example:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{shapes.geometric}}]
+\begin{tikzpicture}
+ \node (start) [draw,shape=ellipse] {start};
+ \foreach \angle in {-90, -80, ..., 90}
+ \draw (node cs:name=start,angle=\angle)
+ .. controls +(\angle:1cm) and +(-1,0) .. (2.5,0);
+ \end{tikzpicture}
+\end{codeexample}
+ \end{key}
+
+ It is possible to provide \emph{neither} the |anchor=| option nor the
+ |angle=| option. In this case, \tikzname\ will calculate an appropriate
+ border position for you. Here is an example:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{shapes.geometric}}]
+\begin{tikzpicture}
+ \path (0,0) node(a) [ellipse,rotate=10,draw] {An ellipse}
+ (3,-1) node(b) [circle,draw] {A circle};
+ \draw[thick] (node cs:name=a) -- (node cs:name=b);
+\end{tikzpicture}
+\end{codeexample}
+
+ \tikzname\ will be reasonably clever at determining the border points that
+ you ``mean'', but, naturally, this may fail in some situations. If
+ \tikzname\ fails to determine an appropriate border point, the center will
+ be used instead.
+
+ Automatic computation of anchors works only with the line-to operations
+ |--|, the vertical/horizontal versions \verb!|-! and \verb!-|!, and with
+ the curve-to operation |..|. For other path commands, such as |parabola| or
+ |plot|, the center will be used. If this is not desired, you should give a
+ named anchor or an angle anchor.
+
+ Note that if you use an automatic coordinate for both the start and the end
+ of a line-to, as in |--(node cs:name=b)--|, then \emph{two} border
+ coordinates are computed with a move-to between them. This is usually
+ exactly what you want.
+
+ If you use relative coordinates together with automatic anchor coordinates,
+ the relative coordinates are computed relative to the node's center, not
+ relative to the border point. Here is an example:
+ %
+\begin{codeexample}[]
+\tikz \draw (0,0) node(x) [draw] {Text}
+ rectangle (1,1)
+ (node cs:name=x) -- +(1,1);
+\end{codeexample}
+
+ Similarly, in the following examples both control points are $(1,1)$:
+ %
+\begin{codeexample}[]
+\tikz \draw (0,0) node(x) [draw] {X}
+ (2,0) node(y) {Y}
+ (node cs:name=x) .. controls +(1,1) and +(-1,1) ..
+ (node cs:name=y);
+\end{codeexample}
+
+ The implicit way of specifying the node coordinate system is to simply use
+ the name of the node in parentheses as in |(a)| or to specify a name
+ together with an anchor or an angle separated by a dot as in |(a.north)| or
+ |(a.10)|.
+
+ Here is a more complete example:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{shapes.geometric}}]
+\begin{tikzpicture}[fill=blue!20]
+ \draw[help lines] (-1,-2) grid (6,3);
+ \path (0,0) node(a) [ellipse,rotate=10,draw,fill] {An ellipse}
+ (3,-1) node(b) [circle,draw,fill] {A circle}
+ (2,2) node(c) [rectangle,rotate=20,draw,fill] {A rectangle}
+ (5,2) node(d) [rectangle,rotate=-30,draw,fill] {Another rectangle};
+ \draw[thick] (a.south) -- (b) -- (c) -- (d);
+ \draw[thick,red,->] (a) |- +(1,3) -| (c) |- (b);
+ \draw[thick,blue,<->] (b) .. controls +(right:2cm) and +(down:1cm) .. (d);
+\end{tikzpicture}
+\end{codeexample}
+ %
+\end{coordinatesystem}
+
+% -----------------------------------------------------------------------------
+% Deprecated:
+% -----------------------------------------------------------------------------
+%
+% \subsubsection{Intersection Coordinate Systems}
+%
+% Often you wish to specify a point that is on the
+% intersection of two lines or shapes. For this, the following
+% coordinate system is useful:
+%
+% \begin{coordinatesystem}{intersection}
+% First, you must specify two objects that should be
+% intersected. These ``objects'' can either be lines or the shapes of
+% nodes. There are two option to specify the first object:
+% \begin{key}{/tikz/cs/first line={\ttfamily\char`\{}|(|\meta{first
+% coordinate}|)--(|\meta{second coordinate}|)|{\ttfamily\char`\}}}
+% Specifies that the first object is a line that goes from
+% \meta{first coordinate} to meta{second coordinate}.
+% \end{key}
+% Note that you have to write |--| between the coordinate, but this
+% does not mean that anything is added to the path. This is simply a
+% special syntax.
+% \begin{key}{/tikz/cs/first node=\meta{node}}
+% Specifies that the first object is a previously defined node named
+% \meta{node}.
+% \end{key}
+%
+% To specify the second object, you use one of the following keys:
+% \begin{key}{/tikz/cs/second line={\ttfamily\char`\{}|(|\meta{first
+% coordinate}|)--(|\meta{second coordinate}|)|{\ttfamily\char`\}}}
+% As above.
+% \end{key}
+% \begin{key}{/tikz/cs/second node=\meta{node}}
+% Specifies that the second object is a previously defined node
+% named \meta{node}.
+% \end{key}
+%
+% Since it is possible that two objects have multiple intersections,
+% you may need to specify which solution you want:
+% \begin{key}{/tikz/cs/solution=\meta{number} (initially 1)}
+% Specifies which solution should be used. Numbering starts with 1.
+% \end{key}
+% The coordinate specified in this way is the \meta{number}th
+% intersection of the two objects. If the objects do not intersect,
+% an error may occur.
+%
+% \begin{codeexample}[]
+% \begin{tikzpicture}
+% \draw[help lines] (0,0) grid (3,2);
+% \draw (0,0) coordinate (A) -- (3,2) coordinate (B)
+% (1,2) -- (3,0);
+%
+% \fill[red] (intersection cs:
+% first line={(A)--(B)},
+% second line={(1,2)--(3,0)}) circle (2pt);
+% \end{tikzpicture}
+% \end{codeexample}
+%
+% The implicit way of specifying this coordinate system is to write
+% \declare{|(intersection |\opt{\meta{number}}| of |\meta{first
+% object}%
+% | and |\meta{second object}|)|}. Here, \meta{first object} either
+% has the form \meta{$p_1$}|--|\meta{$p_2$} or it is just a node
+% name. Likewise for \meta{second object}. Note that there are \emph{no}
+% parentheses around the $p_i$. Thus, you would write
+% |(intersection of A--B and 1,2--3,0)| for the intersection of the
+% line through the coordinates |A| and |B| and the line through the
+% points $(1,2)$ and $(3,0)$. You would write
+% |(intersection 2 of c_1 and c_2)| for the second
+% intersection of the node named |c_1| and the node named
+% |c_2|.
+%
+% \tikzname\ needs an explicit algorithm for computing the
+% intersection of two shapes and such an algorithm is available only
+% for few shapes. Currently, the following intersection will be
+% computed correctly:
+% \begin{itemize}
+% \item a line and a line
+% \item a |circle| node and a line (in any order)
+% \item a |circle| and a |circle|
+% \end{itemize}
+% \begin{codeexample}[]
+% \begin{tikzpicture}[scale=.25]
+% \coordinate [label=-135:$a$] (a) at ($ (0,0) + (rand,rand) $);
+% \coordinate [label=45:$b$] (b) at ($ (3,2) + (rand,rand) $);
+%
+% \coordinate [label=-135:$u$] (u) at (-1,1);
+% \coordinate [label=45:$v$] (v) at (6,0);
+%
+% \draw (a) -- (b)
+% (u) -- (v);
+%
+% \node (c1) at (a) [draw,circle through=(b)] {};
+% \node (c2) at (b) [draw,circle through=(a)] {};
+%
+% \coordinate [label=135:$c$] (c) at (intersection 2 of c1 and c2);
+% \coordinate [label=-45:$d$] (d) at (intersection of u--v and c2);
+% \coordinate [label=135:$e$] (e) at (intersection of u--v and a--b);
+%
+% \foreach \p in {a,b,c,d,e,u,v}
+% \fill [opacity=.5] (\p) circle (8pt);
+% \end{tikzpicture}
+% \end{codeexample}
+% \end{coordinatesystem}
+% -----------------------------------------------------------------------------
+
+
+\subsubsection{Tangent Coordinate Systems}
+
+\begin{coordinatesystem}{tangent}
+ This coordinate system, which is available only when the \tikzname\ library
+ |calc| is loaded, allows you to compute the point that lies tangent to a
+ shape. In detail, consider a \meta{node} and a \meta{point}. Now, draw a
+ straight line from the \meta{point} so that it ``touches'' the \meta{node}
+ (more formally, so that it is \emph{tangent} to this \meta{node}). The
+ point where the line touches the shape is the point referred to by the
+ |tangent| coordinate system.
+
+ The following options may be given:
+ %
+ \begin{key}{/tikz/cs/node=\meta{node}}
+ This key specifies the node on whose border the tangent should lie.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/point=\meta{point}}
+ This key specifies the point through which the tangent should go.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/solution=\meta{number}}
+ Specifies which solution should be used if there are more than one.
+ \end{key}
+
+ A special algorithm is needed in order to compute the tangent for a given
+ shape. Currently, tangents can be computed for nodes whose shape is one of
+ the following:
+ %
+ \begin{itemize}
+ \item |coordinate|
+ \item |circle|
+ \end{itemize}
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw[help lines] (0,0) grid (3,2);
+
+ \coordinate (a) at (3,2);
+
+ \node [circle,draw] (c) at (1,1) [minimum size=40pt] {$c$};
+
+ \draw[red] (a) -- (tangent cs:node=c,point={(a)},solution=1) --
+ (c.center) -- (tangent cs:node=c,point={(a)},solution=2) -- cycle;
+\end{tikzpicture}
+\end{codeexample}
+
+ There is no implicit syntax for this coordinate system.
+\end{coordinatesystem}
+
+
+\subsubsection{Defining New Coordinate Systems}
+
+While the set of coordinate systems that \tikzname\ can parse via their special
+syntax is fixed, it is possible and quite easy to define new explicitly named
+coordinate systems. For this, the following commands are used:
+
+\begin{command}{\tikzdeclarecoordinatesystem\marg{name}\marg{code}}
+ This command declares a new coordinate system named \meta{name} that can
+ later on be used by writing |(|\meta{name}| cs:|\meta{arguments}|)|. When
+ \tikzname\ encounters a coordinate specified in this way, the
+ \meta{arguments} are passed to \meta{code} as argument |#1|.
+
+ It is now the job of \meta{code} to make sense of the \meta{arguments}. At
+ the end of \meta{code}, the two \TeX\ dimensions |\pgf@x| and |\pgf@y|
+ should be have the $x$- and $y$-canvas coordinate of the coordinate.
+
+ It is not necessary, but customary, to parse \meta{arguments} using the
+ key--value syntax. However, you can also parse it in any way you like.
+
+ In the following example, a coordinate system |cylindrical| is defined.
+ %
+\begin{codeexample}[]
+\makeatletter
+\define@key{cylindricalkeys}{angle}{\def\myangle{#1}}
+\define@key{cylindricalkeys}{radius}{\def\myradius{#1}}
+\define@key{cylindricalkeys}{z}{\def\myz{#1}}
+\tikzdeclarecoordinatesystem{cylindrical}%
+{%
+ \setkeys{cylindricalkeys}{#1}%
+ \pgfpointadd{\pgfpointxyz{0}{0}{\myz}}{\pgfpointpolarxy{\myangle}{\myradius}}
+}
+\begin{tikzpicture}[z=0.2pt]
+ \draw [->] (0,0,0) -- (0,0,350);
+ \foreach \num in {0,10,...,350}
+ \fill (cylindrical cs:angle=\num,radius=1,z=\num) circle (1pt);
+\end{tikzpicture}
+\end{codeexample}
+ %
+\end{command}
+
+\begin{command}{\tikzaliascoordinatesystem\marg{new name}\marg{old name}}
+ Creates an alias of \meta{old name}.
+\end{command}
+
+
+\subsection{Coordinates at Intersections}
+\label{section-intersection-coordinates}
+
+You will wish to compute the intersection of two paths. For the special and
+frequent case of two perpendicular lines, a special coordinate system called
+|perpendicular| is available. For more general cases, the |intersections|
+library can be used.
+
+
+\subsubsection{Intersections of Perpendicular Lines}
+
+A frequent special case of path intersections is the intersection of a vertical
+line going through a point $p$ and a horizontal line going through some other
+point $q$. For this situation there is a useful coordinate system.
+
+\begin{coordinatesystem}{perpendicular}
+ You can specify the two lines using the following keys:
+
+ \begin{key}{/tikz/cs/horizontal line through={\ttfamily\char`\{}|(|\meta{coordinate}|)|{\ttfamily\char`\}}}
+ Specifies that one line is a horizontal line that goes through the
+ given coordinate.
+ \end{key}
+ %
+ \begin{key}{/tikz/cs/vertical line through={\ttfamily\char`\{}|(|\meta{coordinate}|)|{\ttfamily\char`\}}}
+ Specifies that the other line is vertical and goes through the given
+ coordinate.
+ \end{key}
+
+ However, in almost all cases you should, instead, use the implicit syntax.
+ Here, you write \declare{|(|\meta{p}\verb! |- !\meta{q}|)|} or
+ \declare{|(|\meta{q}\verb! -| !\meta{p}|)|}.
+
+ For example, \verb!(2,1 |- 3,4)! and \verb!(3,4 -| 2,1)! both yield the
+ same as \verb!(2,4)! (provided the $xy$-co\-or\-di\-nate system has not
+ been modified).
+
+ The most useful application of the syntax is to draw a line up to some
+ point on a vertical or horizontal line. Here is an example:
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \path (30:1cm) node(p1) {$p_1$} (75:1cm) node(p2) {$p_2$};
+
+ \draw (-0.2,0) -- (1.2,0) node(xline)[right] {$q_1$};
+ \draw (2,-0.2) -- (2,1.2) node(yline)[above] {$q_2$};
+
+ \draw[->] (p1) -- (p1 |- xline);
+ \draw[->] (p2) -- (p2 |- xline);
+ \draw[->] (p1) -- (p1 -| yline);
+ \draw[->] (p2) -- (p2 -| yline);
+\end{tikzpicture}
+\end{codeexample}
+
+ Note that in \declare{|(|\meta{c}\verb! |- !\meta{d}|)|} the coordinates
+ \meta{c} and \meta{d} are \emph{not} surrounded by parentheses. If they
+ need to be complicated expressions (like a computation using the
+ |$|-syntax), you must surround them with braces; parentheses will then be %$
+ added around them.
+
+ As an example, let us specify a point that lies horizontally at the middle
+ of the line from $A$ to~$B$ and vertically at the middle of the line from
+ $C$ to~$D$:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \node (A) at (0,1) {A};
+ \node (B) at (1,1.5) {B};
+ \node (C) at (2,0) {C};
+ \node (D) at (2.5,-2) {D};
+
+ \draw (A) -- (B) node [midway] {x};
+ \draw (C) -- (D) node [midway] {x};
+
+ \node at ({$(A)!.5!(B)$} -| {$(C)!.5!(D)$}) {X};
+\end{tikzpicture}
+\end{codeexample}
+ %
+\end{coordinatesystem}
+
+
+\subsubsection{Intersections of Arbitrary Paths}
+
+\begin{tikzlibrary}{intersections}
+ This library enables the calculation of intersections of two arbitrary
+ paths. However, due to the low accuracy of \TeX, the paths should not be
+ ``too complicated''. In particular, you should not try to intersect paths
+ consisting of lots of very small segments such as plots or decorated paths.
+\end{tikzlibrary}
+
+To find the intersections of two paths in \tikzname, they must be ``named''. A
+``named path'' is, quite simply, a path that has been named using the following
+key (note that this is a \emph{different} key from the |name| key, which only
+attaches a hyperlink target to a path, but does not store the path in a way the
+is useful for the intersection computation):
+
+\begin{keylist}{%
+ /tikz/name path=\meta{name},
+ /tikz/name path global=\meta{name}%
+}
+ The effect of this key is that, after the path has been constructed, just
+ before it is used, it is associated with \meta{name}. For |name path|, this
+ association survives beyond the final semi-colon of the path but not the
+ end of the surrounding scope. For |name path global|, the association will
+ survive beyond any scope as well. Handle with care.
+
+ Any paths created by nodes on the (main) path are ignored, unless this key
+ is explicitly used. If the same \meta{name} is used for the main path and
+ the node path(s), then the paths will be added together and then associated
+ with \meta{name}.
+\end{keylist}
+
+To find the intersection of named paths, the following key is used:
+
+\begin{key}{/tikz/name intersections=\marg{options}}
+ This key changes the key path to |/tikz/intersection| and processes
+ \meta{options}. These options determine, among other things, which paths to
+ use for the intersection. Having processed the options, any intersections
+ are then found. A coordinate is created at each intersection, which by
+ default, will be named |intersection-1|, |intersection-2|, and so on.
+ Optionally, the prefix |intersection| can be changed, and the total number
+ of intersections stored in a \TeX-macro.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}[every node/.style={opacity=1, black, above left}]
+ \draw [help lines] grid (3,2);
+ \draw [name path=ellipse] (2,0.5) ellipse (0.75cm and 1cm);
+ \draw [name path=rectangle, rotate=10] (0.5,0.5) rectangle +(2,1);
+ \fill [red, opacity=0.5, name intersections={of=ellipse and rectangle}]
+ (intersection-1) circle (2pt) node {1}
+ (intersection-2) circle (2pt) node {2};
+\end{tikzpicture}
+\end{codeexample}
+
+ The following keys can be used in \meta{options}:
+
+ \begin{key}{/tikz/intersection/of=\meta{name path 1}| and |\meta{name path 2}}
+ This key is used to specify the names of the paths to use for the
+ intersection.
+ \end{key}
+
+ \begin{key}{/tikz/intersection/name=\meta{prefix} (initially intersection)}
+ This key specifies the prefix name for the coordinate nodes placed at
+ each intersection.
+ \end{key}
+
+ \begin{key}{/tikz/intersection/total=\meta{macro}}
+ This key means that the total number of intersections found will be
+ stored in \meta{macro}.
+ \end{key}
+
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}
+ \clip (-2,-2) rectangle (2,2);
+ \draw [name path=curve 1] (-2,-1) .. controls (8,-1) and (-8,1) .. (2,1);
+ \draw [name path=curve 2] (-1,-2) .. controls (-1,8) and (1,-8) .. (1,2);
+
+ \fill [name intersections={of=curve 1 and curve 2, name=i, total=\t}]
+ [red, opacity=0.5, every node/.style={above left, black, opacity=1}]
+ \foreach \s in {1,...,\t}{(i-\s) circle (2pt) node {\footnotesize\s}};
+\end{tikzpicture}
+\end{codeexample}
+
+ \begin{key}{/tikz/intersection/by=\meta{comma-separated list}}
+ This key allows you to specify a list of names for the intersection
+ coordinates. The intersection coordinates will still be named
+ \meta{prefix}|-|\meta{number}, but additionally the first coordinate
+ will also be named by the first element of the \meta{comma-separated
+ list}. What happens is that the \meta{comma-separated list} is passed
+ to the |\foreach| statement and for \meta{list member} a coordinate is
+ created at the already-named intersection.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}
+ \clip (-2,-2) rectangle (2,2);
+ \draw [name path=curve 1] (-2,-1) .. controls (8,-1) and (-8,1) .. (2,1);
+ \draw [name path=curve 2] (-1,-2) .. controls (-1,8) and (1,-8) .. (1,2);
+
+ \fill [name intersections={of=curve 1 and curve 2, by={a,b}}]
+ (a) circle (2pt)
+ (b) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+ You can also use the |...| notation of the |\foreach| statement inside
+ the \meta{comma-separated list}.
+
+ In case an element of the \meta{comma-separated list} starts with
+ options in square brackets, these options are used when the coordinate
+ is created. A coordinate name can still, but need not, follow the
+ options. This makes it easy to add labels to intersections:
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}
+ \clip (-2,-2) rectangle (2,2);
+ \draw [name path=curve 1] (-2,-1) .. controls (8,-1) and (-8,1) .. (2,1);
+ \draw [name path=curve 2] (-1,-2) .. controls (-1,8) and (1,-8) .. (1,2);
+
+ \fill [name intersections={
+ of=curve 1 and curve 2,
+ by={[label=center:a],[label=center:...],[label=center:i]}}];
+\end{tikzpicture}
+\end{codeexample}
+ \end{key}
+
+ \begin{key}{/tikz/intersection/sort by=\meta{path name}}
+ By default, the intersections are simply returned in the order that the
+ intersection algorithm finds them. Unfortunately, this is not
+ necessarily a ``helpful'' ordering. This key can be used to sort the
+ intersections along the path specified by \meta{path name}, which
+ should be one of the paths mentioned in the |/tikz/intersection/of|
+ key.
+ %
+\begin{codeexample}[preamble={\usetikzlibrary{intersections}}]
+\begin{tikzpicture}
+\clip (-0.5,-0.75) rectangle (3.25,2.25);
+\foreach \pathname/\shift in {line/0cm, curve/2cm}{
+ \tikzset{xshift=\shift}
+ \draw [->, name path=curve] (1,1.5) .. controls (-1,1) and (2,0.5) .. (0,0);
+ \draw [->, name path=line] (0,-.5) -- (1,2) ;
+ \fill [name intersections={of=line and curve,sort by=\pathname, name=i}]
+ [red, opacity=0.5, every node/.style={left=.25cm, black, opacity=1}]
+ \foreach \s in {1,2,3}{(i-\s) circle (2pt) node {\footnotesize\s}};
+}
+\end{tikzpicture}
+\end{codeexample}
+ \end{key}
+\end{key}
+
+
+\subsection{Relative and Incremental Coordinates}
+
+\subsubsection{Specifying Relative Coordinates}
+
+You can prefix coordinates by |++| to make them ``relative''. A coordinate such
+as |++(1cm,0pt)| means ``1cm to the right of the previous position, making this
+the new current position''. Relative coordinates are often useful in ``local''
+contexts:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw (0,0) -- ++(1,0) -- ++(0,1) -- ++(-1,0) -- cycle;
+ \draw (2,0) -- ++(1,0) -- ++(0,1) -- ++(-1,0) -- cycle;
+ \draw (1.5,1.5) -- ++(1,0) -- ++(0,1) -- ++(-1,0) -- cycle;
+\end{tikzpicture}
+\end{codeexample}
+
+Instead of |++| you can also use a single |+|. This also specifies a relative
+coordinate, but it does not ``update'' the current point for subsequent usages
+of relative coordinates. Thus, you can use this notation to specify numerous
+points, all relative to the same ``initial'' point:
+
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw (0,0) -- +(1,0) -- +(1,1) -- +(0,1) -- cycle;
+ \draw (2,0) -- +(1,0) -- +(1,1) -- +(0,1) -- cycle;
+ \draw (1.5,1.5) -- +(1,0) -- +(1,1) -- +(0,1) -- cycle;
+\end{tikzpicture}
+\end{codeexample}
+
+There is a special situation, where relative coordinates are interpreted
+differently. If you use a relative coordinate as a control point of a Bézier
+curve, the following rule applies: First, a relative first control point is
+taken relative to the beginning of the curve. Second, a relative second control
+point is taken relative to the end of the curve. Third, a relative end point of
+a curve is taken relative to the start of the curve.
+
+This special behavior makes it easy to specify that a curve should ``leave or
+arrive from a certain direction'' at the start or end. In the following
+example, the curve ``leaves'' at $30^\circ$ and ``arrives'' at $60^\circ$:
+%
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw (1,0) .. controls +(30:1cm) and +(60:1cm) .. (3,-1);
+ \draw[gray,->] (1,0) -- +(30:1cm);
+ \draw[gray,<-] (3,-1) -- +(60:1cm);
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsubsection{Rotational Relative Coordinates}
+
+You may sometimes wish to specify points relative not only to the previous
+point, but additionally relative to the tangent entering the previous point.
+For this, the following key is useful:
+
+\begin{key}{/tikz/turn}
+ This key can be given as an option to a \meta{coordinate} as in the
+ following example:
+ %
+\begin{codeexample}[]
+\tikz \draw (0,0) -- (1,1) -- ([turn]-45:1cm) -- ([turn]-30:1cm);
+\end{codeexample}
+ %
+ The effect of this key is to locally shift the coordinate system so that
+ the last point reached is at the origin and the coordinate system is
+ ``turned'' so that the $x$-axis points in the direction of a tangent
+ entering the last point. This means, in effect, that when you use polar
+ coordinates of the form \meta{relative angle}|:|\meta{distance} together
+ with the |turn| option, you specify a point that lies at \meta{distance}
+ from the last point in the direction of the last tangent entering the last
+ point, but with a rotation of \meta{relative angle}.
+
+ This key also works with curves \dots
+ %
+\begin{codeexample}[]
+\tikz [delta angle=30, radius=1cm]
+ \draw (0,0) arc [start angle=0] -- ([turn]0:1cm)
+ arc [start angle=30] -- ([turn]0:1cm)
+ arc [start angle=60] -- ([turn]30:1cm);
+\end{codeexample}
+\begin{codeexample}[]
+\tikz \draw (0,0) to [bend left] (2,1) -- ([turn]0:1cm);
+\end{codeexample}
+ %
+ \dots and with plots \dots
+ %
+\begin{codeexample}[]
+\tikz \draw plot coordinates {(0,0) (1,1) (2,0) (3,0) } -- ([turn]30:1cm);
+\end{codeexample}
+
+ Although the above examples use polar coordinates with |turn|, you can also
+ use any normal coordinate. For instance, |([turn]1,1)| will append a line
+ of length $\sqrt 2$ that is turns by $45^\circ$ relative to the tangent to
+ the last point.
+ %
+\begin{codeexample}[]
+\tikz \draw (0.5,0.5) -| (2,1) -- ([turn]1,1)
+ .. controls ([turn]0:1cm) .. ([turn]-90:1cm);
+\end{codeexample}
+ %
+\end{key}
+
+
+\subsubsection{Relative Coordinates and Scopes}
+\label{section-scopes-relative}
+
+An interesting question is, how do relative coordinates behave in the presence
+of scopes? That is, suppose we use curly braces in a path to make part of it
+``local'', how does that affect the current position? On the one hand, the
+current position certainly changes since the scope only affects options, not
+the path itself. On the other hand, it may be useful to ``temporarily escape''
+from the updating of the current point.
+
+Since both interpretations of how the current point and scopes should
+``interact'' are useful, there is a (local!) option that allows you to decide
+which you need.
+
+\begin{key}{/tikz/current point is local=\opt{\meta{boolean}} (initially false)}
+ Normally, the scope path operation has no effect on the current point. That
+ is, curly braces on a path have no effect on the current position:
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw (0,0) -- ++(1,0) -- ++(0,1) -- ++(-1,0);
+ \draw[red] (2,0) -- ++(1,0) { -- ++(0,1) } -- ++(-1,0);
+\end{tikzpicture}
+\end{codeexample}
+ %
+ If you set this key to |true|, this behavior changes. In this case, at the
+ end of a group created on a path, the last current position reverts to
+ whatever value it had at the beginning of the scope. More precisely, when
+ \tikzname\ encounters |}| on a path, it checks whether at this particular
+ moment the key is set to |true|. If so, the current position reverts to the
+ value it had when the matching |{| was read.
+ %
+\begin{codeexample}[]
+\begin{tikzpicture}
+ \draw (0,0) -- ++(1,0) -- ++(0,1) -- ++(-1,0);
+ \draw[red] (2,0) -- ++(1,0)
+ { [current point is local] -- ++(0,1) } -- ++(-1,0);
+\end{tikzpicture}
+\end{codeexample}
+ %
+ In the above example, we could also have given the option outside the
+ scope, for instance as a parameter to the whole scope.
+\end{key}
+
+
+\subsection{Coordinate Calculations}
+\label{tikz-lib-calc}
+
+\begin{tikzlibrary}{calc}
+ You need to load this library in order to use the coordinate calculation
+ functions described in the present section.
+\end{tikzlibrary}
+
+It is possible to do some basic calculations that involve coordinates. In
+essence, you can add and subtract coordinates, scale them, compute midpoints,
+and do projections. For instance, |($(a) + 1/3*(1cm,0)$)| is the coordinate
+that is $1/3 \text{cm}$ to the right of the point |a|:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \node (a) at (1,1) {A};
+ \fill [red] ($(a) + 1/3*(1cm,0)$) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsubsection{The General Syntax}
+
+The general syntax is the following:
+%
+\begin{quote}
+ \declare{|(|\opt{|[|\meta{options}|]|}|$|\meta{coordinate computation}|$)|}.
+\end{quote}
+
+As you can see, the syntax uses the \TeX\ math symbol |$| to %$
+indicate that a ``mathematical computation'' is involved. However, the |$| %$
+has no other effect, in particular, no mathematical text is typeset.
+
+The \meta{coordinate computation} has the following structure:
+%
+\begin{enumerate}
+ \item It starts with
+ %
+ \begin{quote}
+ \opt{\meta{factor}|*|}\meta{coordinate}\opt{\meta{modifiers}}
+ \end{quote}
+ \item This is optionally followed by |+| or |-| and then another
+ %
+ \begin{quote}
+ \opt{\meta{factor}|*|}\meta{coordinate}\opt{\meta{modifiers}}
+ \end{quote}
+ \item This is once more followed by |+| or |-| and another of the above
+ modified coordinate; and so on.
+\end{enumerate}
+
+In the following, the syntax of factors and of the different modifiers
+is explained in detail.
+
+
+\subsubsection{The Syntax of Factors}
+
+The \meta{factor}s are optional and detected by checking whether the
+\meta{coordinate computation} starts with a |(|. Also, after each $\pm$ a
+\meta{factor} is present if, and only if, the |+| or |-| sign is not directly
+followed by~|(|.
+
+If a \meta{factor} is present, it is evaluated using the |\pgfmathparse| macro.
+This means that you can use pretty complicated computations inside a factor. A
+\meta{factor} may even contain opening parentheses, which creates a
+complication: How does \tikzname\ know where a \meta{factor} ends and where a
+coordinate starts? For instance, if the beginning of a \meta{coordinate
+computation} is |2*(3+4|\dots, it is not clear whether |3+4| is part of a
+\meta{coordinate} or part of a \meta{factor}. Because of this, the following
+rule is used: Once it has been determined, that a \meta{factor} is present, in
+principle, the \meta{factor} contains everything up to the next occurrence of
+|*(|. Note that there is no space between the asterisk and the parenthesis.
+
+It is permissible to put the \meta{factor} in curly braces. This can be used
+whenever it is unclear where the \meta{factor} would end.
+
+Here are some examples of coordinate specifications that consist of exactly one
+\meta{factor} and one \meta{coordinate}:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \fill [red] ($2*(1,1)$) circle (2pt);
+ \fill [green] (${1+1}*(1,.5)$) circle (2pt);
+ \fill [blue] ($cos(0)*sin(90)*(1,1)$) circle (2pt);
+ \fill [black] (${3*(4-3)}*(1,0.5)$) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsubsection{The Syntax of Partway Modifiers}
+
+A \meta{coordinate} can be followed by different \meta{modifiers}. The first
+kind of modifier is the \emph{partway modifier}. The syntax (which is loosely
+inspired by Uwe Kern's |xcolor| package) is the following:
+%
+\begin{quote}
+ \meta{coordinate}\declare{|!|\meta{number}|!|\opt{\meta{angle}|:|}\meta{second coordinate}}
+\end{quote}
+%
+One could write for instance
+%
+\begin{codeexample}[code only]
+(1,2)!.75!(3,4)
+\end{codeexample}
+%
+The meaning of this is: ``Use the coordinate that is three quarters on the way
+from |(1,2)| to |(3,4)|.'' In general, \meta{coordinate
+x}|!|\meta{number}|!|\meta{coordinate y} yields the coordinate $(1 -
+\meta{number})\meta{coordinate x} + \meta{number} \meta{coordinate y}$. Note
+that this is a bit different from the way the \meta{number} is interpreted in
+the |xcolor| package: First, you use a factor between $0$ and $1$, not a
+percentage, and, second, as the \meta{number} approaches $1$, we approach the
+second coordinate, not the first. It is permissible to use a \meta{number} that
+is smaller than $0$ or larger than $1$. The \meta{number} is evaluated using
+the |\pgfmathparse| command and, thus, it can involve complicated computations.
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \draw (1,0) -- (3,2);
+
+ \foreach \i in {0,0.2,0.5,0.9,1}
+ \node at ($(1,0)!\i!(3,2)$) {\i};
+\end{tikzpicture}
+\end{codeexample}
+
+The \meta{second coordinate} may be prefixed by an \meta{angle}, separated with
+a colon, as in |(1,1)!.5!60:(2,2)|. The general meaning of
+\meta{a}|!|\meta{factor}|!|\meta{angle}|:|\meta{b} is: ``First, consider the
+line from \meta{a} to \meta{b}. Then rotate this line by \meta{angle}
+\emph{around the point \meta{a}}. Then the two endpoints of this line will be
+\meta{a} and some point \meta{c}. Use this point \meta{c} for the subsequent
+computation, namely the partway computation.''
+
+Here are two examples:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,3);
+
+ \coordinate (a) at (1,0);
+ \coordinate (b) at (3,2);
+
+ \draw[->] (a) -- (b);
+
+ \coordinate (c) at ($ (a)!1! 10:(b) $);
+
+ \draw[->,red] (a) -- (c);
+
+ \fill ($ (a)!.5! 10:(b) $) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (4,4);
+
+ \foreach \i in {0,0.125,...,2}
+ \fill ($(2,2) !\i! \i*180:(3,2)$) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+You can repeatedly apply modifiers. That is, after any modifier you can add
+another (possibly different) modifier.
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \draw (0,0) -- (3,2);
+ \draw[red] ($(0,0)!.3!(3,2)$) -- (3,0);
+ \fill[red] ($(0,0)!.3!(3,2)!.7!(3,0)$) circle (2pt);
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsubsection{The Syntax of Distance Modifiers}
+
+A \emph{distance modifier} has nearly the same syntax as a partway modifier,
+only you use a \meta{dimension} (something like |1cm|) instead of a
+\meta{factor} (something like |0.5|):
+%
+\begin{quote}
+ \meta{coordinate}\declare{|!|\meta{dimension}|!|\opt{\meta{angle}|:|}\meta{second coordinate}}
+\end{quote}
+
+When you write \meta{a}|!|\meta{dimension}|!|\meta{b}, this means the
+following: Use the point that is distanced \meta{dimension} from \meta{a} on
+the straight line from \meta{a} to \meta{b}. Here is an example:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \draw (1,0) -- (3,2);
+
+ \foreach \i in {0cm,1cm,15mm}
+ \node at ($(1,0)!\i!(3,2)$) {\i};
+\end{tikzpicture}
+\end{codeexample}
+
+As before, if you use a \meta{angle}, the \meta{second coordinate} is rotated
+by this much around the \meta{coordinate} before it is used.
+
+The combination of an \meta{angle} of |90| degrees with a distance can be used
+to ``offset'' a point relative to a line. Suppose, for instance, that you have
+computed a point |(c)| that lies somewhere on a line from |(a)| to~|(b)| and
+you now wish to offset this point by |1cm| so that the distance from this
+offset point to the line is |1cm|. This can be achieved as follows:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \coordinate (a) at (1,0);
+ \coordinate (b) at (3,1);
+
+ \draw (a) -- (b);
+
+ \coordinate (c) at ($ (a)!.25!(b) $);
+ \coordinate (d) at ($ (c)!1cm!90:(b) $);
+
+ \draw [<->] (c) -- (d) node [sloped,midway,above] {1cm};
+\end{tikzpicture}
+\end{codeexample}
+
+
+\subsubsection{The Syntax of Projection Modifiers}
+
+The projection modifier is also similar to the above modifiers: It also gives a
+point on a line from the \meta{coordinate} to the \meta{second coordinate}.
+However, the \meta{number} or \meta{dimension} is replaced by a
+\meta{projection coordinate}:
+%
+\begin{quote}
+ \meta{coordinate}\declare{|!|\meta{projection coordinate}|!|\opt{\meta{angle}|:|}\meta{second coordinate}}
+\end{quote}
+
+Here is an example:
+%
+\begin{codeexample}[code only]
+(1,2)!(0,5)!(3,4)
+\end{codeexample}
+
+The effect is the following: We project the \meta{projection coordinate}
+orthogonally onto the line from \meta{coordinate} to \meta{second coordinate}.
+This makes it easy to compute projected points:
+%
+\begin{codeexample}[preamble={\usetikzlibrary{calc}}]
+\begin{tikzpicture}
+ \draw [help lines] (0,0) grid (3,2);
+
+ \coordinate (a) at (0,1);
+ \coordinate (b) at (3,2);
+ \coordinate (c) at (2.5,0);
+
+ \draw (a) -- (b) -- (c) -- cycle;
+
+ \draw[red] (a) -- ($(b)!(a)!(c)$);
+ \draw[orange] (b) -- ($(a)!(b)!(c)$);
+ \draw[blue] (c) -- ($(a)!(c)!(b)$);
+\end{tikzpicture}
+\end{codeexample}