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diff --git a/graphics/pgf/base/doc/pgfmanual-en-tikz-coordinates.tex b/graphics/pgf/base/doc/pgfmanual-en-tikz-coordinates.tex new file mode 100644 index 0000000000..bf6047fc1b --- /dev/null +++ b/graphics/pgf/base/doc/pgfmanual-en-tikz-coordinates.tex @@ -0,0 +1,1329 @@ +% 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} |