% Copyright 2006 by Till Tantau
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\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} specified 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[style=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[style=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[style=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{itemize}
  \itemoption{x}|=|\meta{dimension} Distance by which the coordinate
  is to the right of the origin. You can also write things like
  |1cm+2pt| since the |calc| package is used. 
  \itemoption{y}|=|\meta{dimension} Distance by which the coordinate
  is above the origin.
  \end{itemize}
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[style=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 seperated by a comma as in |(0cm,3pt)| or
  |(2cm,\textheight)|. 
\begin{codeexample}[]
\begin{tikzpicture}
  \draw[style=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 $\bigl(-\frac{1}{\sqrt2}
  \textrm{cm},-\frac{1}{\sqrt2}\textrm{cm}\bigr)$.

  To specify the factors by which the vectors should be multiplied
  before being added, you use the following three options:  
  \begin{itemize}
  \itemoption{x}|=|\meta{factor} Factor by which the $x$-vector is
  multiplied. If this option is not given, |0| is used.
  \itemoption{y}|=|\meta{factor} Works like |x|.
  \itemoption{z}|=|\meta{factor} Works like |x|.
  \end{itemize}
\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-seperated 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}


\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. A degree of zero points to the right, a degree of
  90 upward.
  \begin{itemize}
  \itemoption{angle}|=|\meta{degrees} The angle of the coordinate.
  The angle must always be given in degrees and should be between
  $-360$ and $720$.
  \itemoption{radius}|=|\meta{dimension} The distance from the origin.
  \itemoption{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 specifiying identical |x radius| and |y radius| options.
  \itemoption{y radius}|=|\meta{dimension} Works like |x radius|.
  \end{itemize}
\begin{codeexample}[]
\tikz \draw (0,0) -- (canvas polar cs:angle=30,radius=1cm);
\end{codeexample}

  The implicit form for canvas polar coodinates 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 detailedly,
  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. Voilà.
  \begin{itemize}
  \itemoption{angle}|=|\meta{degrees} The angle of the coordinate
  interpreted in the ellipse whose axes are the $x$-vector and the
  $y$-vector. 
  \itemoption{radius}|=|\meta{factor} A factor by which the $x$-vector
  and $y$-vector are multiplied prior to forming the ellipse.
  \itemoption{x radius}|=|\meta{dimension} A specific factor by which
  only the $x$-vector is multiplied.
  \itemoption{y radius}|=|\meta{dimension} works like |x radius|.
  \end{itemize}
\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 x-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 specifiy such coordiantes 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{itemize}
  \itemoption{name}|=|\meta{node name} specifies the node in which you
  which 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.
  \itemoption{anchor}|=|\meta{anchor} specifies an anchor of the
  node. Here is an example: 
\begin{codeexample}[]
\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[-open triangle 90] (node cs:name=rect,anchor=north)
        |- (0,1) -| (node cs:name=shape,anchor=south);
\end{tikzpicture}
\end{codeexample}
  \itemoption{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}[]
\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{itemize}

  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}[]
\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}[]
\begin{tikzpicture}[fill=blue!20]
  \draw[style=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}


           
\subsubsection{Intersection Coordinate Systems}

Often you wish to specify a point that is on the
intersection of two lines. For this, the following coordinate system
is useful:

\begin{coordinatesystem}{intersection}
  To specify the intersection of two line, you provide two lines using
  the following two options:
  \begin{itemize}
  \itemoption{first line}|=(|\meta{first coordinate}|)--(|\meta{second coordinate}|)|
  \itemoption{second line}|=(|\meta{first coordinate}|)--(|\meta{second coordinate}|)|
  \end{itemize}
  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.

  The coordinate specified in this way is the intersection of the two
  lines.  If the lines do not meet or if they are
  identical and arithmetical overflow error will result.

\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 of |\meta{$p_1$}|--|\meta{$p_2$
    }| and |\meta{$q_1$}|--|\meta{$q_2$}|)|}. Note that there are \emph{no}
  parentheses around the $p_i$ and $q_i$. Thus, you would write
  |(intersection of A--B and 1,2--3,0)|.
\end{coordinatesystem}

A frequent special case of 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 another
coordinate system.

\begin{coordinatesystem}{perpendicular}
  This coordinate system works the same way as |intersection|, only
  the lines are specified differently:

  \begin{itemize}
  \itemoption{horizontal line through}|=(|\meta{coordinate}|)|
  Specifies that one line is a horizontal line that goes through the
  given coordinate.  
  \itemoption{vertical line through}|=(|\meta{coordinate}|)|
  Specifies that the other line is vertical and goes through the
  given coordinate.  
  \end{itemize}

  The implicit syntax is to 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 $xy$-coordinate 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}
\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{Relative and Incremental 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.'' 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 one 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 arrives 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}


