Tikz library : command \draw … to… with out and in specification

Question

I would like to ask how the following command works:

\draw [] (0,0) to [out=20,in=120] (50,20) to [out=-60, in=110] (90,20);

I wonder what part .. to [out=20,in=120] .. makes. The result is a curve, but I don't know how to properly understand these parameters.

Answer 1

From the pgfmanual :

If you write (a) to [out=135,in=45] (b) a curve is added to the path, which leaves at an angle of 135° at a and arrives at an angle of 45° at b. This is because the options in and out trigger a special path to be used instead of the straight line.

This actually allows you to bend paths between two points. The angles are specified with respect to the horizontal line, or, if the option relative is passed, with respect to the straight line between the two points.

Consider the simple example below :

The red curve is the result of (a) to [out=20, in=120] (b) as in your example. The angles specified by the out and in parameters are described on the picture. An easy way to remember the parameters is that you specify the angle by which your curve is going out of the source node, and entering in the target node.

To understand the difference induced by the option relative, have a look at the following two examples : first without the option relative, so absolute angles :

And now with relative angles :

Code

\documentclass{article}

\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
  \node[circle,draw] (a) at (0,0) {a};
  \node[circle,draw] (b) at (10,0) {b};
  \draw[dashed] (a) -- (b);
  \draw[dashed] (a) -- +(20:10cm);
  \draw[dashed] (b) -- +(120:4cm);
  \draw[dashed] (b) -- +(0:3cm);

  \draw (a) +(4,0)  arc (0:20:4);
  \draw (a) + (10:4.3) node {$20^{\circ}$};
  \draw (b) + (2,0) arc (0:120:2);
  \draw (b) + (60:2.3) node {$120^{\circ}$};

  \draw[red] (a) to[out=20,in=120] (b);
\end{tikzpicture}

\clearpage

\begin{tikzpicture}
  \node[circle,draw] (a) at (0,0) {a};
  \node[circle,draw] (b) at (0,5) {b};
  \draw[dashed] (a) -- +(0:5cm);
  \draw[dashed] (a) -- +(20:10cm);
  \draw[dashed] (b) -- +(120:4cm);
  \draw[dashed] (b) -- +(0:3cm);

  \draw (a) +(4,0)  arc (0:20:4);
  \draw (a) + (10:4.3) node {$20^{\circ}$};
  \draw (b) + (2,0) arc (0:120:2);
  \draw (b) + (60:2.3) node {$120^{\circ}$};

  \draw[red] (a) to[out=20,in=120] (b);
\end{tikzpicture}

\clearpage

\begin{tikzpicture}[relative]
  \node[circle,draw] (a) at (0,0) {a};
  \node[circle,draw] (b) at (0,5) {b};
  \draw[dashed] (a) -- (b);
  \draw[dashed] (a) -- +(110:10cm);
  \draw[dashed] (b) -- +(210:4cm);
  \draw[dashed] (b) -- +(90:3cm);

  \draw (a) +(0,2)  arc (90:110:2);
  \draw (a) + (100:2.3) node {$20^{\circ}$};
  \draw (b) + (0,2) arc (90:210:2);
  \draw (b) + (120:2.3) node {$120^{\circ}$};

  \draw[red] (a) to[out=20,in=120] (b);
\end{tikzpicture}  


\end{document}

Acknowledgements

Thanks to Paul Gaborit for his remark regarding the option relative.