[Tex/LaTex] Draw curved lines simply

diagramspstrickstikz-pgf

How to draw nearly-arbitrary curved lines simply? For example, how to draw this graph:

Source Introduction to Algorithms (3rd edition) by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein | Page 45 | Figure 3.1

Now I have two ideas of how to draw such figures:

Divide each curved line to many smaller curved lines which can be drawn by a TikZ command, for example
```
\draw (0,0) to [in=80, out=-30] (3,2);
```
This way is quite natural. However, it is difficult to divide, for instance, the lines in the above figure, in this way. Also, when I have already divided, it is very possible that I will get confused with the coordinates.
Find a formula for each of the curved line, and then draw the graph of the curved line.

This way is good for normal graphs, but for the complicated curved lines as in the above figure, it seems to be impossible.

So none of the ways stated is good enough. Do you have any idea to draw such figures? Your help is much appreciated!

Notes

I'm sorry if the question is a duplicate. I have searched for a while without success.
Your help will be great if you can help me draw the above figure. However, I strongly prefer a general solution for this, because the aim of the question is not just to ask how to draw the figure only.
You can see that there is no MWE in my question. As I explained above, all of my idea so far are just bad, so I haven't applied any of them.
I strongly prefer using TikZ. However, answers using PGF, PStricks, etc. are all very useful to me.

Best Answer

This is an attempt to collect some key methods on one page. Clearly, this discussion is not exhaustive, so I am hoping that this post gets complemented by others, who have other methods and/or opinions. (I made zero effort to precisely reproduce your curves, sorry.)

\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{multicol}
\usepackage{tikz}
\usetikzlibrary{hobby}
\begin{document}
\section*{A few ideas to draw smooth curves}

\begin{multicols}{2}
\subsection*{Your idea: use \texttt{out} and \texttt{in}}
This works, and the result looks smooth if you make sure that the  \texttt{in}
of one point and the \texttt{out} of the next point differ by 180. In the
following pic, we have \texttt{\dots to[out=50,in=\textbf{210}] (1,2) circle(1pt)
to[\textbf{out=30},in=240] \dots} to illustrate this. Notice that the
\texttt{looseness} key can be of great help here.

\begin{tikzpicture}
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] (0,0) circle(1pt) to[out=50,in=210] (1,2) circle(1pt)
to[out=30,in=240,looseness=0.5] (3,3) circle(1pt);
\end{tikzpicture}

\subsection*{Your idea: find an analytic formula}
This is IMHO often the simplest way. One only needs to keep in mind a few basic
functions like $\sin$ and $\tanh$.

\begin{tikzpicture}
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] plot[variable=\x,domain=0:6,smooth]
({\x},{1+0.3*(tanh(3-\x)+1)*sin(180*\x)+0.2*\x});
\end{tikzpicture}

\subsection*{Use \texttt{plot[smooth] coordinates}}

A very convenient option, mentioned by Torj{\o}rn T.\ in the comments, is to use
\texttt{plot[smooth] coordinates}. You may want to play with the
\texttt{tension} key.

\begin{tikzpicture}
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] plot[smooth] coordinates
{(0,0) (1,2) (2,2) (3,3) (4,3)};
\draw[red] plot[smooth,tension=2] coordinates
{(0,0) (1,2) (2,2) (3,3) (4,3)};
\end{tikzpicture}

\subsection*{Use the \texttt{hobby} library}

The \texttt{hobby} library is sort of an el Dorado for smooth curve
constructors. It has way too many options to be listed here. One thing I find
very useful is the \texttt{tangent} style from the manual. 

\begin{tikzpicture}[tangent/.style={%
 in angle={(180+#1)},Hobby finish ,
designated Hobby path=next , out angle=#1,
}]
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] plot[smooth,hobby,tension=0.3] coordinates
{(0,0) (1,1.2) (2,2) (3,3) (4,3)};
\draw [red,use Hobby shortcut] 
   (0,0)  .. ([tangent=30]1.5,1) ..  ([tangent=-10]3,2) .. (5,1);
\end{tikzpicture}

I'd also like to mention that, when it comes to decorations, in my experience
the Hobby paths are advantageous. They often do not lead to \texttt{dimension too
large} errors when their almost identically looking \texttt{plot[smooth]}
counterparts do.

\subsection*{Not sure about \texttt{controls}}

There is also the possibility of using Bezier curves (without \texttt{hobby}).
Many users are able to use that to obtain great results. Unfortunately, I am not
one of those since IMHO the relation between position of the control points and
the outcome is not too obvious. But this is of course a very subjective
statement.
\end{multicols}
\end{document}

Related Solutions

[Tex/LaTex] Curved waved lines with TikZ

Actually the library snakes has been superseded by decorations, but for your need I think you are just looking for decorations.pathmorphing.

Thus, your code revised, is like:

\documentclass{beamer}
\usepackage{lmodern}
\usepackage{tikz}
\usetikzlibrary{decorations.pathmorphing}

\tikzset{snake it/.style={decorate, decoration=snake}}

\begin{document}
\vspace*{2cm}
\begin{center}
\begin{tikzpicture}[thick]
  \path [draw=blue,snake it]
    (-4,0) -- (-2,0) -- (2,0) -- (4,0);
  \draw[draw=blue, snake it] (2,0) arc (0:180:2cm);
\end{tikzpicture}
\end{center}
\end{document}

Notice that I grouped in a single style snake it the keys that really set up the decoration decorate and the one that sets the type of decoration decoration=snake.

For further details, you can have a look to section 72 Decorations and more specifically to section 30.2 Path Morphing Decorations on the pgfmanual. (section 48.2 in pgfmanual 3.0)

[Tex/LaTex] Tikz Graphics: Curved arrow drawn parallel to curved line

The curve drawn by a to[bend left] is probably a bezier curve, and thus is not an arc nor a parabola. Unfortunately this syntas hides the control points used.

Even if the control points were known, they won't be useful for your problem, since basically you want to get another bezier curve parallel to a "segment" or portion of another given bezier curve. I have no idea of the equations that should be solved to find those control points.

But not all hope is lost. I had a crazy idea. Once a bezier path is computed by tikz, any of its intermediate points are available via [pos] key, so it is possible to find a number of those points, between the desired start and end point, and connect all of them via straight lines. If the points are close enough, the straight segments will not be visible and they will look like a curve.

So this is the proof of concept:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\thispagestyle{empty}
\usetikzlibrary{calc}

\begin{tikzpicture}

  % First, define nodes
  \draw (0,0) node[circle, inner sep=0.8pt, fill=black, label={below:{$E$}}] (E) {};  
  \draw (5,0) node[circle, inner sep=0.8pt, fill=black, label={below:{$F$}}] (F) {}; 

  % Draw the curved line. No to[bend] is allowed, only explicit control points
  \draw  (E) .. controls +(1.9,1) and +(-1.9,1).. (F);
  % Now, repeat the same curve, shifted up, and define 20 inner points named
  % p1, p2, p3, etc.. for positions 0.15, 0.16, 0.17, etc up to 0.35 inside that curve
  \path  ($(E)+(0,0.2)$) .. controls +(1.9,1) and +(-1.9,1) ..  ($(F)+(0, 0.2)$)
     {\foreach \t [count=\i] in {0.15,0.16,...,0.35} {  coordinate[pos=\t] (p\i) } };

  % Finally draw the "curve" (polygonal line indeed) through those 20 inner points
  \draw[->>] (p1) { \foreach \i in {1,...,20} {-- (p\i) } };

  % A second example, with the same ideas but different path
  \draw[red]  (E) .. controls +(2,-2) and +(-3,-1).. (F);
  \path  ($(E)+(0,0.2)$) .. controls +(2,-2) and +(-3,-1)..  ($(F)+(0, 0.2)$) 
     {\foreach \t [count=\i] in {0.15,0.16,...,0.55} {  coordinate[pos=\t] (p\i) } };

  \draw[red, ->>] (p1) { \foreach \i in {1,...,40} {-- (p\i) } };
\end{tikzpicture}
\end{document}

This is the result:

Result

Update

A little refinement can be done in above code. Instead of specifying the foreach loop as from 0.15 to 0.55 in steps of 0.1, it looks more natural to simply specify the desired number of inner points (i.e: the "resolution" of the parallel curve), and let to some other expression to compute the pos for each of those points. This is done like this:

\path  ..curve specification..
     {\foreach \i in {1,...,40} {  coordinate[pos=0.15+0.55*\i/40] (p\i) } };

So, in this example, 40 intermediate points are computed, and the formula for the pos key specifies 0.15 and 0.55 as the extremes of the parallel curve.

However, this approach does not use a "true" parallel line. The curve used to compute the points is simply the same orginal curve, only shifted in the Y axis. This could work for smooth arcs which are mostly horizontal, but it will fail for parts of the curve where the tangent is not so horizontal. Consider for example:

\begin{tikzpicture}
  % First, define nodes
  \draw (0,0) node[circle, inner sep=0.8pt, fill=black, label={below:{$E$}}] (E) {};  
  \draw (5,2) node[circle, inner sep=0.8pt, fill=black, label={below:{$F$}}] (F) {}; 

  \draw[red]  (E) .. controls +(5,-3) and +(-4,1).. (F);
  \path  ($(E)+(0,0.2)$) .. controls +(5,-3) and +(-4,1)..  ($(F)+(0,0.2)$) 
     {\foreach \i in {1,...,40} {  coordinate[pos=0.15+0.75*\i/40] (p\i) } };

  \draw[blue, ->>] (p1) { \foreach \i in {1,...,40} {-- (p\i) } };
\end{tikzpicture}

This produces:

Bad Result

It can be seen that the distance between the red and blue lines is not constant. How can this be solved?

The following idea works: instead of shifting the curve, compute the sequence of points (p1), (p2) etc. on the original curve. But then, when drawing the parallel line, do not use those points, but points which are at a fixed distance (eg: 0.2cm) from each of these points in the direction perpendicular to the curve. This direction can be computed as the vector from current point (p_i) to next point (p_j), rotated 90 degrees.

Fortunately, tikz has the interpolated coordinates expression which allows for this with a simple formula: ($(p\i)!0.2cm!90:(p\j)$).

So, using this idea:

\begin{tikzpicture}
  % First, define nodes
  \draw (0,0) node[circle, inner sep=0.8pt, fill=black, label={below:{$E$}}] (E) {};  
  \draw (5,2) node[circle, inner sep=0.8pt, fill=black, label={below:{$F$}}] (F) {}; 

  % Draw curved path
  \draw[red]  (E) .. controls +(5,-3) and +(-4,1).. (F);
  % Compute points on the same curve
  \path  (E) .. controls +(5,-3) and +(-4,1)..  (F) 
     {\foreach \i in {1,...,40} {  coordinate[pos=0.15+0.75*\i/40] (p\i) } };
  % Draw parallel curve
  % (note that first and last points are specified out of the loop)
  \draw[blue, ->>] ($(p1)!.2cm!90:(p2)$) 
     { \foreach \i [count=\j from 3] in {2,...,39} {-- ($(p\i)!.2cm!90:(p\j)$) } }
     -- ($(p40)!.2cm!-90:(p39)$);
\end{tikzpicture}

And this gives a perfect result!

Result2