Easy curves in TikZ

Question

I find myself needing to draw lots of elegantly curved paths in TikZ. Ideally, I'd just specify a series of points, and TikZ would calculate the extra data itself to draw a nice series of curves passing smoothly through these points, perhaps with an optional "looseness" parameter that I could specify. But the only way I can find to draw nice curves is by explicitly giving control points, or by manually specifying in and out angles.

I can think up a simple algorithm to do this, which would certainly be within TikZ's power to perform: just choose the in and out angles in a simple fashion based on the relative angles between each adjacent pair of line segments.

Is something like this already built-in? Or can someone cook something up that does the job?

Edit: Jake has provided an answer using the plot [smooth] functionality. This is almost perfect! But it can't do what I need, because it doesn't let me specify tangent angles manually where needed, which is especially important at the beginning and end of the curve. I would have thought this would be a natural and straightforward addition to the existing plot [smooth] algorithm: for every coordinate, an optional angle should be able to be specified as an argument, which if supplied is treated as the tangent angle for the curve at that point. And while we're at it, it wouldn't hurt to also allow the tension to be modified along the path.

A minimal extension to the algorithm would just accept two optional parameters, for the curve tangent at the beginning and end.

Answer 1

You can use the \draw plot [smooth] coordinates {<coordinate1> <coordinate2> <coordinate3> ...}; syntax, which uses an algorithm similar to the one you described.

The looseness is controlled by the tension parameter. If you want to close the line, you can use [smooth cycle] instead of smooth:

\documentclass{article}

\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
\draw [gray!50]  (0,0) -- (1,1) -- (3,1) -- (1,0)  -- (2,-1) -- cycle;
\draw [red] plot [smooth cycle] coordinates {(0,0) (1,1) (3,1) (1,0) (2,-1)};

\draw [gray!50, xshift=4cm]  (0,0) -- (1,1) -- (2,-2) -- (3,0);
\draw [cyan, xshift=4cm] plot [smooth, tension=2] coordinates { (0,0) (1,1) (2,-2) (3,0)};
\end{tikzpicture}
\end{document}

The smooth algorithm is quite simple: it sets the support points so that the tangent at each corner is parallel to the line from the previous to the next corner. The distance of the support points to the corner is the same in either direction, and proportional to the distance from the previous corner to the next corner. The tension is used as a multiplier for the support point distance. It can not be changed along the curve, and neither can the starting and finishing angles of the line be specified. The algorithm can be found in pgflibraryplothandlers.code.tex as \pgfplothandlercurveto.

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,shapes.misc}

\begin{document}
\begin{tikzpicture}
\tikzset{
    show curve controls/.style={
        decoration={
            show path construction,
            curveto code={
                \draw [blue, dashed]
                    (\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
                    node [at end, cross out, draw, solid, red, inner sep=2pt]{};
                \draw [blue, dashed]
                    (\tikzinputsegmentsupportb) -- (\tikzinputsegmentlast)
                    node [at start, cross out, draw, solid, red, inner sep=2pt]{};
            }
        }, decorate
    }
}

\draw [gray!50]  (0,0) -- (1,1) -- (3,1) -- (1,0)  -- (2,-1) -- cycle;
\draw [show curve controls] plot [smooth cycle] coordinates {(0,0) (1,1) (3,1) (1,0) (2,-1)};
\draw [red] plot [smooth cycle] coordinates {(0,0) (1,1) (3,1) (1,0) (2,-1)};

\draw [gray!50, xshift=4cm]  (0,0) -- (1,1) -- (3,-1) -- (5,1) -- (7,-2);
\draw [cyan, xshift=4cm] plot [smooth, tension=2] coordinates { (0,0) (1,1) (3,-1) (5,1) (7,-2)};
\draw [show curve controls,cyan, xshift=4cm] plot [smooth, tension=2] coordinates { (0,0) (1,1) (3,-1) (5,1) (7,-2)};
\end{tikzpicture}
\end{document}

---

Here is a slightly modified version of the plothandler, which allows you to specify the first and last support point using the TikZ key first support={<point>} and last support={<point>}, where <point> can be any TikZ coordinate expression, such as(1,2), (1cm,2pt), (A.south west), ([xshift=1cm] A.south west) (thanks to Andrew Stacey's wonderful answer to Extract x, y coordinate of an arbitrary point in TikZ).

By default, the points are assumed to refer to coordinates relative to the first/last point of the path. You can specify that the support points are given as absolute coordinates by using the keys absolute first support, absolute last support, or absolute supports.

 \documentclass{article}

\usepackage{tikz}
\usetikzlibrary{decorations.pathreplacing,shapes.misc}

\begin{document}
\begin{tikzpicture}
\tikzset{
    show curve controls/.style={
        decoration={
            show path construction,
            curveto code={
                \draw [blue, dashed]
                    (\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
                    node [at end, cross out, draw, solid, red, inner sep=2pt]{};
                \draw [blue, dashed]
                    (\tikzinputsegmentsupportb) -- (\tikzinputsegmentlast)
                    node [at start, cross out, draw, solid, red, inner sep=2pt]{};
            }
        }, decorate
    }
}

\makeatletter
\newcommand{\gettikzxy}[3]{%
  \tikz@scan@one@point\pgfutil@firstofone#1\relax
  \edef#2{\the\pgf@x}%
  \edef#3{\the\pgf@y}%
}

\newif\iffirstsupportabsolute
\newif\iflastsupportabsolute

\tikzset{
    absolute first support/.is if=firstsupportabsolute,
    absolute first support=false,
    absolute last support/.is if=lastsupportabsolute,
    absolute last support=false,
    absolute supports/.style={
        absolute first support=#1,
        absolute last support=#1
    },
    first support/.code={
        \gettikzxy{#1}{\pgf@plot@firstsupportrelx}{\pgf@plot@firstsupportrely}
    },
    first support={(0pt,0pt)},
    last support/.code={
        \gettikzxy{#1}{\pgf@plot@lastsupportrelx}{\pgf@plot@lastsupportrely}
    },
    last support={(0pt,0pt)}
}

\def\pgf@plot@curveto@handler@initial#1{%
  \pgf@process{#1}%
  \pgf@xa=\pgf@x%
  \pgf@ya=\pgf@y%
  \pgf@plot@first@action{\pgfqpoint{\pgf@xa}{\pgf@ya}}%
  \xdef\pgf@plot@curveto@first{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}%
  \iffirstsupportabsolute
    \pgf@xa=\pgf@plot@firstsupportrelx%
    \pgf@ya=\pgf@plot@firstsupportrely%
  \else
    \advance\pgf@xa by\pgf@plot@firstsupportrelx%
    \advance\pgf@ya by\pgf@plot@firstsupportrely%
  \fi
  \xdef\pgf@plot@curveto@firstsupport{\noexpand\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}%
  \global\let\pgf@plot@curveto@first@support=\pgf@plot@curveto@firstsupport%
  \global\let\pgf@plotstreampoint=\pgf@plot@curveto@handler@second%
}

\def\pgf@plot@curveto@handler@finish{%
  \ifpgf@plot@started%
    \pgf@process{\pgf@plot@curveto@second}
    \pgf@xa=\pgf@x%
    \pgf@ya=\pgf@y%
    \iflastsupportabsolute
      \pgf@xa=\pgf@plot@lastsupportrelx%
      \pgf@ya=\pgf@plot@lastsupportrely%
    \else
      \advance\pgf@xa by\pgf@plot@lastsupportrelx%
      \advance\pgf@ya by\pgf@plot@lastsupportrely%
    \fi
    \pgfpathcurveto{\pgf@plot@curveto@first@support}{\pgfqpoint{\the\pgf@xa}{\the\pgf@ya}}{\pgf@plot@curveto@second}%
  \fi%
}
\makeatother

\coordinate (A) at (2,-1);

\draw [gray!50]  (-1,-0.5) -- (1.5,1) -- (3,0);
\draw [
    cyan,
    postaction=show curve controls
] plot [
    smooth, tension=2,
    absolute supports,
    first support={(A)},
    last support={(A)}] coordinates { (-1,-0.5) (1.5,1) (3,0)};

\draw [
    yshift=-3cm,
    magenta,
    postaction=show curve controls
] plot [
    smooth, tension=2,
    first support={(-0.5cm,1cm)},
    last support={(0.5cm,1cm)}] coordinates { (-1,-0.5) (1.5,1) (3,0)};

\end{tikzpicture}
\end{document}

Answer 2

Another option could be the to operation. With this you can specify the angles but it’s maybe less automatic than the plot using solution that Jake presented.

\documentclass{minimal}

\usepackage{tikz}

\begin{document}
\begin{tikzpicture}
    \draw [ultra thick,red] (-2,2) to[out=45,in=115] (1,1) to[out=-180+115,in=10] (-5,-3);
\end{tikzpicture}
\end{document}

Answer 3

I was surprised that no one used the "bend" option. Here is the code:

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{tikz}


\begin{document}

\begin{tikzpicture}
  \coordinate (O) at (0,0,0);
  \coordinate (A) at (3,0,0);

  \draw[] (O)--(A);
  \draw[color=red] (O) to [bend left=10] (A);
  \draw[color=red] (O) to [bend right=10] (A);
  \draw[color=blue] (O) to [bend left=30] (A);
  \draw[color=blue] (O) to [bend right=30] (A);
  \draw[color=green] (O) to [bend left=50] (A);
  \draw[color=green] (O) to [bend right=50] (A);
  \draw[color=yellow] (O) to [bend left=70] (A);
  \draw[color=yellow] (O) to [bend right=70] (A);
  \draw[color=orange] (O) to [bend left=90] (A);
\end{tikzpicture}


 \end{document}

Here the figure:

I ran this with "lualatex"

This is an extremely useful function. There are many occasions when we need to connect two points and there is not a simple equation or arc function to use between points. This comes quite handy in this context. For example check the following post

curve triangles on a sphere

The red and blue triangles in the bottom picture are faked with the "bend" function. It is hard to find an analytic equation that represent them. Without the "bend" function you would obtain something like the equivalent figure in this post: flat edged triangles on a sphere

While it is true that the connection of two curves using "bend" is not as smooth as a spline (unless the curvature is preserved) this is precisely a point of favor when we want to do certain type of curves which do not require smoothness. The best example I can think of are lunes in a sphere or triangles in a sphere as shown in the link above.

H.

Answer 4

Solution with Rounded Corners

It may have occured that you wanted to draw a smooth line in tikz, by using coordinates and by repeatedly improving the outcome. This can be achieved with \draw[rounded corners].

At some point I wanted to draw a smooth looking line, see the picture below. I first tried using tikz's \draw[smooth] option which was mentioned in earlier answers here. But it often created loops and it wasn't obvious to me how I could improve the intermediate results.

I came across the \draw[rounded corners] option here.

Defining a path

The following code produces a path:

\documentclass[border=5]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=0.4]
\draw[thick,densely dotted, blue,rounded corners=0.4mm] (0,1)--(-0.1,1)--(-0.4,0.87)--(-0.6,0.83)--(-0.8,0.87)--(-1,1.04)--(-1.4,0.9)--(-1.6,1.1)--(-1.8,1.1)--(-1.8,0.8)--(-1.66,0.7)--(-1.68,0.4)--(-1.84,0.24)--(-1.8,0)--(-1.6,0)--(-1.4,0.11)--(-1.2,0.15)--(-1,0.11)--(-0.8,0)--(-0.6,-0.05)--(-0.4,0.02)--(-0.2,0.07)--(0,0);
\fill [black] (0,1) circle (0.1);
\fill [black] (0,0) circle (0.1);
\end{tikzpicture} 
\end{document}

Scaling

After making a path this way, there are two parameters that can be changed, scale and the rounded corner's width. To both scale the image and keep it smooth, it turns out neccesary to change both the scaling factor and the corner width. In this case after increasing the scale more dots are created.

For instance, in the images below I set scale=1 and rounded corners=0.4mm, alongside scale=1 and rounded corners=1mm.

Conclusion

Making curves this way could take substantial time. With a picture in mind (maybe drawn on paper) and having chosen a suitable corner width, you will have to define and adjust the coordinates so as to obtain a smooth, good looking curve. For those who have to draw a lot of these curves this method might not be very practical.

If you have to draw just a single curved line this method will work just fine.

Answer 5

This is a link-only answer because, frankly, this link does not deserve to be buried in comments as it is now. Since this question was asked, a similar question at a later date stimulated the creation of a new package. You specify the points, optionally with angles, tension changes etc., and the library calculates a smooth curve using the Hobby algorithm. It does a better job than TikZ's smooth, is much easier than figuring out control points etc. by hand and generally just makes something quite tricky easily tractable.

Examples and introduction to the hobby library are covered in responses to Curve through a sequence of points with Metapost and TikZ.