How does one pick control points to control Bézier curves in TikZ?

Question

[Note: the system didn't like the word 'you' in my question. It prefers 'one'. Go figure.]

When I first used TikZ, I struggled to draw curves using control points.

\documentclass[tikz,border=10pt,multi]{standalone}

\begin{document}

\begin{tikzpicture}

  \draw (0,0) .. controls (1,1) and (2,-1) .. (3,0);

\end{tikzpicture}

\end{document}

This works great if I happen to want a curve like this

But if I want a curve like this

I have no idea how to figure out how to substitute for the question marks

  \draw (0,0) .. controls (??,??) and (??,??) .. (3,0);

Note that I know the two images look identical. That is because they are the same image. That's my point.

When I discovered TikZ offered other ways to draw curves, I essentially gave up on Bézier curves altogether.

But in many cases, Bézier curves look nicer and Wikipedia assures me that they can be controlled 'intuitively' by control points which is why they are so popular in drawing computer graphics.

I understand that what Wikipedia probably has in mind is a GUI, where the curves can be controlled more-or-less intuitively by adjusting control points. (I take it this is what I do in GIMP, for example.)

But how can I figure out which control points to specify in order to draw a particular curve without doing it by mere trial-and-error?

Indeed, how can I decide when to draw a curve using this method rather than one of the other curve-drawing methods TikZ provides?

I'm not asking how to calculate precise values. I want to know how to get an intuitive idea of roughly what might be about right. I want to have a sense of when doing it this way rather than some other way might make sense.

Answer 1

The analogy that I've always used is to think of control points as points that "pull the path towards them". In the examples below, control points are drawn using the same color than the corresponding path in which they were used; when there are two control points, the number below the point indicates which one goes first in the code.

The example with the loop is particularly good to explain why I meant with "pull the curve towards them": the path starts in (1,0) then the path is pulled towards (10,2) and then it goes backwards when it is pulled towards (-2,2) and finally it heads towards (7,0). Immediately you can see the loop forming in your head.

The code for the image:

\documentclass[parskip=full]{scrartcl}
\usepackage[margin=2cm,paperheight=40cm]{geometry}
\usepackage{tikz}

\newcommand\DrawControl[3]{
  node[#2,circle,fill=#2,inner sep=2pt,label={above:$#1$},label={[black]below:{\footnotesize#3}}] at #1 {}
}

\pagestyle{empty}

\begin{document}
\centering

One control point:\\
\begin{tikzpicture}[baseline]
\draw[help lines] (0,0) grid (8,5);
\draw[ultra thick] 
  (1,0) 
    .. controls (4,0) .. 
  (7,0) \DrawControl{(4,0)}{black}{};  
\draw[ultra thick,blue] 
  (1,0) 
    .. controls (4,2) .. 
  (7,0) \DrawControl{(4,2)}{blue}{};  
\draw[ultra thick,red] 
  (1,0) 
    .. controls (4,6) .. 
  (7,0) \DrawControl{(4,6)}{red}{};  
\end{tikzpicture}\hfill
\begin{tikzpicture}[baseline]
\draw[help lines] (0,0) grid (8,5);
\draw[ultra thick] 
  (1,0) 
    .. controls (2,0) .. 
  (7,0) \DrawControl{(2,0)}{black}{};  
\draw[ultra thick,blue] 
  (1,0) 
    .. controls (2,2) .. 
  (7,0) \DrawControl{(2,2)}{blue}{};  
\draw[ultra thick,red] 
  (1,0) 
    .. controls (2,6) .. 
  (7,0) \DrawControl{(2,6)}{red}{};  
\end{tikzpicture}

\rule{\textwidth}{2pt}

Two control points:\\
\begin{tikzpicture}[baseline]
\draw[help lines] (0,0) grid (8,3);
\draw[ultra thick,blue] 
  (1,0) 
    .. controls (3,2) and (5,2) .. 
  (7,0) \DrawControl{(3,2)}{blue}{1}\DrawControl{(5,2)}{blue}{2} ;  
\draw[ultra thick,red] 
  (1,0) 
    .. controls (3,4) and (5,4) .. 
  (7,0) \DrawControl{(3,4)}{red}{1}\DrawControl{(5,4)}{red}{2};  
\end{tikzpicture}\hfill
\begin{tikzpicture}[baseline]
\draw[help lines] (0,0) grid (8,3);
\draw[ultra thick,blue] 
  (1,0) 
    .. controls (2,2) and (6,2) .. 
  (7,0) \DrawControl{(2,2)}{blue}{1}\DrawControl{(6,2)}{blue}{2};  
\draw[ultra thick,red] 
  (1,0) 
    .. controls (2,4) and (6,4) .. 
  (7,0) \DrawControl{(2,4)}{red}{1}\DrawControl{(6,4)}{red}{2};  
\end{tikzpicture}

\rule{\textwidth}{2pt}

\vspace{3cm}

\begin{tikzpicture}[baseline]
\draw[help lines] (0,0) grid (8,3);
\draw[overlay,ultra thick,blue] 
  (1,0) 
    .. controls (10,2) and (-2,2) .. 
  (7,0) \DrawControl{(10,2)}{blue}{1}\DrawControl{(-2,2)}{blue}{2};  
\draw[overlay,ultra thick,red] 
  (1,0) 
    .. controls (12,4) and (-4,4) .. 
  (7,0) \DrawControl{(12,4)}{red}{1}\DrawControl{(-4,4)}{red}{2};  
\end{tikzpicture}

\rule{\textwidth}{2pt}

\begin{tikzpicture}[baseline]
\draw[help lines] (0,-2) grid (8,2);
\draw[ultra thick,blue] 
  (1,0) 
    .. controls (3,2) and (5,-2) .. 
  (7,0) \DrawControl{(3,2)}{blue}{1}\DrawControl{(5,-2)}{blue}{2};  
\draw[ultra thick,red] 
  (1,0) 
    .. controls (-1,5) and (8,-5) .. 
  (7,0) \DrawControl{(-1,5)}{red}{1}\DrawControl{(8,-5)}{red}{2};  
\end{tikzpicture}

\rule{\textwidth}{2pt}

\end{document}

With the examples and, hopefully with the help of the "pull towards" analogy, it should become clear that, in order to get a path like the one in your question, you should use two control points with these characteristics:

• Both control points should have same absolute value for the y-coordinate (given the symmetry with respect to the x-axis). The y-coordinate for the first control point should be positive (to pull the path upwards), whilst the y-coordinate for the second control point should be negative (to pull the path downwards).

• The distance between the x-coordinates for the initial point and the first control point should be equal to the one between the x-coordinates for the second control point and the final point (given the symmetry with respect to the "middle" point on the path).

So

\draw (0,0) .. controls (??,??) and (??,??) .. (3,0);

will become something like

\draw (0,0) .. controls (1,2) and (2,-2) .. (3,0);

Answer 2

Just for fun, here is an animation of some Bézier curves and of their control points.

To get this picture, compile the document below then call convert from imagemagick:

convert -density 150 tikz-bezier-animation.pdf tikz-bezier-animation.gif

The document (tikz-bezier-animation.tex):

\documentclass[tikz]{standalone}

\usetikzlibrary{decorations.pathreplacing,backgrounds}

\tikzset{
  show curve controls/.style={
    decoration={
      show path construction,
      curveto code={
        \draw[#1!50]
        (\tikzinputsegmentfirst)
        -- (\tikzinputsegmentsupporta)
        -- (\tikzinputsegmentsupportb)
        -- (\tikzinputsegmentlast)
        ;
        \fill[#1!50] (\tikzinputsegmentsupporta) circle(1pt);
        \fill[#1!50] (\tikzinputsegmentsupportb) circle(1pt);
        \draw[#1,line width=1pt]
        (\tikzinputsegmentfirst)
        .. controls (\tikzinputsegmentsupporta)
                and (\tikzinputsegmentsupportb) ..
        (\tikzinputsegmentlast);
      }
    },decorate
  }
}

\begin{document}
\foreach \p in {0,10,...,360} {
  \begin{tikzpicture}
    \begin{scope}
      \path (-4,-2) rectangle (4,2.1);
      \coordinate (a) at (-2,0);
      \coordinate (b) at (2,0);
      \path (a) ++(1,0) ++(\p:0 and 2) coordinate (a1);
      \path (b) ++(-1,0) ++({180-\p}:0 and 2) coordinate (b1);
      \draw[show curve controls={red}] (a) .. controls (a1) and (b1) .. (b);
    \end{scope}
    \begin{scope}[yshift=-4.5cm]
      \path (-4,-1) rectangle (4,4);
      \coordinate (a) at (-2,0);
      \coordinate (b) at (2,0);
      \path (a) ++(45:3) ++(\p:3 and 0) coordinate (a1);
      \path (b) ++(90+45:3) ++(180-\p:3 and 0) coordinate (b1);
      \draw[show curve controls={blue}] (a) .. controls (a1) and (b1) .. (b);
    \end{scope}
    \begin{scope}[yshift=-6cm]
      \path (-4,-3) rectangle (4,4);
      \coordinate (a) at (-2,0);
      \coordinate (b) at (2,0);
      \path (a) ++(1,0) [rotate=45] ++(\p:0 and 2) coordinate (a1);
      \path (b) ++(-1,0) [rotate=45] ++({180+\p}:0 and 2) coordinate (b1);
      \draw[show curve controls={green!50!black}]
        (a) .. controls (a1) and (b1) .. (b);
    \end{scope}
    \begin{pgfonlayer}{background}
      \fill[white] (current bounding box.south west)
         rectangle (current bounding box.north east);
    \end{pgfonlayer}
  \end{tikzpicture}
}
\end{document}

Answer 3

In addition to other answers, it is worth noting that it is possible to specify the control points relative to the ends points. When used in combination with polar coordinates, this approach gives a slightly more intuitive approach (I think) to visualising how a Bézier curve will turn out.

In particular, it makes it a bit easier to make consecutive Bézier curves appear to be smoothly joined: in most cases it is possible to just add 180 degrees to the angle of the second control point of the previous curve or (as pointed out by Paul Gaborit) use the same angle and reverse the sign of the distance:

\documentclass[tikz,border=5]{standalone}
\usetikzlibrary{decorations.pathreplacing}
\tikzset{%
  show curve controls/.style={
    postaction={
      decoration={
        show path construction,
        curveto code={
          \draw [blue] 
            (\tikzinputsegmentfirst) -- (\tikzinputsegmentsupporta)
            (\tikzinputsegmentlast) -- (\tikzinputsegmentsupportb);
          \fill [red, opacity=0.5] 
            (\tikzinputsegmentsupporta) circle [radius=.5ex]
            (\tikzinputsegmentsupportb) circle [radius=.5ex];
        }
      },
      decorate
}}}
\begin{document}
\begin{tikzpicture}
\draw [help lines] (-4, -1) grid (4, 5);
\draw [show curve controls]
  (-3, 4) .. controls ++(135:-1) and ++(135:1) .. (0, 4);
\draw [show curve controls] (0, 0) 
  .. controls ++(165:-1) and ++(240: 1) .. ( 3, 2)
  .. controls ++(240:-1) and ++(165:-1) .. ( 2, 4)
  .. controls ++(165: 1) and ++(175:-2) .. (-1, 2)
  .. controls ++(175: 2) and ++(165: 1) .. ( 0, 0);
\end{tikzpicture}
\end{document}

Answer 4

In addition to Gonzalo Medina's answer, I would like to point out that another criterion you could base your drawings on is that the Bézier curve is tangent to the lines going from the first control point to the second and from the second last to the last. This could be something that gives you an idea of the initial and final path of the curve (something like the to [out=...,in=...] notation).

In the following examples I borrowed some of GM's points for comparison purposes.

\documentclass[border=5pt,tikz]{standalone}
\tikzset{
  ctrlpoint/.style={%
    draw=gray,
    circle,
    inner sep=0,
    minimum width=1ex,
  }
}
\newcommand\Bezier[4]{% \bezier (lowercase 'b') was already defined elsewhere
  \node (p1) [ctrlpoint,label=90:$P_1$] at (#1) {};
  \node (p2) [ctrlpoint,label=90:$P_2$] at (#2) {};
  \node (p3) [ctrlpoint,label=90:$P_3$] at (#3) {};
  \node (p4) [ctrlpoint,label=90:$P_4$] at (#4) {};
  \draw [gray] (p1) -- (p2) -- (p3) -- (p4);
  \draw [blue] (#1) .. controls (#2) and (#3) .. (#4);
}
\begin{document}
\begin{tikzpicture}
  \Bezier{0,0}{1,1}{2,-1}{3,0}
  \begin{scope}[xshift=4cm]
    \Bezier{0,0}{9,2}{-2,2}{7,0}
  \end{scope}
  \begin{scope}[yshift=-5cm]
    \Bezier{0,0}{1,3}{2,3}{7,0}
  \end{scope}
  \begin{scope}[xshift=8cm,yshift=-5cm]
    \Bezier{0,0}{-2,4}{4,-1}{5,0}
  \end{scope}
\end{tikzpicture}
\end{document}

If you want to see a naive implementation of a 5 points-based Bézier curve, I posted a question a short while ago; it has a link to a Wikipedia page explaining the algorithm used and demonstrating the "tangency" aforementionend.

Answer 5

I found picking control points to be such a terrible experience that I wrote some code to construct a Bézier spline through some given points.

You load the library in the usual way: \usetikzlibrary{spline}.

Some examples from the manual (the pictures are taken from the manual so the background color is different):

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{spline}
\begin{document}
\begin{tikzpicture}[thick]
  \draw[help lines] (0,0) grid (3,2);
  \node[draw=red, circle] (A) at (1,0) {A};
  \node[draw=blue, rectangle] (B) at (2,2) {B};
  \node[draw=green!60!black, circle] (C) at (3,1) {C};
  \draw (A) to[spline through={(0,1)(B)(2,1)}] (C);
\end{tikzpicture}
\end{document}

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{spline}
\begin{tikzpicture}[thick]
  \draw[help lines] (0,0) grid (3,2);
  \coordinate (A) at (1,0);
  \coordinate (B) at (2,2);
  \coordinate (C) at (3,1);

  \draw (A) to[spline coordinates=S,
               closed spline through={(0,1)(B)(2,1)(C)}] (A);
  \foreach \i [evaluate=\i as \j using \i+1] in {1,2,...,\tikzsplinesegments} {
    \draw[semithick]  (S K-\i) -- (S P-\i)
                      (S K-\j) -- (S Q-\i);
    \path[fill=green!60!black] (S P-\i) circle (2pt);
    \path[fill=red] (S Q-\i) circle (2pt);
  }
\end{tikzpicture}
\end{document}

So my answer is: I don't! I'd much rather let the computer pick the control points.

Answer 6

Chỉ cho vui thôi. (Just for fun)

Compile with Asymptote.

import graph;
pair Bezier(pair P[], real t)
{
pair Bezi;
for (int k=0; k <= P.length-1; ++k)
{  Bezi=Bezi+choose(P.length-1,k)*(1-t)^(P.length-1-k)*t^k*P[k]; }
return Bezi;
}
pair BezierRecursion(pair P[], real t){
int n = P.length - 1;
pair Q[][]=new pair[P.length][P.length];
for (int i=0; i <= n; ++i){ Q[i][0]=P[i]; }
for (int j=1; j <=n; ++j){
for (int i=j; i<=n; ++i){
Q[i][j]=(1-t)*Q[i-1][j-1]+t*Q[i][j-1];
}
}
return Q[n][n];
}
unitsize(1cm);
pair[] P={(0,0),(4,1),(5,3),(2,5),(3.8,6.5)}; // 5 points
pair F(real t){return Bezier(P,t);}
pair G(real t){return BezierRecursion(P,t);}
write(F(0.76)); write(G(0.76));
draw(graph(F,0,1),red);
dot(P); 
draw(operator --(... P));
draw(shift(4,0)*graph(G,0,1),red);
dot(shift(4,0)*P);  draw(shift(4,0)*operator --(... P));

An animation: https://en.wikipedia.org/wiki/B%C3%A9zier_curve#/media/File:B%C3%A9zier_4_big.gif

import animate;
settings.tex="pdflatex"; 
settings.outformat="pdf"; 

animation Ani;
import graph;
size(400);

pair Bezier(pair P[], real t)
{
pair Bezi;
// real choose(int n, int k); // Mathematical functions (page 69)
for (int k=0; k <= P.length-1; ++k)
{
// https://en.wikipedia.org/wiki/B%C3%A9zier_curve
Bezi=Bezi+choose(P.length-1,k)*(1-t)^(P.length-1-k)*t^k*P[k];
}
return Bezi;
}
pair[] P={(0,0),(-.5,3),(4,3),(6,0),(9,2.6)};
pair F(real t){return Bezier(P,t);}
real t=0.5;
pair A[]=new pair[4], B[]=new pair[3], C[]=new pair[2], D;
for (int k=0; k <= 50; ++k)
{
t=k/50;
save();
// A array has 4 points
for (int i : sequence(A.length)) { A[i] = interp(P[i],P[i+1],t); }
// B array has 3 points
for (int i : sequence(B.length)) { B[i] = interp(A[i],A[i+1],t); }
// C array has 2 points
C[0]=interp(B[0],B[1],t);
C[1]=interp(B[1],B[2],t); 
// D array has 1 points
D=interp(C[0],C[1],t);

draw(operator --(... P),magenta+0.7bp);
draw(operator --(... A),(t != 0) ? green+0.7bp : invisible);
draw(operator --(... B),(t != 0) ? orange+0.7bp : invisible);
draw(operator --(... C),(t != 0) ? blue+0.7bp : invisible);
draw(graph(F,0,t),(t != 0) ? red+1.3bp : invisible);
dot(P,magenta);
dot(A,(t != 0) ? green : invisible);
dot(B,(t != 0) ? orange : invisible);
dot(C,(t != 0) ? blue : invisible);
dot(D,pink);
label(Label(format("t=%.2f",t),Fill(white)),(3,0));
Ani.add();
restore();
}
erase();
Ani.movie(BBox(3mm,Fill(black)));