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}

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

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

newcommandDrawControl[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}

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

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

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}

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}

documentclass[border=5pt,tikz]{standalone}
tikzset{
  ctrlpoint/.style={%
    draw=gray,
    circle,
    inner sep=0,
    minimum width=1ex,
  }
}
newcommandBezier[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}

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}

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}