What is the easiest method to construct a bezier curve for the half part of a symmetric bezier curve?

Consider the following symmetric (about vertical axis) bezier curve constructed by 4 points A, B, C, and D. Now I want to construct another bezier curve (named L) for just either the left or right part of it.

\documentclass[pstricks,border=24pt]{standalone}
\usepackage{pst-eucl}
\begin{document}

\begin{pspicture}[showgrid=false](-3,-1)(3,3)
\pstGeonode
(3,3){A}
(1,1){B}
(-1,1){C}
(-3,3){D}
\psbezier(A)(B)(C)(D)
\psline[linecolor=red](0,3)(0,-1)
\end{pspicture}

\end{document} Without solving a polynomial equation first, how can I construct the bezier curve L? Answers with PSTricks (preferred) or TikZ or Asymptote are welcome.

Note that I hate a trial and error method!

• Is it specifically this curve and only the half of it or more general? – percusse Dec 29 '13 at 19:56
• @percusse: \psbezier can only accept 4 points at most so @g.kov solution is enough. No need to expand for higher-order bezier curves consisting of more than 4 points. – kiss my armpit Dec 29 '13 at 20:04
• PDF doesn't support higher order anyway. I meant is this specific to these curves or arbitrary curves? – percusse Dec 29 '13 at 20:09
• @percusse: I don't understand you. The given 4 points can be changed to any numbers as long as its curve is symmetrical about vertical axis. :-) – kiss my armpit Dec 29 '13 at 20:11
• See, we have a constraint of symmetricity? Do you have others that's what I asked. – percusse Dec 29 '13 at 20:21 Here is the Asymptote version, which does not use the general Asymptote commands to split a path (guide), it just performs a simple subdivision of the cubic Bezier segment:

// split.asy:

size(5cm);
pair A,B,C,D;
A=(3,3); B=(1,1); C=(-1,1); D=(-3,3);

pair P,B1,C1,B2,C2;

P=(A+D+3(B+C))/8;

B1 = (A+B)/2;
C1 = ((A+C)/2+B)/2;
C2 = (D+C)/2;
B2 = ((D+B)/2+C)/2;

guide g=A..controls B and C..D;
guide gl=A..controls B1 and C1..P;
guide gr=P..controls B2 and C2..D;
draw(g);
draw(gl,deepgreen+1.6bp+opacity(0.5));
draw(gr,red+1.6bp+opacity(0.5));
draw((0,3)--(0,-1),red);

dot(A--B--C--D--P,UnFill);
dot(B1--C1,deepgreen,UnFill);
dot(B2--C2,red,UnFill);
label("$A$",A,SE);
label("$B$",B,S);
label("$C$",C,S);
label("$D$",D,SW);
label("$P$",P,NE);
label("$B_1$",B1,E,deepgreen);
label("$C_1$",C1,E,deepgreen);
label("$B_2$",B2,W,red);
label("$C_2$",C2,W,red);

To get a standalone split.pdf, run asy -f pdf split.asy.

• I really need the converting formulae. – kiss my armpit Dec 29 '13 at 19:29
• @Stiff Jokes: Btw, the Bezier curve is a very interesting object, you might enjoy studying it in a spare time. – g.kov Dec 29 '13 at 19:43

A translated version for @g.kov's Asymptote answer in PSTricks.

\documentclass[pstricks,border=24pt]{standalone}
\usepackage{pst-eucl}
\begin{document}

\begin{pspicture}[showgrid=true](-3,-1)(3,3)
\pstGeonode
(3,3){A}
(1,1){B}
(-1,1){C}
(-3,3){D}
\psbezier(A)(B)(C)(D)
\psline[linecolor=red](0,3)(0,-1)
\nodexn{.125(A)+.125(D)+.375(B)+.375(C)}{R'}
\nodexn{.5(A)+.5(B)}{P'}
\nodexn{.25(A)+.5(B)+.25(C)}{Q'}
\pstGeonode
(P'){P}
(Q'){Q}
(R'){R}
\psbezier[linecolor=red](A)(P)(Q)(R)
\end{pspicture}

\end{document} 