Is it possible to have a Bézier curve with 3 control points in TikZ?

Question

In TikZ, it is possible to draw a Bézier curve of degree 3 with 2 control points p0, p1, p2, and p3 as follows:

\draw(p0)..controls (p1) and (p2)..(p3);

I am wondering whether there is a way to draw a Bézier curve with 5 points where 3 of them work as control points. That would then look like:

\draw(p0)..controls (p1) and (p2) and (p3)..(p4);

It looks like it is not available.

I tried to give a brief explanation in this answer for the regular case. If you can tell us what the extra point will do, then, maybe we can come up with something. But I also didn't understand your original intention. — percusse, Mar 11 '12 at 11:00
@percusse: Bézier curves are not limited to 4 points and a polynomial of degree 3 (what you call regular case) and I am wondering how I can get with Tikz a Bézier curve with 3 control points and 2 end points (instead of two control points and 2 end points). — pluton, Mar 11 '12 at 14:12
I understand that but you can do it with a series of ..controls.. commands instead of a one shot. — percusse, Mar 11 '12 at 14:14
yes but you will not get the same curve. You will have two polynomials of degree 3 instead of one of degree 4. — pluton, Mar 11 '12 at 14:18
Not sure why? http://en.wikipedia.org/wiki/B%C3%A9zier_curve — pluton, Mar 11 '12 at 14:49
Bezier curves are a family of curves of arbitrary degree so the question makes sense. However, TikZ/PGF only implements cubic beziers because the output formats that it uses do not do higher degree curves. — Andrew Stacey, Mar 11 '12 at 20:06
so that would be a limitation of the postscript language for instance? — pluton, Mar 11 '12 at 21:40
@pluton A quick search shows evidence that postscript only guarantees cubic beziers. I'm pretty sure that PDF only goes up to cubics. SVG certainly goes no further than cubics. So, yes, it is a "limitation" of, for example, the postscript language. But it seems to be a fairly standard limitation. — Andrew Stacey, Apr 13 '12 at 07:56

score 11 · Accepted Answer · answered Apr 13 '12 at 08:07

The fact that TikZ/PGF only has capabilities for cubic beziers is a facet of the output formats that it produces: those only support cubic beziers.

For postscript, see page 564 (internal numbering, 578 total) of PostScript® Language Reference, Third Edition, available from http://partners.adobe.com/public/developer/ps/index_specs.html.
For PDFs, you can check Adobe's version before it was converted to an ISO (Adobe's version is free) Adobe PDF Reference, Sixth Edition available from http://www.adobe.com/devnet/pdf.html. The relevant information starts on page 226.
For SVGs, see http://www.w3.org/TR/SVG11/paths.html#PathDataCurveCommands

So to produce anything of higher degree, you would have to produce an approximation (indeed, TikZ/PGF approximates arcs by cubic beziers since PDF does not have a native arc command (Postscript does)). There are methods that could be used for implementing one, for example using a plot command, but as it would be an approximation and not a common feature, there is not - to my knowledge - an already inbuilt implementation. If you found an algorithm for approximating higher degree curves by cubic beziers, I can think of no reason why it would not be straightforward to implement one via some sort of to path magic.

score 5 · Answer 2 · edited Nov 06 '12 at 10:30

Here is some code to create a Bezier's curve with three control points. If you require more or less points to draw the curve, just modify the argument {0.05,0.1,...,1} on the "foreach" line. With the provided values the curve is drawn with 20 points. P0 and P4 are the end points, while p1, p2, and p3 are the control points.

\newcommand {\bezierq}[5]{
%\bezierq{p0}{p1}{p2}{p3}{p4};
\newdimen\pxa
\newdimen\pya
\newdimen\pxb
\newdimen\pyb
\newdimen\pxc
\newdimen\pyc
\newdimen\pxd
\newdimen\pyd
\newdimen\pxe
\newdimen\pye
\pgfextractx{\pxa}{\pgfpointanchor{#1}{center}}
\pgfextracty{\pya}{\pgfpointanchor{#1}{center}}
\pgfextractx{\pxb}{\pgfpointanchor{#2}{center}}
\pgfextracty{\pyb}{\pgfpointanchor{#2}{center}}
\pgfextractx{\pxc}{\pgfpointanchor{#3}{center}}
\pgfextracty{\pyc}{\pgfpointanchor{#3}{center}}
\pgfextractx{\pxd}{\pgfpointanchor{#4}{center}}
\pgfextracty{\pyd}{\pgfpointanchor{#4}{center}}
\pgfextractx{\pxe}{\pgfpointanchor{#5}{center}}
\pgfextracty{\pye}{\pgfpointanchor{#5}{center}}
%\def\pxi{4}
%\def\pyi{4}
\foreach \t in {0.05,0.1,...,1}{
  \pgfmathsetmacro{\pxf}{\pxa*(1-\t)^4 + \pxb*4*\t*(1-\t)^3 + \pxc*6*\t^2*(1-\t)^2 + \pxd*4*(1-\t)*\t^3 + \pxe*\t^4}
  \pgfmathsetmacro{\pyf}{\pya*(1-\t)^4 + \pyb*4*\t*(1-\t)^3 + \pyc*6*\t^2*(1-\t)^2 + \pyd*4*(1-\t)*\t^3 + \pye*\t^4}
  \pgfmathsetmacro{\q}{\t-0.05}
  \pgfmathsetmacro{\pxi}{\pxa*(1-\q)^4 + \pxb*4*\q*(1-\q)^3 + \pxc*6*\q^2*(1-\q)^2 + \pxd*4*(1-\q)*\q^3 + \pxe*\q^4}
  \pgfmathsetmacro{\pyi}{\pya*(1-\q)^4 + \pyb*4*\q*(1-\q)^3 + \pyc*6*\q^2*(1-\q)^2 + \pyd*4*(1-\q)*\q^3 + \pye*\q^4}
  \draw (\pxi pt,\pyi pt)--(\pxf pt,\pyf pt);
}

Welcome to TeX.sx! Your code looks like a useful starting point, thanks for the contribution. — Andrew Stacey, Nov 06 '12 at 10:33
@Juan yes interesting, thank you. By the way, on how many points does the usual command \draw(p0)..controls (p1) and (p2)..(p3); draw the curve? — pluton, Nov 06 '12 at 19:56

Is it possible to have a Bézier curve with 3 control points in TikZ?

2 Answers2