Rotation transformation of a parametrized plot

Question

I'm working on this plot with pgfplots:

The image is from our sister-site Mathematica.SE and there's also a nice explanation how to do it with Mathematica.

I follow these steps:

1. Parametrize the 2d cut
2. Embed it in 3d
3. Rotate both in a circle and around itself

I originally posted the question on the TeXwelt LaTeX forum in German, my first steps which I wrote in "Drehtransformation mit pgfplots" are:

1. The 2d cut:

   \documentclass[border=10pt]{standalone}
   \usepackage{pgfplots}
   \usepgfplotslibrary{polar}
   \begin{document}
   \begin{tikzpicture}
     \begin{polaraxis}
       \addplot[mark=none, domain=0:360, samples=100] 
         {sin(3*x) + 1.25}; 
     \end{polaraxis}
   \end{tikzpicture}
   \end{document}
   

   

2. Embedding in 3d for better visualizing with some temporary filling:

   \documentclass[border=10pt]{standalone}
   \usepackage{pgfplots}
   \begin{document}
   \begin{tikzpicture}
     \begin{axis}
       \addplot3 [domain=0:360, samples=60, fill=blue!30, opacity=0.8]
         ( {cos(x)*(sin(3*x) + 1.25)},
           {sin(x)*(sin(3*x) + 1.25)}, 0 );
     \end{axis}
   \end{tikzpicture}  
   \end{document}
   

   

   No problem also to get a surface with simple 3d expansion:

   \documentclass[border=10pt]{standalone}
   \usepackage{pgfplots}
   \begin{document}
   \begin{tikzpicture}
     \begin{axis}
       \addplot3 [
           surf,
           domain    = 0:360,
           y domain  = 0:360,
           samples   = 50,
           samples y = 20,
         ]
         ( {cos(x)*(sin(3*x) + 1.25)},
           {y}, {sin(x)*(sin(3*x) + 1.25)} );
     \end{axis}
   \end{tikzpicture}  
   \end{document}
   

   

3. That's my question, because I still need to bend and twist it:

   How can I rotate that 2d plot around a circle, while rotating it at the same time around its origin, to get a surface plot like in the image at the top?

Answer 1

When I saw your first picture, it immediately made me think of a torus; of course, it's a bit of a twisted torus, but it does have quite a lot in common. As detailed in How to draw a torus (for example), a torus can be parametrized as a surface using (for example)

x(t,s) = (2+cos(t))*cos(s+pi/2) 
y(t,s) = (2+cos(t))*sin(s+pi/2) 
z(t,s) = sin(t)

where t and s both lie on the interval [0,2\pi].

So, how can we apply this to your shape?

Let's start by trying to plot the curve given by the following parametrization in three dimensions:

x(t) = sin(3t)cos(t)
y(t) = sin(3t)sin(t)

A little bit of thought, and application of the torus idea tells us that in your example we can use the following, for example,

  x(t,s) = (4+(sin(3*(t))+1.25)*cos(t))*cos(s) 
  y(t,s) = (4+(sin(3*(t))+1.25)*cos(t))*sin(s) 
  z(t,s) = ((sin(3*(t))+1.25)*sin(t))

Which gives, for a fixed value of s and t\in[0,2\pi]

From here, we can allow s to vary - for example, if we allow s\in[0,\pi] then we achieve:

This is close, but it doesn't twist enough - for that we can choose, for example

  x(t,s) = (4+(sin(3*(t+Ns))+1.25)*cos(t))*cos(s) 
  y(t,s) = (4+(sin(3*(t+Ns))+1.25)*cos(t))*sin(s) 
  z(t,s) = ((sin(3*(t+Ns))+1.25)*sin(t))

The number N can be changed to increase/decrease the 'twistyness'; for example with N=10 and a fairly low resolution, then we achieve:

Finally, you crank up the resolution as much as you can before exhausting memory (running LuaLaTeX really helps - thanks Stefan), and you get the following:

This can be improved by using, for example, shader=interp. Here's the complete code to play with :)

% arara: lualatex
% !arara: indent: {overwrite: yes}
\documentclass[border=10pt]{standalone}
\usepackage{pgfplots}

\begin{document}

\begin{tikzpicture}
    \begin{axis}[
            axis equal,
        ]
        \addplot3[
            surf,
            samples=100, samples y=70,
            colormap/cool,
            domain=0:360,y domain=0:360,
            z buffer=sort,
        ]
        ({(4+(sin(3*(x+2*y))+1.25)*cos(x))*cos(y)},
        {(4+(sin(3*(x+2*y))+1.25)*cos(x))*sin(y)},
        {((sin(3*(x+2*y))+1.25)*sin(x))});
    \end{axis}
\end{tikzpicture}

\end{document}

---

Once we have the template, we can play with it a little more; for example, changing the parametrisation from sin(3(x+2y)) to sin(4(x+2y)) and using the hot2 colormap as follows

    \addplot3[
        surf,
        samples=100, samples y=70,
        colormap/hot2,
        domain=0:360,y domain=0:360,
        z buffer=sort,
    ]
    ({(4+(sin(4*(x+2*y))+1.25)*cos(x))*cos(y)},
    {(4+(sin(4*(x+2*y))+1.25)*cos(x))*sin(y)},
    {((sin(4*(x+2*y))+1.25)*sin(x))});

gives a pleasing picture with a few more twists: