3D helix torus with hidden lines

Question

Torus wrapped with helix

How can I generate the above image using TikZ (possibly other methods will work if they produce better looking results)?
I would like to have a nicely shaded (not so dark image) without gridlines and possibly enough opacity on the torus to see the hidden part of the helix.

The image given is very close to what I want though and I would just try to make it visually a little better.

Just in case it might be useful, here's the Maple worksheet that I used to generate that image.

Jake · Accepted Answer · 2012-09-10 16:04:10Z

Here's a Sketch/TikZ approach.

Running sketch on this file:

def helix {
    def n_segs 600
  sweep [draw=orange] { n_segs, rotate(24*360 / n_segs, (1.5,0,0), [0,0,1]), rotate(1*360/n_segs, (0,0,0), [0,1,0]) } (2.01,0,0)
}

def torus {
    def n_segs 60
    sweep [draw=none, fill=cyan, fill opacity=0.75] {n_segs, rotate(360/n_segs, (0,0,0), [0,1,0])}
        sweep {n_segs, rotate(360/n_segs, (1.5,0,0), [0,0,1])}
        (2,0,0)
}

put { view((10,4,2)) } {{helix} {torus}}

global { language tikz }

generates a .tex file which can be compiled using pdflatex.

The helix winding around the helix winding around a torus

can be generated using

def helix {
    def n_segs 10000
  sweep [draw=orange] {
    n_segs,
    rotate(1000*360 / n_segs, (2,0,0), [0,1,0]),
    rotate(24*360 / n_segs, (1.5,0,0), [0,0,1]),
    rotate(1*360/n_segs, (0,0,0), [0,1,0])
  } (2.04,0,0)
}

def torus {
    def n_segs 50
    sweep [draw=none, fill=cyan, fill opacity=0.75] {n_segs, rotate(360/n_segs, (0,0,0), [0,1,0])}
        sweep {n_segs, rotate(360/n_segs, (1.5,0,0), [0,0,1])}
        (1.9,0,0)
}

put { view((10,4,2)) } {{torus} {helix}}

global { language tikz }

@JosephWright: Hehe, I tried posting it, but the system wouldn't let me. It's about 250kB, so not much fun to edit by hand. The resulting PDF is 47kB.
OK. Thanks for your effort. I am still waiting for the pure TikZ. If it is impossible then I will assign you the bounty.
Wow. The helix winding around the other winding around the torus is incredible. This is most likely impossible in a 'normal' graphics program such as Illustrator or Inkscape and quite hard even in Blender or other 3D programs.

Herbert · Answer 2 · 2012-09-19 13:22:49Z

it shows the function

 x(u,v)=(R1 + (R0 +RL*sin(u))*sin(k*v))*cos(v)-RL*cos(u)*sin(v)
 y(u,v)=(R1 + (R0 +RL*sin(u))*sin(k*v))*sin(v)+RL*cos(u)*cos(v)
 z(u,v)=(R0 + RL*sin(u))*cos(k*v)

with the parameter setting shown in the example. RL: radius of the coil line; R1: Torus outer; R0: Torus inner radius; k:number of coils

run it with xelatex or latex>dvips>ps2pdf (takes some time to run!)

\documentclass{minimal}
\usepackage{pst-solides3d}
\pagestyle{empty}
\begin{document}

\begin{pspicture}[solidmemory](-6,-4)(6,4)
\psset{viewpoint=30 0 15 rtp2xyz,Decran=30,lightsrc=viewpoint}
\psSolid[object=tore,r1=5,r0=1,ngrid=36 36,
    fillcolor=blue!30,action=none,name=Torus]%
%\axesIIID(4.5,4.5,0)(5,5,4)
\codejps{/R1 5 def /RL 0.05 def /R0 1.1 def /k 25 def}%
\defFunction[algebraic]{helix}(u,v)
    {(R1 + (R0 +RL*sin(u))*sin(k*v))*cos(v)-RL*cos(u)*sin(v)}
    {(R1 + (R0 +RL*sin(u))*sin(k*v))*sin(v)+RL*cos(u)*cos(v)}
    {(R0 + RL*sin(u))*cos(k*v)}
\psSolid[object=surfaceparametree,
       base=0 6.2831853 0 6.2831853,
       linecolor=blue,linewidth=0.01,fillcolor=yellow,
       ngrid=0.8 0.01,function=helix,action=none,name=Helix]%
\psSolid[object=fusion,base=Torus Helix,grid=false]
%\gridIIID[Zmin=-3,Zmax=3,showAxes=false](-2,2)(-2,2)
\end{pspicture}

\begin{pspicture}[solidmemory](-6,-6)(6,6)
\psset{viewpoint=30 0 90 rtp2xyz,Decran=30,lightsrc=viewpoint}
\psSolid[object=tore,r1=5,r0=1,ngrid=36 36,
    fillcolor=blue!30,action=none,name=Torus]%
%\axesIIID(4.5,4.5,0)(5,5,4)
\codejps{/R1 5 def /RL 0.05 def /R0 1.1 def /k 25 def}%
\defFunction[algebraic]{helix}(u,v)
    {(R1 + (R0 +RL*sin(u))*sin(k*v))*cos(v)-RL*cos(u)*sin(v)}
    {(R1 + (R0 +RL*sin(u))*sin(k*v))*sin(v)+RL*cos(u)*cos(v)}
    {(R0 + RL*sin(u))*cos(k*v)}
\psSolid[object=surfaceparametree,
       base=0 6.2831853 0 6.2831853,
       linecolor=blue,linewidth=0.01,fillcolor=yellow,
       ngrid=0.8 0.01,function=helix,action=none,name=Helix]%
\psSolid[object=fusion,base=Torus Helix,grid=false]
%\gridIIID[Zmin=-3,Zmax=3,showAxes=false](-2,2)(-2,2)
\end{pspicture}

\end{document}

enter image description here

an animation is here: http://tug.org/PSTricks/main.cgi?file=Animation/gif/gif

With \psSolid[object=fusion,base=Torus Helix,grid=false,opacity=0.5]  (setting transparency) and a thinner helix (decrease /RL) one gets

enter image description here

and just for fun with

\documentclass{minimal}
\usepackage{pst-solides3d}
\begin{document}
\begin{pspicture}[solidmemory](-6.5,-3.5)(6.5,3)
\psset{viewpoint=30 0 15 rtp2xyz,Decran=30,lightsrc=viewpoint}
\psSolid[object=tore,r1=5,r0=1,ngrid=36 36,tablez=0 0.05 1 {} for,
          zcolor= 1 .5 .5 .5 .5 1,action=none,name=Torus]
\pstVerb{/R1 5 def /R0 1.2 def /k 20 def /RL 0.15 def /kRL 40 def}%
\defFunction[algebraic]{helix}(t)
     {(R1+R0*cos(k*t))*sin(t)+RL*sin(kRL*k*t)}
     {(R1+R0*cos(k*t))*cos(t)+RL*cos(kRL*k*t)}
     {R0*sin(k*t)+RL*sin(kRL*k*t)}
\psSolid[object=courbe,
        resolution=7800,
        fillcolor=black,incolor=black,
        r=0,
        range=0 6.2831853,
        function=helix,action=none,name=Helix]%
\psSolid[object=fusion,base=Torus Helix,grid]
\end{pspicture}
\end{document}

enter image description here

Is there any way to fix the artifacts where the yellow helix seems to be missing in the first picture? I would also like to use a parameteric curve instead of a solid(I tried to modify the code but always get errors). I'd like the yellow helix to look more like a grid line on the torus rather than a separate object.
\pstVerb{/R1 5 def /R0 1.2 def /k 20 def /RL 0.1 def /kRL 40 def}%
The submitted code for the nested helix does not produce the same result as your uploaded image.
then you have not the current pst-solides3d.tex|pro files. Compare with the ones from texnik.dante.de

Tom Bombadil · Answer 3 · 2012-09-11 11:04:06Z

Here's a work-in-progress. It's missing the most important thing: the visibility of the helix. At first I thought that the points where it becomes (in)visible are evenly spaced, but they are not quite. I think I'll have to do some vector algebra to (hopefully) find a solution. The torus is not a 3D object, it's made from many almost transparent circular rings.

Code

\documentclass[tikz,border=5mm]{standalone}

\begin{document}

\newcommand{\xangle}{-30}
\newcommand{\yangle}{210}
\newcommand{\zangle}{90}

\newcommand{\xlength}{1}
\newcommand{\ylength}{1}
\newcommand{\zlength}{1}

\pgfmathsetmacro{\xx}{\xlength*cos(\xangle)}
\pgfmathsetmacro{\xy}{\xlength*sin(\xangle)}
\pgfmathsetmacro{\yx}{\ylength*cos(\yangle)}
\pgfmathsetmacro{\yy}{\ylength*sin(\yangle)}
\pgfmathsetmacro{\zx}{\zlength*cos(\zangle)}
\pgfmathsetmacro{\zy}{\zlength*sin(\zangle)}

\begin{tikzpicture}
[   x={(\xx cm,\xy cm)},
    y={(\yx cm,\yy cm)},
    z={(\zx cm,\zy cm)},
]

\pgfmathsetmacro{\RO}{1.1}
\pgfmathsetmacro{\RI}{5}
\pgfmathsetmacro{\K}{10}

\pgfmathsetmacro{\doubleRO}{2*\RO}

\foreach \h in {0,0.01,...,\doubleRO}
{   \pgfmathsetmacro{\pm}{sqrt(\h*(\doubleRO-\h))}
    \fill[opacity=0.007,blue,even odd rule] (0,0,\h-\RO) circle (\RI+\pm) (0,0,\h-\RO) circle (\RI-\pm);
}

\xdef\initialx{\RI}
\xdef\initialy{0}
\xdef\initialz{\RO}

\foreach \v in {0.1,0.2,...,360.1}
{   \pgfmathsetmacro{\newx}{(\RI + \RO*sin(\K*\v))*cos(\v)}
    \pgfmathsetmacro{\newy}{(\RI + \RO*sin(\K*\v))*sin(\v)}
    \pgfmathsetmacro{\newz}{\RO*cos(\K*\v)}
    \pgfmathsetmacro{\mycolor}{cos(\v)*50+50}
    \draw[red!\mycolor!green,thick] (\initialx,\initialy,\initialz) -- (\newx,\newy,\newz);
    \xdef\initialx{\newx}
    \xdef\initialy{\newy}
    \xdef\initialz{\newz}

}

\end{tikzpicture}

\end{document}

(insufficient) Result

enter image description here

if you want hidden lines and surfaces then you have to build small line or polygon segments of all objects, then build the direction vector and sort all calculated vectors. And in the end you have to fill the canvas with the line and polygon segments depending to the sorted list of direction vectors.
see my deleted answer for the same limitations using pgfplots
The torus is nice and you got the windings. Maybe put a solid black torus behind the blue one so it looks shaded rather than glowing

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3D helix torus with hidden lines

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3D helix torus with hidden lines

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