How to create horn torus in latex?

Question

This is a horn torus. How can I use latex to generate it ?

Answer 1

\documentclass[tikz,border=3.14pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\usepgfplotslibrary{colormaps}
\begin{document}
\foreach \X in {0,...,14}
{\begin{tikzpicture}
     \begin{axis}[view/h=45,axis lines = none,colormap={fake}{
rgb255(0cm)=(255,255,255); rgb255(1cm)=(255,255,255); rgb255(2cm)=(255,0,0);
rgb255(3cm)=(255,0,0)},unit vector ratio=1 1 1]
        \addplot3[domain=0:360,domain y=0:360,point meta=x-y+z,mesh,
    z buffer=sort]
       ({(1+cos(x+\X))*cos(y+\X)}, 
        {(1+cos(x+\X))*sin(y+\X)}, 
        {sin(x+\X)});
      \end{axis}
      \path (1,1) rectangle (5.9,4.7); % <- Thanks to J Leon V. !
\end{tikzpicture}}
\end{document}

UPDATE: Removed the flickers, big thanks to J Leon V.! And removed the amsmath package and slightly changed the view.

Answer 2

Just for discover, an option in Asymptote, I tried to find a way to control the color of the spline pen, but I found nothing, the manual is very dark, with this code you get the render according to the question, which is to draw the torus, the file that creates asymptote, allows to rotate the generated graphic, as long as it is through pdf readers that support, 3d, like Acrobat.

The animation is a separate work, because it does not work the same as with tikz the method of using foreach in the stand alone environment, so the animation was done manually by generating 15 files and putting them together in a single pdf, then processing them with the imagemagic converter.

RESULT:

MWE:

% arara: pdflatex: {synctex: yes, action: nonstopmode, shell: yes}
% arara: asymptote
% arara: pdflatex: {synctex: yes, action: nonstopmode, shell: yes}
\documentclass[border=20pt]{standalone}
\usepackage{asymptote}
\begin{document}
    \begin{asy}
        import graph3;      
        size(400,0);
        currentprojection=orthographic(4,0,2);

        real R=10;
        real a=10;
        real D=15*pi/180;

        triple f(pair t) {
        return ((R+a*cos(t.y+D))*cos(t.x+D),(R+a*cos(t.y+D))*sin(t.x+D),a*sin(t.y+D));
        }

        pen p=rgb(1,0,0)+thick();
        surface s=surface(f,(0,0),(2pi,2pi),24,24,Spline);

        draw(s,surfacepen=material(white+opacity(0.8),
        ambientpen=white),meshpen=p);


    \end{asy}
\end{document}

PSD: I could not avoid the flickering since I used the standalone document class, which cuts the canvas according to what is rendered, and it happens that when the rotation is made it varies, I think it can be solved with the geometry package.

UPDATE: AVOID FLICKERING IN ANIMATION...

% PROCESADOR ARARA V3.0
% arara: pdflatex: {synctex: yes, action: nonstopmode, shell: yes}
% arara: asymptote
% arara: pdflatex: {synctex: yes, action: nonstopmode, shell: yes}
\documentclass[a5paper]{article}
\usepackage{pdflscape}
\usepackage[
    left=60pt,
    right=0pt,
    top=0pt,
    bottom=0pt,
    foot=0pt
    ]{geometry}
\usepackage{asymptote}
\begin{document}
\pagestyle{empty}
\begin{landscape}
    \centering
    \begin{asy}
        import graph3;      
        size(500,300);
        currentprojection=orthographic(4,0,2);

        real R=10;
        real a=10;
        real D=15*pi/180;

        triple f(pair t) {
        return ((R+a*cos(t.y+D))*cos(t.x+D),(R+a*cos(t.y+D))*sin(t.x+D),a*sin(t.y+D));
        }

        pen p=rgb(1,0,0)+thick();
        surface s=surface(f,(0,0),(2pi,2pi),24,24,Spline);

        draw(s,surfacepen=material(white+opacity(0.8),
        ambientpen=white),meshpen=p);


    \end{asy}
\end{landscape}
\end{document}

UPDATED ANIMATION:

---

About Colors and Opacity for meshpen:

Using palette module: For surface we have s.colors(palette(s.map(xpart),Gradient(color1,....colorx)) That change the surface color according x position; for pen ¿will not have any similar properties? like p=unknown_macro(palette(p.map(xpart), Gradient(color1,..., colorx)); instead for pgplots colormap in the marmot's answer , if it is possible to do it with the mesh, the problem there is that the mesh is not curved type as splines in Asymptote. For the surfaces it works even with the opacity, as the following test example:

TEST RESULT:

Testing MWE:

% PROCESADOR ARARA V3.0
% arara: pdflatex: {synctex: yes, action: nonstopmode, shell: yes}
% arara: asymptote
% arara: pdflatex: {synctex: yes, action: nonstopmode, shell: yes}
\documentclass[a5paper]{article}
\usepackage{pdflscape}
\usepackage[
    left=60pt,
    right=0pt,
    top=0pt,
    bottom=0pt,
    foot=0pt
    ]{geometry}
\usepackage{asymptote}
\begin{document}
\pagestyle{empty}
\begin{landscape}
    \centering
    \begin{asy}
        import graph3;
        import palette;     
        size(500,300);
        currentprojection=orthographic(0,-2,1);

        real R=10;
        real a=10;
        real D=15*pi/180;

        triple f(pair t) {
        return ((R+a*cos(t.y+D))*cos(t.x+D),(R+a*cos(t.y+D))*sin(t.x+D),a*sin(t.y+D));
        }

        pen p=rgb(0,0,0)+thick();
        surface s=surface(f,(0,0),(2pi,2pi),24,24,Spline);
        s.colors(palette(s.map(xpart),Gradient(red,purple,blue,green+opacity(0.7),green+opacity(0.2))));

        draw(s,meshpen=p,render(merge=true));


    \end{asy}
\end{landscape}
\end{document}

Answer 3

I normally do not post two answers, but somehow I felt that it should be noted that there is no problem making the torus fade in asymptote, nor to do the animation. I'll be happy to remove this if @J.LeonV adds the elements to her/his answer. I just did not want the readers to get the impression that asymptote has major deficiencies, which I don't think it has.

\documentclass[border=5pt]{standalone}
% using Charles Staats' tutorial: https://math.uchicago.edu/~cstaats/Charles_Staats_III/Notes_and_papers_files/asymptote_tutorial.pdf 
% https://tex.stackexchange.com/a/339046/121799 for the plane
% https://tex.stackexchange.com/a/339046/121799 for the animation
% and https://tex.stackexchange.com/a/435341/121799 for the torus parametrization
\usepackage{filecontents}
\begin{filecontents*}{horntorus.asf}
\begin{asypicture}{name=horntorus}
settings.render = 8;
settings.prc = false;
settings.outformat = "pdf";

import graph3;
import contour;
size3(8cm);

real rotangle = @myangle;

currentprojection = orthographic(10,1,4);
defaultrender = render(merge = true);

        real R=2;
        real a=2;
        real D=15*pi/180;

triple f(pair t) {
        return ((R+a*cos(t.y+D))*cos(t.x+D),(R+a*cos(t.y+D))*sin(t.x+D),a*sin(t.y+D));
        }

path3 vcircle(real vx){
 triple fx(real t) {
    return f((t,vx*D));
  }
  return graph(fx, 0, 2*pi, operator ..);
}

path3 hcircle(real vx){
 triple fy(real t) {
    return f((vx*D,t));
  }
  return graph(fy, 0, 2*pi, operator ..);
}

for (int irun=1; irun<=24;irun+=1)
{
draw(vcircle(irun+rotangle), p=red);
draw(hcircle(irun+rotangle), p=red);
}

real Rad=2*R; 
path3 plane =  Rad*sqrt(2)*Y+Rad*Z -- Rad*sqrt(2)*Y-Rad*Z -- 
-Rad*sqrt(2)*Y-Rad*Z -- -Rad*sqrt(2)*Y+Rad*Z -- cycle;
surface cheatplane = surface(plane);
for (int irun=0; irun<=10;irun+=1)
{
draw(shift((0.2-0.25*irun)*(X))*rotate(-45,Y)*cheatplane,white+opacity(0.1*irun),light=nolight);
}
\end{asypicture}
\end{filecontents*}
\usepackage{pgffor}
\usepackage{asypictureB}
\standaloneenv{asypicture}
\begin{document}
\foreach \X [count=\n,evaluate={\myangle=\X/24)}] in {1,...,24}
{
\RequireAsyRecompile
\input{horntorus.asf}
}
\end{document}

The way I made the torus fade away is putting white planes with a certain opacity at the right (?) places. The really cool thing about asymptote is that it has a 3D engine, such that this is really easy to implement.