Kdtyjb

I am trying to turn a planar grid diagram into a 3D toroidal graph by identifying the top edge with the bottom and the left with the right, i.e:

So the grid below on the left would be transformed into a right-hand trefoil knot on a torus:

I would like to draw a torus which looks like this but with the grid structure (including the grid itself, the o's, the x's, and the red lines). Could I please have some help. Many thanks in advance!

This is a starting point. The functions defined can be used to distinguish the visible from the hidden patches, but they are not used here.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{tikz-3dplot}

tikzset{declare function={%

torusx(u,v,R,r)=cos(u)*(R + r*cos(v)); 

torusy(u,v,R,r)=(R + r*cos(v))*sin(u);

torusz(u,v,R,r)=r*sin(v);

vcrit1(u,th)=atan(tan(th)*sin(u));% first critical v value

vcrit2(u,th)=180+atan(tan(th)*sin(u));% second critical v value

thetacritA(R,r)=atan(sqrt(R/r-1));

thetacritB(R,r)=acos(r/R);

ucritA(R,r,th)=180+(90/pi)*sqrt(abs(-(R^2*pow(cot(th),2))+4*pow(r,2)/pow(sin(2*th),2)))/R; 

ucritB(R,r,th)=540-ucritA(R,r,th);

umaxA(R,r,th)=asin(sqrt(abs(-pow(cot(th),2)+4*pow(r,2)/(pow((sin(2*th)*R),2)))));

umaxB(R,r,th)=180-umaxA(R,r,th);}}

begin{document} 

tdplotsetmaincoords{65}{0}

begin{tikzpicture}[tdplot_main_coords]

     pgfmathsetmacro{RadiusA}{3}

     pgfmathsetmacro{RadiusB}{1}

     pgfmathsetmacro{rprime}{1.25}

     % all v curves

     foreach X in {0,10,...,350}

     {draw

        plot[variable=x,domain=0:360,smooth]

    ({torusx(X,x,RadiusA,RadiusB)},{torusy(X,x,RadiusA,RadiusB)},{torusz(X,x,RadiusA,RadiusB)});

     }

     % all u curves

     foreach X in {0,30,...,330}

     {draw plot[variable=x,domain=0:360,smooth]

        ({torusx(x,X,RadiusA,RadiusB)},{torusy(x,X,RadiusA,RadiusB)},{torusz(x,X,RadiusA,RadiusB)});

     }

  end{tikzpicture}

end{document}

They can be used to discern hidden from visible stretches of something wrapping the torus, as illustrated in this answer where the functions are explained. In case you find it to cumbersome to patch things together you way want to consider switching to asymptote.

documentclass[tikz,border=3.14mm]{standalone}

usepackage{tikz-3dplot}

tikzset{declare function={%

torusx(u,v,R,r)=cos(u)*(R + r*cos(v)); 

torusy(u,v,R,r)=(R + r*cos(v))*sin(u);

torusz(u,v,R,r)=r*sin(v);

vcrit1(u,th)=atan(tan(th)*sin(u));% first critical v value

vcrit2(u,th)=180+atan(tan(th)*sin(u));% second critical v value

thetacritA(R,r)=atan(sqrt(R/r-1));

thetacritB(R,r)=acos(r/R);

ucritA(R,r,th)=180+(90/pi)*sqrt(abs(-(R^2*pow(cot(th),2))+4*pow(r,2)/pow(sin(2*th),2)))/R; 

ucritB(R,r,th)=540-ucritA(R,r,th);

umaxA(R,r,th)=asin(sqrt(abs(-pow(cot(th),2)+4*pow(r,2)/(pow((sin(2*th)*R),2)))));

umaxB(R,r,th)=180-umaxA(R,r,th);}}

tikzset{3d torus/.style n

args={2}{/utils/exec=pgfmathsetmacro{DDA}{int(sign(sin(thetacritA(#1,#2))-sin(tdplotmaintheta)))}

  pgfmathsetmacro{DDB}{int(sign(sin(thetacritB(#1,#2))-sin(tdplotmaintheta)))},

  insert path={

  plot[variable=x,domain=1:359,smooth cycle,samples=71]

  ({torusx(x,vcrit1(x,tdplotmaintheta),#1,#2)},

 {torusy(x,vcrit1(x,tdplotmaintheta),#1,#2)},

 {torusz(x,vcrit1(x,tdplotmaintheta),#1,#2)}) 

   ifnumDDA=1

    plot[variable=x,domain=0:360,smooth cycle,samples=71]

    ({torusx(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusy(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusz(x,vcrit2(x,tdplotmaintheta),#1,#2)})    

   else

   ifnumDDB=1 

    plot[variable=x,domain={umaxA(#1,#2,tdplotmaintheta)}:{umaxB(#1,#2,tdplotmaintheta)},smooth,samples=71]

    ({torusx(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusy(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusz(x,vcrit2(x,tdplotmaintheta),#1,#2)})    --

    plot[variable=x,domain={180+umaxA(#1,#2,tdplotmaintheta)}:{180+umaxB(#1,#2,tdplotmaintheta)},smooth,samples=71]

    ({torusx(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusy(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusz(x,vcrit2(x,tdplotmaintheta),#1,#2)})  -- cycle  

    fi 

   fi

  }},3d torus stretch/.style n args={2}{/utils/exec=pgfmathsetmacro{DDA}{int(sign(thetacritA(#1,#2)-tdplotmaintheta))},

  insert path={ifnumDDA=-1

   plot[variable=x,domain={ucritA(#1,#2,tdplotmaintheta)}:{ucritB(#1,#2,tdplotmaintheta)},smooth,samples=71]

    ({torusx(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusy(x,vcrit2(x,tdplotmaintheta),#1,#2)},

    {torusz(x,vcrit2(x,tdplotmaintheta),#1,#2)}) 

  fi 

}}}

begin{document} 

tdplotsetmaincoords{65}{0}

begin{tikzpicture}[tdplot_main_coords]

     pgfmathsetmacro{RadiusA}{3}

     pgfmathsetmacro{RadiusB}{1}

     pgfmathsetmacro{rprime}{1.25}

     foreach X/Y in {105/195,245/335}

     {draw[line width=2mm,blue] plot[variable=x,domain=X:Y,smooth]

    ({torusx(x,2*x,RadiusA,rprime)},{torusy(x,2*x,RadiusA,rprime)},{torusz(x,2*x,RadiusA,rprime)});}

     draw[thick,samples=71,fill=gray,fill opacity=0.7,even odd

    rule,3d torus={RadiusA}{RadiusB}] ;

    draw[thick,samples=71,3d torus stretch={RadiusA}{RadiusB}];

     foreach X/Y in {-27/107,193/247}

     {draw[line width=2mm,blue] plot[variable=x,domain=X:Y,smooth]

    ({torusx(x,2*x,RadiusA,rprime)},{torusy(x,2*x,RadiusA,rprime)},{torusz(x,2*x,RadiusA,rprime)});}

  end{tikzpicture}

end{document}