\( (*\ 
      This\ program\ draws\ the\ original\ CMC\ tori\ found\ by\ Wente . \ 
          Which\ of\ these\ 
\ \ tori\ one\ draws\ depends\ on\ the\ choice\ of\ \(\(θ\)\(.\)\)\ \ *) 
\ 
\(Clear[u, \ v, \ H, θ, \ θ\&_, \ Z, \ w, \ 
        j, \ tgj, \ x3];\)

 (*\ 
      W_ {4/3}\ *) \)
\(	\(θ = \ N[\(12.7898\ Pi\)\/180];\)\)
\(	\(g\  = \ 
        3;\)


 (*\ 
      W_ {3/2}\ *) 
	 (*\ θ = \ 
        N[\(17.7324\ Pi\)\/180]; 
	
      g\  = \ 4;\ *) 

 (*\ 
      W_ {6/5}\ *) 
	 (*\ θ = \ 
        N[\(8.0983\ Pi\)\/180]; 
	
      g\  = \ 5;\ *) 

\)
H  =  (1/2);
\(\(θ\&_\  = \ 
        N[\(65.354955\ Pi\)\/180];\)\)
k  =  Sin[θ];
\(\(k\&_\  = \ Sin[θ\&_];\)\)
γ  =  (Sqrt[Tan])[θ];
\(\(γ\&_\  = \ \@Tan[θ\&_];\)\)
\(\(T\  = \ γ\ γ\&_;\)\)
\(\(α\  = \ \@\(\(4\ H\ Sin[2\ θ\&_]\)\/Sin[2\ \((θ\  + \ θ\&_)\)]\);\)\)
\(\(α\&_\  = \ \@\(\(4\ H\ Sin[2\ θ]\)\/Sin[2\ \((θ\  + \ θ\&_)\)]\);\)\)
\(\(a\  = \ \(\(-\ 4\)\ H\ Cos[2\ θ]\ Sin[2\ θ\&_]\)\/Sin[2\ \((θ\  + \ θ\&_)\)];\)\)
\(\(b\  = \ \(\(-\ 4\)\ H\ Sin[2\ θ]\ Cos[2\ θ\&_]\)\/Sin[2\ \((θ\  + \ θ\&_)\)];\)\)
\(\(p\  = \ \(4\ H\ Sin[2\ θ]\ Sin[2\ θ\&_]\)\/Sin[2 \((θ\  + \ θ\&_)\)];\)

\)
\(\(Z\  = \ \((\@2\/\(\(\@H\) \((α\&_^2)\)\))\)\ \((\ \((\((\((\((α\&_^2)\)\  - \ b)\)\ \((\((γ\ Cos[u])\)^2)\)\  + \ 
                        p)\)\ γ\&_\ Cos[
                      v]\  - \ \((p\ \((\((γ\ Cos[
                                    u])\)^2)\)\  + \ \((α\&_^2)\)\  + \ b)\)\ γ\ Cos[u])\)/\((\((1\  - \ 
                      T\ Cos[u]\ Cos[
                          v])\)\ \@\(p\  - \ 2\ b\ \((\((γ\ Cos[u])\)^2)\)\  - \ p\ \((\((γ\ Cos[u])\)^4)\)\))\)\ )\);\)\)
w  =  ((2 (Sqrt[H]))/α) NIntegrate[((1  +  ((T Cos[t])^2))/((1  -  ((T Cos[t])^2)) (Sqrt[1  -  ((k Sin[t])^2)]))),  {t,  0,  u}];
tgj  =  ((Tan[u] α)/(2 Sqrt[H]))*
          Sqrt[1 - k^2 (Sin[u])^2];
j  =  
        Which[u < 0.5 Pi, ArcTan[tgj], 0.5 Pi ≤ u < 1.5 Pi, 
          ArcTan[tgj] + Pi, 1.5  Pi ≤ u < 2.5  Pi, 
          ArcTan[tgj] + 2  Pi, 2.5  Pi ≤ u < 3.5  Pi, 
          ArcTan[tgj] + 3  Pi, 3.5  Pi ≤ u, 
          ArcTan[tgj] + 4  Pi];
\(\(x3\  = \ \((1\/\(α\&_\ \@H\))\)\ \((\(2\ T\ Cos[u]\ Sin[v]\ \@\(1\  - \ \((\((k\&_\ Sin[v])\)^2)\)\)\)\/\(1\  - \ T\ Cos[u]\ Cos[v]\)\  + \ 
              1\/γ\&_\ NIntegrate[\(1\  - \ 2\ \((\((k\&_\ Sin[t])\)^2)\)\)\/\@\(1\  - \ \((\((k\&_\ Sin[t])\)^2)\)\), \ {t, \ 0, \ 
                    v}])\);\)\)
wenteO  =  
        ParametricPlot3D[{Z Cos[w  -  j]   +  Cos[w]/(2 H),  
            Z Sin[w  -  j]  +  Sin[w]/(2 H),  
            x3},  {u,  - Pi / 2,  ((2 g  -  1) Pi) / 
              2},  {v,  - Pi,  Pi}, 
          PlotPoints -> 41];

<< JavaView`RunThrough`
RunJavaView[wenteO]