\( (*\ 
      This\ section\ will\ draw\ a\ Delaunay\ unduloid\ and\ its\ undulary\ profile\ curve . \ 
\ \ It\ also\ draws\ the\ parallel\ H - 
        surface\ and\ parallel\ K - 
        surface . \ \ 
\ \ \ \ \ Which\ Delaunay\ unduloid\ is\ computed\ will\ depend\ on\ the\ choices\ for\ the\ 
\ \ \ \ \ \ \ \ free\ parameters\ "a"\ and\ \(\("b"\)\(.\)\)\ \ *) 

\(Clear[a, \ b, \ H, \ PF[1], \ 
        PF[2], \ sn, \ cs, \ x, \ y, \ Df, \ 
        Nlength];\)\)
a  =  1.25;
b  =  1;
H  =  1 / (2 a);
PF[1]  =  
        N[a  +  Cos[u] Sqrt[a^2  -  b^2]];
PF[2]  =  
        N[a  -  Cos[u] Sqrt[a^2  -  b^2]];
sn  =  
        N[b / Sqrt[(a Sin[u])^2  +  (b Cos[
                        u])^2]];
cs  =  
        N[Sin[u] Sqrt[a^2  -  b^2]/ 
              Sqrt[(a Sin[u])^2  +  (b Cos[
                          u])^2]];
x  =  
        N[(∫_0^u) 
                  Sqrt[(a Sin[alpha])^2  +  (b Cos[
                            alpha])^2] \[DifferentialD]alpha]  +  
          PF[1] cs;
y  =  PF[1] sn;
Df  =  D[y,  u] / D[x,  u];
Nlength  =  
        Sqrt[1 +  Df^2];

  
undularyO  =  
        ParametricPlot[{x,  y},  {u,  0,  4 Pi}, 
          AspectRatio -> True];
undularyH  =  
        ParametricPlot[{x  +  (Df / Nlength) / H,  
            y  -  (1 / Nlength) / H},  {u,  0,  4 Pi}, 
          AspectRatio -> True];
undularyK  =  
        ParametricPlot[{x  +  (Df / Nlength) / (2 H),  
            y  -  (1 / Nlength) / (2 H)},  {u,  0,  4 Pi}, 
          AspectRatio -> True];
Show[undularyO,  undularyH,  
        undularyK];

  
unduloidO  =  
        ParametricPlot3D[{x,  y Cos[v],  
            y Sin[v]}, 
{u,  Pi,  
            4 Pi}, {v,  - Pi / 2,  Pi / 2}, 
          AspectRatio -> True, 
          PlotPoints -> {31, 10}];
unduloidH  =  
        ParametricPlot3D[{x  +  (Df / Nlength) / 
                H,  (y  -  (1 / Nlength) / H) Cos[
                v],  (y  -  (1 / Nlength) / H) Sin[
                v]}, 
{u,  Pi,  4 Pi}, {v,  Pi / 2,  
            3 Pi / 2}, AspectRatio -> True, 
          PlotPoints -> {31, 10}];
unduloidK  =  
        ParametricPlot3D[{x  +  (Df / 
                    Nlength) / (2 H),  (y  -  (1 / 
                        Nlength) / (2 H)) Cos[
                v],  (y  -  (1 / Nlength) / (2 H)) Sin[
                v]}, 
{u,  Pi,  
            4 Pi}, {v,  - Pi / 2,  Pi / 2}, 
          AspectRatio -> True, 
          PlotPoints -> {31, 10}];

<< JavaView`RunThrough`
RunJavaView[unduloidO]

Clear[a,  b,  H,  
          beta];
a  =  
        1;
b  =  
        1;
H  =  
        1 / (2 a);
beta  =  b^2 / a^2;

  

nodaryO  =  
        ParametricPlot[{a (1  -  Cos[u]  +  
                  N[(∫_0^u) 
                                Sin[α]^(2/(Sqrt[Sin[α]^2 + beta])) \[DifferentialD]α]),  
            a (Sin[u]  +  (Sqrt[Sin[u]^2 + beta]))}, {u,  0,  
            4 Pi}, 
AspectRatio -> 
            True];
nodaryH  =  
        ParametricPlot[{a (1  +  Cos[u]  +  
                  N[(∫_0^u) 
                                Sin[α]^(2/(Sqrt[Sin[α]^2 + beta])) \[DifferentialD]α]),  
            a (-Sin[u]  +  (Sqrt[Sin[u]^2 + beta]))}, {u,  0,  
            4 Pi}, AspectRatio -> 
            True];
nodaryK  =  
        ParametricPlot[{a (1  +  
                  N[(∫_0^u) 
                                Sin[α]^(2/(Sqrt[Sin[α]^2 + beta])) \[DifferentialD]α]), a ((Sqrt[Sin[u]^2 + beta]))}, {u,  0,  
            4 Pi}, AspectRatio -> True];
Show[nodaryO,  nodaryH,  nodaryK]

  
nodoidO  =  
          ParametricPlot3D[
{a (1  -  Cos[u]  +  
                    N[(∫_0^u) 
                                  Sin[α]^(2/(Sqrt[Sin[α]^2 + beta])) \[DifferentialD]α]),  
              a Cos[v] (Sin[u]  +  (Sqrt[Sin[u]^2 + beta])),  
              a Sin[v] (Sin[
                      u]  +  (Sqrt[Sin[u]^2 + beta]))}, 
{u,  - 5 Pi / 2,  
              3 Pi / 2}, {v,  - Pi / 2,  Pi / 2}, 
            AspectRatio -> True, PlotPoints -> {31, 10}];

nodoidH  =  
          ParametricPlot3D[
{a (1  +  Cos[u]  +  
                    N[(∫_0^u) 
                                  Sin[α]^(2/(Sqrt[Sin[α]^2 + beta])) \[DifferentialD]α]),  
              a Cos[v] (-Sin[u]  +  (Sqrt[Sin[u]^2 + beta])),  
              a Sin[v] (-Sin[
                        u]  +  (Sqrt[Sin[u]^2 + beta]))}, 
{u,  - 5 Pi / 2,  
              3 Pi / 2}, {v,  - Pi / 2,  Pi / 2}, 
            AspectRatio -> True, PlotPoints -> {31, 10}];

nodoidK  =  
        ParametricPlot3D[
{a (1  +  
                  N[(∫_0^u) 
                                Sin[α]^(2/(Sqrt[Sin[α]^2 + beta])) \[DifferentialD]α]),  a Cos[v] ((Sqrt[Sin[u]^2 + beta])),  
            a Sin[v] ((Sqrt[Sin[u]^2 + beta]))}, 
{u,  
            0,  4 Pi}, {v,  - Pi,  Pi / 2}, 
          AspectRatio -> True, 
          PlotPoints -> {31, 
              10}];
Show[{nodoidO,  nodoidH}, 
        Axes -> False];

<< JavaView`RunThrough`
RunJavaView[nodoidO]