Superficies de revolucion con curvatura media constante |H|=1

Las superficies de revolución de curvatura media constante se llaman superficies de Delaunay (s. XIX). La curva generatriz es la traza del foco de una cónica cuando ésta rueda (sin deslizamiento) a lo largo de una recta. El eje de revolución de la superficie es esta recta.

Si la cónica es una parábola, la superficie es minimal, y por tanto, es la catenoide. 

Si la cónica es una elipse, la superficie obtenida se llama onduloide. En el caso de ser un círculo, la curva generatriz es una recta paralela al eje de revolución y se obtiene un cilindro. Si es un segmento, la curva generatriz es un semicírculo, obteniendo una esfera.

Si la cónica es un hipérbola, la superficie obtenida se llama nodaira.

Ecuación de la curva generatriz

La superficie tiene parametrización X(u,v)={r[u] cos[v],r[u] Sin[v],u}. 
La ecuación de H es: 1+r'^2-r r''-2 H r (1+r'^2)^(3/2) =0. 
Pondremos condiciones iniciales r[0]=r0, r'[0]=0.

X[u_, v_] := {r[u] Cos[v], r[u] Sin[v], u}

Nodoides

Son las superficies con H=1 y -1<r0<1 (La esfera es obtenida con r0=1; el cilindro, con r0=1/2).

r0 = 1; 
    solu1 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -1, 1}]

u0 =  .999479409;
c1 = Plot[r[u] /. solu1, {u, -u0, u0}]

u0 =  .999479409;
nod1 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu1], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 1/2; 
    solu2 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -3, 3}]

u0 = 3;
c2 = Plot[r[u] /. solu2, {u, u0, -u0}]

u0 = 1;
nod2 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu2], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 3/4; 
    solu3 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -3, 3}]

u0 = 3;
c3 = Plot[r[u] /. solu3, {u, u0, -u0}]

u0 = 3;
nod3 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu3], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 1/4; 
    solu4 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -3, 3}]

u0 = 3;
c4 = Plot[r[u] /. solu4, {u, u0, -u0}]

u0 = 3;
nod4 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu4], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 1/10; 
    solu5 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -3, 3}]

u0 = 3;
c5 = Plot[r[u] /. solu5, {u, u0, -u0}]

u0 = 3;
nod5 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu5], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

Show[c1, c2, c3, c4, c5, AspectRatio ->  .2]

Show[nod1, nod2, nod3, nod4, nod5, 
      ViewPoint -> {1, 2, 1}]

Nodairas

Son las superficies con H=1 y |r0|>1, o H=-1.

r0 = 2; 
    solu6 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -1, 1}]

u0 =  .6327875;
c6 = Plot[r[u] /. solu6, {u, u0, -u0}]

u0 =  .6327875;
nod6 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu6], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 4; 
    solu7 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] - 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -1, 1}]

u0 =  .556234;
c7 = Plot[r[u] /. solu7, {u, u0, -u0}]

u0 =  .556234;
nod7 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu7], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 1/2; 
    solu8 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] + 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -1, 1}]

u0 =  .296846;
c8 = Plot[r[u] /. solu8, {u, u0, -u0}]

u0 =  .296846;
nod8 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu8], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]

r0 = 1; 
    solu9 = NDSolve[{1 + r'[u]^2 - r[u] r''[u] + 
              2 r[u] (1 + r'[u]^2)^(3/2)  == 0, 
          r[0] == r0, r'[0] == 0}, 
        r[u], {u, -1, 1}]

u0 =  .367196;
c8 = Plot[r[u] /. solu9, {u, u0, -u0}]

u0 =  .367196;
nod9 = 
      ParametricPlot3D[
        Evaluate[X[u, v] /. solu9], {u, -u0, u0}, {v, 0, 2  Pi}, 
        PlotPoints -> {60, 25}, ViewPoint -> {1, 2, 1}]