Schläuche und andere geometrischen Formen 2011

Kugel

x[u_, v_, r_] := Cos[u] Sin[v] r;
y[u_, v_, r_] := Sin[u] Sin[v] r;
z[u_, v_, r_] := Cos[v] r;

w[u_, v_, r_] := {x[u, v, r], y[u, v, r], z[u, v, r]}

w[u, v, r]

Transpose[{w[u, v, r]}] // MatrixForm

J[u_, v_, r_] := {D[w[u, v, r], u], D[w[u, v, r], v], 
          D[w[u, v, r], r]};

J[u, v, r] // Transpose // MatrixForm

Det[J[u, v, r]]

Det[J[u, v, r]] // Abs // Simplify

Det[J[u1, v1, r1]] // Abs // Simplify

Evaluate[Simplify[Abs[Det[J[u, v, r]]]]]

Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]]

Evaluate[
      Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]] /. {r1 -> r, 
          v1 -> v, u1 -> u}]

Integrate[
      Evaluate[Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]] /. {r1 -> r, 
            v1 -> v, u1 -> u}], {u, 0, 2  Pi}, {v, 0, Pi}, {r, 0, 
        R}]

Schlauch

Vektorfunktion

Remove["Global`*"]

?AspectRatio

kreis[φ_, r_] := {r  Cos[φ], 
          r  Sin[φ]};
ParametricPlot[kreis[φ, 1], {φ, 0, 2  Pi}, 
        AspectRatio -> Automatic];

Schlauch[α_, φ_, r_, 
        RS_] := {RS Cos[α], RS Sin[α], 
          0} + {Cos[α] r  Cos[φ], 
          Sin[α] r  Cos[φ], 
          r  Sin[φ]}

ParametricPlot3D[
        Schlauch[α, φ, 1, 5], {α, 0, 
          2  Pi}, {φ, 0, 2  Pi}];

Schlauch[α_, φ_, r_] := 
        Schlauch[α, φ, r, 5];

    
Schlauch[α, φ, r]

x[u_, v_, 
          r_] := Schlauch[u, v, r][[1]];
y[u_, v_, 
          r_] := Schlauch[u, v, r][[2]];
z[u_, v_, 
          r_] := Schlauch[u, v, r][[3]];
w[u_, v_, r_] := {x[u, v, r], y[u, v, r], z[u, v, r]}
w[u, v, r]

Transpose[{w[u, v, r]}] // MatrixForm

Volumen

J[u_, v_, r_] := {D[w[u, v, r], u], D[w[u, v, r], v], 
          D[w[u, v, r], r]};

J[u, v, r] // Transpose // MatrixForm

Det[J[u, v, r]]

Det[J[u, v, r]] // Abs // Simplify

Det[J[u1, v1, r1]] // Abs // Simplify

Evaluate[Simplify[Abs[Det[J[u, v, r]]]]]

Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]]

Evaluate[
      Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]] /. {r1 -> r, 
          v1 -> v, u1 -> u}]

Integrate[
      Evaluate[Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]] /. {r1 -> r, 
            v1 -> v, u1 -> u}], {u, 0, 2  Pi}, {v, 0, 2  Pi}, {r, 
        0, 1}]

NIntegrate[
      Evaluate[Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]] /. {r1 -> r, 
            v1 -> v, u1 -> u}], {u, 0, 2  Pi}, {v, 0, 2  Pi}, {r, 
        0, 1}]

Integrate[
        Evaluate[
          Evaluate[Simplify[Abs[Det[J[u1, v1, r1]]]]] /. {r1 -> r, 
              v1 -> v, u1 -> u}], {u, 0, 2  Pi}, {v, 0, 2  Pi}, {r, 
          0, 1}] // N

Oberfläche

aussen[u_, v_] := w[u, v, 1];  aussen[u, v]

Norm[Cross[{a1, a2, a3}, {b1, b2, b3}]]

norm[v_] := Sqrt[v . v];  norm[{v1, v2, v3}]

norm[Cross[{a1, a2, a3}, {b1, b2, b3}]]

norm[
      Cross[Evaluate[D[aussen[u, v], u]], 
        Evaluate[D[aussen[u, v], v]]]]

Evaluate[
      norm[Cross[Evaluate[D[aussen[u, v], u]], 
          Evaluate[D[aussen[u, v], v]]]]]

Evaluate[
      Evaluate[norm[
            Cross[Evaluate[D[aussen[u1, v1], u1]], 
              Evaluate[D[aussen[u1, v1], v1]]]]] /. {u1 -> u, 
          v1 -> v}]

Integrate[
      Evaluate[Evaluate[
            norm[Cross[Evaluate[D[aussen[u1, v1], u1]], 
                Evaluate[D[aussen[u1, v1], v1]]]]] /. {u1 -> u, 
            v1 -> v}], {u, 0, 2  Pi}, {v, 0, 2  Pi}]

% // N

Evaluate[
        Evaluate[
            norm[Cross[Evaluate[D[aussen[u1, v1], u1]], 
                Evaluate[D[aussen[u1, v1], v1]]]]] /. {u1 -> u, 
            v1 -> v}] // Simplify

Eierschachtel

f[x_, y_] := Sin[x]  +  Sin[y];
Plot3D[f[x, y], {x, 0, 4  Pi}, {y, 0, 4  Pi}];

f[x_, y_] := Sin[x] Sin[y];
Plot3D[f[x, y], {x, 0, 4  Pi}, {y, 0, 4  Pi}, 
        PlotPoints -> 70];

4  Pi 4  Pi // N

x[u_, v_, r_] := u;
y[u_, v_, r_] := v;
z[u_, v_, r_] := 
        f[x[u, v, r], y[u, v, r]];
w[u_, v_, r_] := {x[u, v, r], y[u, v, r], z[u, v, r]}
w[u, v, r]

aussen[u_, v_] := w[u, v, 1];  aussen[u, v]

norm[v_] := Sqrt[v . v];  norm[{v1, v2, v3}]

Evaluate[
      Evaluate[norm[
            Cross[Evaluate[D[aussen[u1, v1], u1]], 
              Evaluate[D[aussen[u1, v1], v1]]]]] /. {u1 -> u, 
          v1 -> v}]

NIntegrate[
      Evaluate[Evaluate[
            norm[Cross[Evaluate[D[aussen[u1, v1], u1]], 
                Evaluate[D[aussen[u1, v1], v1]]]]] /. {u1 -> u, 
            v1 -> v}], {u, 0, 4  Pi}, {v, 0, 4  Pi}]

Schläuche um beliebige Kurven

Remove["Global`*"]

v[t_] := {(t^2), (t^3), (t^4)};  v[t]

ParametricPlot3D[v[t], {t, -1.1, 1.1}, 
        ViewPoint -> {-2.809,  -1.552,  -1.073}];

ParametricPlot3D[v[t], {t, -0.1, 0.1}, 
        ViewPoint -> {-2.809,  -1.552,  -1.073}];

Remove["Global`*"]

v[t_] := {Sin[t], (t^3), Cos[t^2]};  v[t]

v'[t]

norm[v_] := Sqrt[v . v];  norm[{v1, v2, v3}]

Evaluate[norm[v'[t]]]

Plot[
      Evaluate[norm[v'[t]]] /. t -> t1, {t1, -2 Pi, 
        2  Pi}]

Plot[
      Evaluate[norm[v'[t]]] /. t -> t1, {t1, -0.5, 
        0.5}]

ParametricPlot3D[v[t], {t, -1.1, 1.1}, 
        ViewPoint -> {-2.809,  -1.552,  -1.073}];

vTgNorm[t_] := 1/norm[v'[t]]   v'[t] ;    vTgNorm[
      t]

κ[t_] := 
      1/norm[v'[t]]^4 
norm[v''[t] norm[v'[t]]^2  - 
v'[
            t] (v'[t] . v''[t]) ];  κ[t]

normalV[t_] := 
        Simplify[ 
          1/κ[t]  1/norm[v'[t]] D[v'[t]/norm[v'[t]] , 
              t ]];
normalV[t]

normVTg[t_] := v'[t]/norm[v'[t]];  normVTg[t]

normalNormV[t_] := normalV[t]/norm[normalV[t]];  
    normalNormV[t]

biNormV[t_] := Cross[ normVTg[t], normalNormV[t]]; 
    biNormV[t]

w[t_, φ_, rad_] := 
        Evaluate[(Evaluate[(v[t1] + 
                    rad Cos[φ] normalNormV[t1] + 
                    rad Sin[φ] biNormV[t1])] /. 
              t1 -> 
                t)];

ParametricPlot3D[
        w[t, φ, 0.15], {t, 0.1, 1}, {φ, 0,  
          2  Pi}];

w[t, φ, 
        1] /. {t -> 2, φ -> 0.3}