Schläuche

Rechnung

Remove["Global`*"];

\(\(from[uhu_]\  := \ 
        Module[{}, 
 (*Rechnung*) 
v[
              t_] := {x[t], y[t], z[t]}; 
κ[t_] := 
            Sqrt[\((\((\(v''\)[t] . \(v''\)[t])\) \((\(v'\)[t] . \(v'\)[
                            t])\) - \((\(v'\)[t] . \(v''\)[
                            t])\)^2)\)/\((\((\(v'\)[t])\) . \((\(v'\)[
                          t])\))\)^3]; 

          tT[t_] := \(v'\)[t]/Sqrt[\(v'\)[t] . \(v'\)[t]]; 

          nN[t_] := 1/κ[t]\ 1/
                Sqrt[\(v'\)[t] . \(v'\)[t]]\ D[\(v'\)[t]/
                  Sqrt[\(v'\)[t] . \(v'\)[t]], t]; 

          bB[t_] := Cross[tT[t], nN[t]]; 

          w[t_, φ_] := 
            v[t] + a[t, φ]\ Cos[φ]\ nN[t] + 
              b[t, φ]\ Sin[φ]\ bB[
                  t]; 
 (*Plot*) 
ParametricPlot3D[
            Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 
              2  Pi}]];\)\)

Achtung Rechenzeit!

Demos Schlauchdefinitionen

1

x[t_] = 
      4 Cos[t]; y[t_] := 4 Sin[t]; z[t_] := 0;

    a[t_, φ_] := 1; b[t_, φ_] := 1;

    from[uhu];

2

x[t_] = 
      4 Cos[t]; y[t_] := 4 Sin[t]; z[t_] := 0;

    a[t_, φ_] := 1 + Sin[6  φ]/5; 
    b[t_, φ_] := 1 + Sin[6  φ]/5;

    from[uhu];

    ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 4  Pi}, {φ, 0, 2  Pi}, 
        PlotPoints -> 80];

2

x[t_] = 
      4 Cos[t]; y[t_] := 4 Sin[t]; z[t_] := t;

    a[t_, φ_] := 1; b[t_, φ_] := 1;

    from[uhu];

ParametricPlot3D[Evaluate[w[t,φ]],{t,0,4Pi},{φ,0,2Pi},PlotPoints->80];

2

x[t_] = 4 Cos[t]; 
    y[t_] := 4 Sin[t]; z[t_] := t;

    a[t_, φ_] := (Abs[Cos[3  t]] + 0.2); 
    b[t_, φ_] := (Abs[Cos[3  t]] + 0.2)/2;

    from[uhu];

    ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 4  Pi}, {φ, 0, 2  Pi}, 
        PlotPoints -> 80];

Ergebnisse von Untersuchungen der Klasse M2p: Schlauchdefinitionen für interessante Formen

1

x[t_] = 
      4 Cos[t]; y[t_] := 4 Sin[t]; z[t_] := t;

    a[t_, φ_] := (Abs[Cos[3  t φ]] + 0.2); 
    b[t_, φ_] := (Abs[Cos[3  t φ]] + 0.2)/
        2;

    from[uhu];

    ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-1.735, -0.867, 0.664}, 
        PlotPoints -> 60];

2

x[t_] = 4 Cos[t]; y[t_] := 4 Sin[t]; 
    z[t_] := 4  t;
a[t_, φ_] := (Abs[Cos[3  t]] + 0.5  φ); 
    b[t_, φ_] := (Abs[Cos[3  t]] + φ)/
        2;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-1.730,  -2.722,  -1.025}];

3

x[t_] = 5 Cos[t]; y[t_] := 5 Sin[t]; 
    z[t_] := t;
a[t_, φ_] := φ^2; 
    b[t_, φ_] := 2  t;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 3  Pi}, {φ, 0, 
          2  Pi}];

4

x[t_] = 4 Cos[t]; y[t_] := 4 Sin[t]; 
    z[t_] := t;
a[t_, φ_] := 2; 
    b[t_, φ_] := Pi;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-1.730,  -2.722,  -1.025}];

5

x[t_] = 3 Cos[t]; y[t_] := 4 Sin[t]; 
    z[t_] := t;
a[t_, φ_] := 4; 
    b[t_, φ_] := 1;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-1.730,  -2.722,  -1.025}, 
        PlotPoints -> 60];

6

x[t_] = 10 Cos[t]; y[t_] := 5 Sin[t]; 
    z[t_] := t;
a[t_, φ_] := φ^2; 
    b[t_, φ_] := φ^2;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-1.730,  -2.722,  -1.025}];

7  Bleistiftspitze

x[t_] = 4 Cosh[t]; y[t_] := 4  Sin[t]; 
    z[t_] := t;
a[t_, φ_] := Sinh[t] + 1; 
    b[t_, φ_] := Sinh[t] + 1;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-2.540, 1.173, 1.904}];

8  Wirrwar

x[t_] = 4 Cos[t]; y[t_] := 4  Sin[t]; 
    z[t_] := t;
a[t_, φ_] := Sin[t] + 1; 
    b[t_, φ_] := Sin[t] + 1;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-2.540, 1.173, 1.904}];

9

x[t_] = 4 Cos[t]; y[t_] := 4  Sin[t]; 
    z[t_] := t;
a[t_, φ_] := Sin[t] + 1; 
    b[t_, φ_] := Sin[t] + 1;
from[uhu];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 5  Pi}, {φ, 0, 2  Pi}, 
        ViewPoint -> {-2.540, 1.173, 1.904}, 
        PlotPoints -> 80];

10

x[t_] = 4 Cos[t] Sin[2  t];  y[t_] := 4 Sin[t + 0.01];  
    z[t_] := t;
a[t_, φ_] := 1; 
    b[t_, φ_] := 1;
from[uhu_];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 2  Pi}, {φ, 0, 2  Pi}, 
        PlotPoints -> 80];

11

x[t_] = 10 Cos[t]; y[t_] := 5 Sin[t]; 
    z[t_] := t;
a[t_, φ_] := 3  φ*2; 
    b[t_, φ_] := 3  φ*2;
from[uhu_];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, Pi}, {φ, 0, 
          Pi}];

12

x[t_] = 10 Cos[t]; y[t_] := 5 Sin[t]; 
    z[t_] := t;
a[t_, φ_] := φ*-2; 
    b[t_, φ_] := 3  φ*2; 
    from[uhu_];
ParametricPlot3D[
        Evaluate[w[t, φ]], {t, 0, 5/4  Pi}, {φ, 0, 
          5/4  Pi}];