Konzept und Darstellung der nachfolgenden Ergebnisse sind Resultate von durch Studenten ausgeführten Arbeiten - WIR

Spiel mit räumlichen räumlichen  Kurven

Kurven im (R^(2 ))

Berechnung von Krümmung und Anschmiegekreis bei ebenen Kurven

Definition der Raumkurve

Remove["Global`*"]

x[t_] := 2 Cos[t];   y[t_] :=  2 Sin[t];   z[t_] := 
      0.7  t;    v[t_] := {x[t], y[t], z[t]};

Definition des Bereiches der Variablen

tmin = 0;   tmax = 6 Pi;  step = 0.3;  gu = tmin;  
    go = tmax;

Plot Ausgangsfunktion

origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t],  {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic];

Begleitendes Dreibein

vLen[v_] := Sqrt[v . v]; 
normVec[v_] := v/vLen[v];
tg[t_] = normVec[v'[t]];
n[t_] = 
        normVec[Cross[
            Cross[v'[t], v''[t]], v'[
              t]]];
b[t_] := Cross[tg[t], n [t]];

{tg[t], n[t], b[t]}

tVec[t_] := 
        Line[{v[t], v[t] + 3  tg[t]}];
nVec[t_] := 
        Line[{v[t], v[t] + 3  n[t]}];
bVec[t_] := Line[{v[t], v[t] + 3  b[t]}];

dreibein[t_] := 
      Graphics3D[{RGBColor[1, 0, 1], tVec[t], RGBColor[0.01, 0.6, 0.62], 
          nVec[t], RGBColor[0.54, 0.17, 0.89], bVec[t]}]

Show[origPlot, dreibein[5]];

Krümmung

ϰ[
          t_] = Sqrt[(((v''[t] . v''[t])*(v'[
                              t] . v'[t]) - (v''[t] . v'[
                              t])^2)/vLen[v'[t]]^6)] // N // 
          Chop;
ϰ[t]

Plot[Evaluate[ϰ[t]], {t, tmin, tmax}];

Krümmungsradius

ρ[t_] := Abs[1/ϰ[t]];   ρ[t]

Plot[Evaluate[ρ[t]], {t, tmin, tmax}, 
        PlotPoints -> 100];

Krümmungskreismittelpunkt (Evolute)

Remove[evolute]

m[t_] := v[t] +  ρ[t]*n[t];   m[t]

evolute = 
        ParametricPlot3D[
          Evaluate[Join[m[t], {RGBColor[0, 0, 1]}]], {t, tmin, tmax}, 
          AspectRatio -> 3];

Show[origPlot, evolute];

Zeichnen von Kurven und den Schmiegekreisen

Linie zum Krümmungskreismittelpunkt

line[t_, s_] := m[t] + s (v[t] - m[t]);

tab1 = 
        ParametricPlot3D[
          Evaluate[Table[line[t, s], {t, gu, go, step}]], {s, 0, 
            1}];

Show[origPlot, evolute, tab1];

Darstellung der Tangentenvektoren

Länge der Tangente

lenTg = 1;
line1[t_, s_] := v[t] + (s tg[t])
tab2 = 
        ParametricPlot3D[
          Evaluate[Table[line1[t, s], {t, gu, go, step}]], {s, 0, 
            lenTg}];

Show[origPlot, evolute, tab1, tab2];

Show[origPlot, tab2];

Darstellung von Krümmungskreisen entlang der Kurve

Abstand der Kreise

step1 = 0.3;
xNew[t_, w_] := 
        v[t] + (Cos[w] tg[t] + Sin[w] n[t]) ϰ[
              t] + ϰ[t] n[t];
tab3 = 
        ParametricPlot3D[
          Evaluate[Table[xNew[t, w], {t, gu, go, step1}]], {w, 0, 
            2  Pi}];

Show[origPlot, tab3];

Darstellung von Kreisen als Einhüllung der Kurve

Abstand der Kreise, Schlauchradius

step1 = 0.2;  rad = 0.5;

yNew[t_, u_] := 
        v[t] + (Cos[u] n[t] + Sin[u] b[t]) rad;

tab4 = 
        ParametricPlot3D[
          Evaluate[Table[yNew[t, u], {t, gu, go, step1}]], {u, 0, 
            2  Pi}];

Show[origPlot, tab4];

Schlauch

ParametricPlot3D[Evaluate[yNew[t, u]], {t, gu, go}, {u, 0, 2  Pi}, 
        PlotPoints -> {70, 30}];

Programm für Ringdarstellung und Schlauch

Input

Remove["Global`*"]

x[t_] := 2 Cos[t];   y[t_] :=  
      2 Sin[t];   z[t_] := 
      Cos[2  t];   

\(\((v[t_] := {x[t], y[t], z[t]}; 
tmin = 0; \ \ tmax = 
        6\ Pi; \ step = 0.3; \ gu = tmin; \ go = tmax; rad = 0.5; 
      step1 = 0.2; step = 0.3; 
      origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t], \ {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic, 
          DisplayFunction -> Identity]; 
vLen[v_] := 
        Sqrt[v . v]; \ 
normVec[v_] := v/vLen[v]; 

      tg[t_] = normVec[\(v'\)[t]]; 

      n[t_] = normVec[Cross[Cross[\(v'\)[t], \(v''\)[t]], \(v'\)[t]]]; 

      b[t_] := Cross[tg[t], n\ [t]]; ϰ[
          t_] = \(Sqrt[\((\((\((\(v''\)[t] . \(v''\)[t])\)*\((\(v'\)[
                              t] . \(v'\)[t])\) - \((\(v''\)[t] . \(v'\)[
                              t])\)^2)\)/vLen[\(v'\)[t]]^6)\)] // N\) // 
          Chop; ρ[t_] := 
        Abs[1/ϰ[t]]; 
m[t_] := 
        v[t] + \ ρ[t]*n[t]; \ \ 
yNew[t_, u_] := 
        v[t] + \((Cos[u] n[t] + Sin[u]\ b[t])\) 
            rad; 
tab4 = 
        ParametricPlot3D[
          Evaluate[Table[yNew[t, u], {t, gu, go, step1}]], {u, 0, 2  Pi}, 
          DisplayFunction -> Identity]; 
Show[origPlot, 
        tab4, DisplayFunction -> $DisplayFunction]; 
ParametricPlot3D[
        Evaluate[yNew[t, u]], {t, gu, go}, {u, 0, 2  Pi}, 
        PlotPoints -> {70, 30}];)\)\)

x[t_] := 2 Cos[t];   y[t_] :=  
      2 Sin[t];   z[t_] := 
      Cos[t];   

\(\((v[t_] := {x[t], y[t], z[t]}; 
tmin = 0; \ \ tmax = 
        6\ Pi; \ step = 0.3; \ gu = tmin; \ go = tmax; rad = 0.5; 
      step1 = 0.2; step = 0.3; 
      origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t], \ {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic, 
          DisplayFunction -> Identity]; 
vLen[v_] := 
        Sqrt[v . v]; \ 
normVec[v_] := v/vLen[v]; 

      tg[t_] = normVec[\(v'\)[t]]; 

      n[t_] = normVec[Cross[Cross[\(v'\)[t], \(v''\)[t]], \(v'\)[t]]]; 

      b[t_] := Cross[tg[t], n\ [t]]; ϰ[
          t_] = \(Sqrt[\((\((\((\(v''\)[t] . \(v''\)[t])\)*\((\(v'\)[
                              t] . \(v'\)[t])\) - \((\(v''\)[t] . \(v'\)[
                              t])\)^2)\)/vLen[\(v'\)[t]]^6)\)] // N\) // 
          Chop; ρ[t_] := 
        Abs[1/ϰ[t]]; 
m[t_] := 
        v[t] + \ ρ[t]*n[t]; \ \ 
yNew[t_, u_] := 
        v[t] + \((Cos[u] n[t] + Sin[u]\ b[t])\) 
            rad; 
tab4 = 
        ParametricPlot3D[
          Evaluate[Table[yNew[t, u], {t, gu, go, step1}]], {u, 0, 2  Pi}, 
          DisplayFunction -> Identity]; 
Show[origPlot, 
        tab4, DisplayFunction -> $DisplayFunction]; 
ParametricPlot3D[
        Evaluate[yNew[t, u]], {t, gu, go}, {u, 0, 2  Pi}, 
        PlotPoints -> {70, 30}];)\)\)

x[t_] := 2 Cos[t];   y[t_] :=  
      2 Sin[t];   z[t_] := 
      Cos[4  t];   

\(\((v[t_] := {x[t], y[t], z[t]}; 
tmin = 0; \ \ tmax = 
        6\ Pi; \ step = 0.3; \ gu = tmin; \ go = tmax; rad = 0.5; 
      step1 = 0.2; step = 0.3; 
      origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t], \ {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic, 
          DisplayFunction -> Identity]; 
vLen[v_] := 
        Sqrt[v . v]; \ 
normVec[v_] := v/vLen[v]; 

      tg[t_] = normVec[\(v'\)[t]]; 

      n[t_] = normVec[Cross[Cross[\(v'\)[t], \(v''\)[t]], \(v'\)[t]]]; 

      b[t_] := Cross[tg[t], n\ [t]]; ϰ[
          t_] = \(Sqrt[\((\((\((\(v''\)[t] . \(v''\)[t])\)*\((\(v'\)[
                              t] . \(v'\)[t])\) - \((\(v''\)[t] . \(v'\)[
                              t])\)^2)\)/vLen[\(v'\)[t]]^6)\)] // N\) // 
          Chop; ρ[t_] := 
        Abs[1/ϰ[t]]; 
m[t_] := 
        v[t] + \ ρ[t]*n[t]; \ \ 
yNew[t_, u_] := 
        v[t] + \((Cos[u] n[t] + Sin[u]\ b[t])\) 
            rad; 
tab4 = 
        ParametricPlot3D[
          Evaluate[Table[yNew[t, u], {t, gu, go, step1}]], {u, 0, 2  Pi}, 
          DisplayFunction -> Identity]; 
Show[origPlot, 
        tab4, DisplayFunction -> $DisplayFunction]; 
ParametricPlot3D[
        Evaluate[yNew[t, u]], {t, gu, go}, {u, 0, 2  Pi}, 
        PlotPoints -> {70, 30}];)\)\)

x[t_] := 2 Cos[t];   y[t_] :=  
      2 Sin[t];   z[t_] := 
      Cos[3  t];   

\(\((v[t_] := {x[t], y[t], z[t]}; 
tmin = 0; \ \ tmax = 
        6\ Pi; \ step = 0.3; \ gu = tmin; \ go = tmax; rad = 0.2; 
      step1 = 0.2; step = 0.3; 
      origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t], \ {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic, 
          DisplayFunction -> Identity]; 
vLen[v_] := 
        Sqrt[v . v]; \ 
normVec[v_] := v/vLen[v]; 

      tg[t_] = normVec[\(v'\)[t]]; 

      n[t_] = normVec[Cross[Cross[\(v'\)[t], \(v''\)[t]], \(v'\)[t]]]; 

      b[t_] := Cross[tg[t], n\ [t]]; ϰ[
          t_] = \(Sqrt[\((\((\((\(v''\)[t] . \(v''\)[t])\)*\((\(v'\)[
                              t] . \(v'\)[t])\) - \((\(v''\)[t] . \(v'\)[
                              t])\)^2)\)/vLen[\(v'\)[t]]^6)\)] // N\) // 
          Chop; ρ[t_] := 
        Abs[1/ϰ[t]]; 
m[t_] := 
        v[t] + \ ρ[t]*n[t]; \ \ 
yNew[t_, u_] := 
        v[t] + \((Cos[u] n[t] + Sin[u]\ b[t])\) 
            rad; 
tab4 = 
        ParametricPlot3D[
          Evaluate[Table[yNew[t, u], {t, gu, go, step1}]], {u, 0, 2  Pi}, 
          DisplayFunction -> Identity]; 
Show[origPlot, 
        tab4, DisplayFunction -> $DisplayFunction]; 
ParametricPlot3D[
        Evaluate[yNew[t, u]], {t, gu, go}, {u, 0, 2  Pi}, 
        PlotPoints -> {70, 30}];)\)\)

x[t_] := 2 Cos[t];   y[t_] :=  
      2 Sin[2  t];   z[t_] := 
      Cos[3  t];   

\(\((v[t_] := {x[t], y[t], z[t]}; 
tmin = 0; \ \ tmax = 
        6\ Pi; \ step = 0.3; \ gu = tmin; \ go = tmax; rad = 0.2; 
      step1 = 0.2; step = 0.3; 
      origPlot = 
        ParametricPlot3D[
          Evaluate[Join[v[t], \ {RGBColor[1, 0, 0]}]], {t, tmin, tmax}, 
          PlotRange -> All, AxesLabel -> {x, y, z}, 
          AspectRatio -> Automatic, 
          DisplayFunction -> Identity]; 
vLen[v_] := 
        Sqrt[v . v]; \ 
normVec[v_] := v/vLen[v]; 

      tg[t_] = normVec[\(v'\)[t]]; 

      n[t_] = normVec[Cross[Cross[\(v'\)[t], \(v''\)[t]], \(v'\)[t]]]; 

      b[t_] := Cross[tg[t], n\ [t]]; ϰ[
          t_] = \(Sqrt[\((\((\((\(v''\)[t] . \(v''\)[t])\)*\((\(v'\)[
                              t] . \(v'\)[t])\) - \((\(v''\)[t] . \(v'\)[
                              t])\)^2)\)/vLen[\(v'\)[t]]^6)\)] // N\) // 
          Chop; ρ[t_] := 
        Abs[1/ϰ[t]]; 
m[t_] := 
        v[t] + \ ρ[t]*n[t]; \ \ 
yNew[t_, u_] := 
        v[t] + \((Cos[u] n[t] + Sin[u]\ b[t])\) 
            rad; 
tab4 = 
        ParametricPlot3D[
          Evaluate[Table[yNew[t, u], {t, gu, go, step1}]], {u, 0, 2  Pi}, 
          DisplayFunction -> Identity]; 
Show[origPlot, 
        tab4, DisplayFunction -> $DisplayFunction]; 
ParametricPlot3D[
        Evaluate[yNew[t, u]], {t, gu, go}, {u, 0, 2  Pi}, 
        PlotPoints -> {70, 30}];)\)\)