Studien zu Plots, Beispiele

Ideen von Studenten

Möglichkeit einer Schnecke

ParametricPlot3D[{u Cos[u] (4 + Cos[v + u]), 
          u Sin[u] (4 + Cos[v + u]), u Sin[v + u]}, {u, 0, 4  Pi}, {v, 
          0, 2  Pi}, PlotPoints -> {60, 12}, 
        ViewPoint -> {-2.979,  -0.328,  -1.642}];

ParametricPlot3D[{u Cos[u] (4 + Cos[v + u]), 
          u Sin[u] (4 + Cos[v + u]), u Sin[v + u]}, {u, 0, 4  Pi}, {v, 
          0, 2  Pi}, PlotPoints -> {60, 12}, 
        ViewPoint -> {-1.187, -0.610, 2.191}];

ParametricPlot3D[{u Cos[u] (4 + Cos[v + u]), 
          u Sin[u] (4 + Cos[v + u]), u Sin[v + u]}, {u, 0, 4  Pi}, {v, 
          0, 2  Pi}, PlotPoints -> {60, 12}, 
        ViewPoint -> {2.191,  -1.806,  1.662}];

ParametricPlot3D[{u Cos[u] (4 + Cos[v + u]), 
          u Sin[u] (4 + Cos[v + u]), u Sin[v + u]}, {u, 0, 4  Pi}, {v, 
          0, 2  Pi}, PlotPoints -> {60, 12}, 
        ViewPoint -> {1.969,  0.917,  2.471}];

ParametricPlot3D[{u Cos[u] (4 + Cos[v + u]), 
          u Sin[u] (4 + Cos[v + u]), u Sin[v + u]}, {u, 0, 4  Pi}, {v, 
          0, 2  Pi}, PlotPoints -> {60, 12}, 
        ViewPoint -> {10, 10, 1}];

ParametricPlot3D[{u Cos[u] (4 + Cos[v + u]), 
          u Sin[u] (4 + Cos[v + u]), u Sin[v + u]}, {u, 0, 4  Pi}, {v, 
          0, 2  Pi}, PlotPoints -> {60, 12}, 
        ViewPoint -> {-0.018,  -0.025,  -2.328}];

Zeta-Funktion

Plot3D[Abs[Zeta[x + I y]], {x, -4, 4}, {y, -10, 40}, 
        PlotPoints -> {70, 90}, PlotRange -> {0, 5}, 
        ViewPoint -> {8, 1, 3}, Boxed -> False, 
        Axes -> False];

Plot3D[Abs[1/Zeta[x + I y]], {x, -4, 4}, {y, -10, 40}, 
        PlotPoints -> {70, 90}, PlotRange -> {0, 5}, 
        ViewPoint -> {8, 1, 3}, Boxed -> False, 
        Axes -> False];

Hutartige

Plot3D[
        E^-(0.3 (x^2 + y^2)) Sin[
            1.5 (x^2 + y^2)], {x, -3, 3}, {y, -3, 3}, 
        PlotPoints -> 100, Mesh -> False, Boxed -> False, 
        Axes -> False];

Plot3D[
        Re[E^-(0.3 (x^2 + y^2)) 
            Cos[1.5 (x^2 + y^2)]], {x, -3, 3}, {y, -3, 3}, 
        PlotPoints -> 100, Mesh -> False, Boxed -> False, 
        Axes -> False];

Animation

Table[
        Plot3D[Re[
            E^-(0.3 (x^2 + y^2)) 
              Cos[1.5 (x^2 + y^2) + t]], {x, -3, 3}, {y, -3, 3}, 
          PlotPoints -> 100, Mesh -> False, Boxed -> False, 
          Axes -> False], {t, 0, Pi, Pi/15}];

Knackig

ParametricPlot3D[{Tan[t], Cos[2  t] Sin[3  u], 
          Cos[2  t] Cos[u]}, {t, -Pi/2, Pi/2}, {u, 0, 2  Pi}, 
        Ticks -> None];

Remove["Global\.b4*"]

Möbius und Co

a = 1; b = 0.5;
r[t_, v_] := a + b v Cos[t/2];
x[t_, v_] := r[t, v] Cos[t];
y[t_, v_] := r[t, v] Sin[t];
z[t_, v_] := b v  Sin[t/2];
ParametricPlot3D[{x[t, v], y[t, v] , z[t, v] }, {t, 0, 
          2  Pi}, {v, -1, 1}, Axes -> False, 
        Boxed -> False];

Plot3D[Cos[x + Sin[y]], {x, -3, 3}, {y, -6, 6}, 
        Axes -> False, Boxed -> False, 
        PlotPoints -> 100];

ContourPlot[Cos[x + Sin[y]], {x, -10, 10}, {y, -6, 6}, 
        PlotPoints -> 100];

ParametricPlot3D[{u Cos[t], u Sin[t] , t/3 }, {t, 0, 
          3}, {u, -1, 1}, Axes -> False, 
        Boxed -> False];

plo1 = 
        Plot3D[Sin[
              Sqrt[x^2 + 
                  y^2]] E^-(0.1  Sqrt[x^2 + y^2] ), {x, -3 Pi, 
            3  Pi}, {y, -3 Pi, 3  Pi}, Axes -> False, 
          Boxed -> False, 
          PlotPoints -> 100];
plo2 = 
        Plot3D[x y/(x^2 + y^2) , {x, -3 Pi, 3  Pi}, {y, -3 Pi, 
            3  Pi}, Axes -> False, Boxed -> False, 
          PlotPoints -> 100];
Show[%, %%];

Rose und Stern

MaurerRose[n_Integer, d_Integer, 
          z_Integer:  
            300] := 
Block[{zz = 
              If[! EvenQ[n]  &&  z == 360, 180, z], 
            cosets}, 
rect[theta_, 
              r_] := {r Cos[theta], r Sin[theta]} // 
              N; 
rose[i_] := 
            rect[i 2 Pi/z,  
              Sin[n i 2 Pi/z]]; 
Attributes[
              rose] = {Listable}; 
cosets = 
            Table[Line[rose[Range[m, m + LCM[d, zz], d]]], {m, 0, 
                GCD[d, zz] - 1}]; 
Show[
            Graphics[{Thickness[0.0005], cosets}], 
            AspectRatio -> 1]];
MaurerRose[360, 360, 334];

Remove["Global\.b4*"]

Nachstehende Figur vergrössern!

f[t_, a_, b_, 
          c_] := (2 + Sin[a t]/2) {Cos[tt = (t + Sin[b t]/c)], 
            Sin[tt]};
Attributes[f] = Listable;
stern[a_, b_, c_, 
          n_] := (values = 
            f[Range[0, 2  Pi, 2  Pi/n // N], a, b, 
              c]; 
Show[
            Graphics[{Thickness[0.0005], 
                Map[Line[{{0, 0}, #}] &, values], 
Table[
                  Line[values r^0.2], {r, 0, 1, 
                    0.05}]}], 
PlotRange -> {{-2.5, 2.5}, {-2.5, 2.5}}, 
            AspectRatio -> 1]);
stern[6, 18, 12, 720];

Zweidimensionale Plots

Beispiele

Plot[{Cos[x]^2 - x Sin[x],Cos[x]^2 - Sin[x]},{x,-5,5}];

Plot[2 Sin[x]^2 - Sin[x + Tan[x]^2],{x,-8,8}];

Plot[Sin[x^21] Cos[2x],{x,-5,5},PlotPoints->70];

Plot[{Sin[x]-Floor[x],-Floor[x],Sin[x]},{x,-2Pi,2Pi},
	PlotStyle->{{Thickness[0.006]},{Thickness[0.001]},
				{Thickness[0.001]}}];

Plot[Sin[x]+Sin[13x]/7+Sin[7x]/13,{x,-10,10},PlotPoints->100];

Plot[{Sin[x],Sinh[x]},{x,-2Pi,2Pi},
PlotStyle->{{Thickness[0.005],Dashing[{0.05,0.03}]},
				{Thickness[0.008]}}];

Plot[Cos[x]/x,{x,-10,10},PlotRange->{-5,1}];

Plot[x^3+Cos[10x+5]/x,{x,-2.99,3.1}];

Plot[x^3+Cos[10x+5]/x,{x,-3,3}];

Plot[(Sin[x-4]^2 E^Sin[x]-1/8x)/(x-2)^2,{x,-6.05,6.11},
PlotRange->{1/6,1/6},PlotStyle->Thickness[0.015],
AxesLabel->{"Hallo","Velo"}];

Dreidimensionale Plots

Beispiele

Plot3D[ArcTan[x y],{x,-5,5},{y,-5,5},Lighting->False];

Plot3D[Sin[3x y]-Sin[5y] E^x,{x,-5,5},{y,-5,5},PlotPoints->70];

Plot3D[Tan[x+y]-Cos[x+y]+Sin[x-y],{x,-5,5},{y,-5,5},PlotPoints->70];

Plot3D[Sin[Cos[y]+x],{x,-Pi,2Pi},{y,-Pi,Pi},
	ViewPoint->{-1,3,2},PlotPoints->30,
	AxesLabel->{Achse X, Achse Y, Achse Z}];

Plot3D[Sin[x]^2 Cos[y]^2 E^(-(x/4)^8),{x,0,Pi},{y,-1/2 Pi, 3/2 Pi},
	PlotPoints->30];

Plot3D[Tan[x^7]-Sin[y^7],{x,0,Pi},{y,0,Pi}];

Plot3D[Tan[x^7]-Sin[y^7],{x,0,Pi},{y,0,Pi},PlotPoints->70];

Plot3D[y Sin[x]^2+ x Cos[y]^2,{x,0,2Pi},{y,0,2Pi}];

Plot3D[1/Sin[x y]^2,{x,0.1,Pi-0.1},{y,0.1,2Pi},BoxRatios->{1,1,4}];

Plot3D[Sin[x y]^2,{x,0,4},{y,0,4}];

Plot3D[Sin[x y]^2,{x,0,4},{y,0,4},PlotPoints->70];

Plot3D[Sin[x y]^2,{x,0,4},{y,0,1.5},PlotPoints->70];

Plot3D[Sin[x y]^2,{x,0,4},{y,0,1.5},PlotPoints->70];

Plot3D[x Cos[y],{x,-3,3},{y,-3,3}];

Plot3D[6 Sin[x-4] E^Cos[y],{x,-6,6},{y,-6,6},PlotPoints->25];

List Plots

Beispiele

ListPlot[N[Table[{Random[],Random[]},{10000}]],AspectRatio->1];

ListPlot[N[Table[{Random[],Random[]},{5000}]],AspectRatio->1];

messpkt=N[Table[{x,5x+2x Random[]},{x,0,5,0.2}]];
poly=Plot[Evaluate[InterpolatingPolynomial[messpkt,x],{x,0,5},
PlotRange->{-1,30},PlotStyle->RGBColor[0,1,1]]];
pkte=ListPlot[messpkt,PlotStyle->PointSize[0.02]];
Show[{poly,pkte},PlotRange->{-1,30}];

Parametric Plots

Beispiele

ParametricPlot[{x Cos[x], Sin[x]^x},{x,0.001,3.1},AspectRatio->1/2];

ParametricPlot[{Sin[x]^3,  Cos[x^2]},{x,0,10},PlotPoints->70];

ParametricPlot[{2Cos[-4t/6]+Cos[t], 4Sin[-2t/6]-2Sin[t]},
				{t,0,6Pi},PlotPoints->70];

ParametricPlot[{3Cos[t]+Cos[t/2], Sin[t]-2Sin[t/2]},{t,0,4Pi}];

ParametricPlot[{9Sin[x],4Cos[-11x]+7Cos[x+1]},{x,0,4Pi}];

ParametricPlot[{Cos[x^2+5],5Sin[x^3-2]},{x,0,2Pi}];

ParametricPlot[{Cos[x^2+5],5Sin[x^3-2]},{x,0,4Pi}];

Diverse Plots

Interpolationspolynom

Plot[(Sin[x-4]^2 E^Sin[x]-1/8x)/(x-2)^2,{x,-6,2Pi}

fkplot=Plot[(Sin[x-4]^2 E^Sin[x]-1/8x)/(x-2)^2,{x,-6,2Pi},
		DisplayFunction->Identity,PlotPoints->7];
Show[Graphics[{Thickness[0.0001],
	Map[Line[{{#[[1]],0},#}]&,
	Nest[First,fkplot,4]]}],Axes->Automatic];

Apfelmännchen

iter[x_,y_,lim_]:=Module[{c,z,ct},
	c=x+I y;
	z=c;
	ct=0;
	While[(Abs[z]<2.0) &&(ct<lim),++ct;z=z z+c];
	Return[ct]
						];
DensityPlot[-iter[x,y,50],{x,-2.0,1.0},{y,-1.5,1.5},
	PlotPoints->120,Mesh->False];

?iter

Table[iter[x/100,0,50],{x,0,100}]

Ideen von Studenten

Plot3D[x^2-y^2,{x,-4,4},{y,-4,4}];

ContourPlot[x^2-y^2,{x,-4,4},{y,-4,4}];

Plot3D[Sqrt[x^2+y^2],{x,-4,4},{y,-4,4}];

Plot3D[x y,{x,-4,4},{y,-4,4}];

Plot3D[Sin[x y],{x,-4,4},{y,-4,4},PlotPoints->50];

Ringe

Plot3D[
        Sqrt[(9 - x^2 - y^2) (x^2 + y^2 - 4)], {x, -3, 
          3}, {y, -3, 3}, PlotPoints -> 100];

ParametricPlot3D[{r Cos[φ],  r Sin[φ],  
          Sqrt[(9 - r^2) (r^2 - 4)]}, {r, 2, 3}, {φ, 0, 
          2  Pi}, PlotPoints -> 100];

?ParametricPlot3D

WIR93

Etwas Graphik

Die Lady von Herrn Senti

(Ob er sie selbst kreiert oder nur kopiert hat, hat er nicht mitgeteilt...)

f[x_] := Re[Sqrt[44^2 - x^2]/2]; 
    g[x_, y_] := 
      Re[Sqrt[44^2 - 
              4 ( x - 22)^2 - (y + 44)^2] (y/
                88)^2];
h[x_, y_] := 
        Re[Sqrt[44^2 - 
                4 ( x + 22)^2 - (y + 44)^2] (y/
                  88)^2];
k[x_, y_] := 
      Re[Sqrt[30 -  ( x - 22)^2 - (y + 74)^2] /5  + 
          Sqrt[10 - (x - 22)^2 - (y + 74)^2]]; 
    l[x_, y_] := 
      Re[Sqrt[30 -  ( x + 22)^2 - (y + 74)^2] /5  + 
          Sqrt[10 - (x + 22)^2 - (y + 
                    74)^2]];
z[x_, y_] := 
        f[x] + g[x, y] + h[x, y] + k[x, y] + 
          l[x, y];
Plot3D[z[x, y], {x, -44, 44}, {y, -95, 0}, 
        PlotPoints -> 100, ViewVertical -> {0, 1, 0}, 
        ViewPoint -> {1.5, -2, 1.5}, Axes -> False, 
        Boxed -> False];

Gratulation för das Spirit-III-Team: Signet

Package laden

<<Graphics`Polyhedra`

Sternform des Ikosaeders

Show[Polyhedron[GreatIcosahedron]]

?ParametricPlot3D

??ParametricPlot3D

??PlotPoints

ParametricPlot3D[{1.5 Sin[u] Cos[v], 1.5 Sin[u] Sin[v],
    2.5 Cos[u] },
   {u,0,Pi},{v,0,Pi},BoxRatios->{1,1,1}]

ParametricPlot3D[{u Sin[17 u^2], u Cos[17 u^2], u^2 -2},
{u,0,1},BoxRatios->{1,1,1},PlotPoints->500,Boxed->False]

Show[%2,%4,%5,Boxed->False]

Einige Contour Plots

Einige Ideen von Studenten

(a)

ContourPlot[-(y^2E^(x y)+3x^2 y)/((x^3+(1+x y)E^(x y))),
	{x,-1,1},{y,-1,1}];

ContourPlot[-(y^2E^(x y)+3x^2 y)/((x^3+(1+x y)E^(x y))),
	{x,-2,1},{y,-2,2}];

ContourPlot[-(y^2E^(x y)+3x^2 y)/((x^3+(1+x y)E^(x y))),
	{x,-2,2},{y,-2,2}];

ContourPlot[-(y^2E^(x y)+3x^2 y)/((x^3+(1+x y)E^(x y))),
	{x,-20,20},{y,-20,20}];

ContourPlot[-(y^2E^(x y)+3x^2 y)/((x^3+(1+x y)E^(x y))),
	{x,-200,200},{y,-200,200}];

(b)

Plot3D[-(y^2E^(x y)+3x^2 y)/((x^3+(1+x y)E^(x y))),
	{x,-2,2},{y,-2,2},PlotRange->{-50,50}];

Halbleiterphysik

Anwendungsbeispiel: Zwei parallele, unendlich lange Leiter.

Strom in Gegenrichtung.

a=0.05;
i1=40;
i2=-40;
h1= i1/(2Pi Sqrt[(x-a)^2 + y^2]);
h2= i2/(2Pi Sqrt[(x+a)^2 + y^2]);
Plot3D[Abs[h1+h2],{x,-0.1,0.1},{y,0,0.1},
				PlotRange->{0,500}, PlotPoints->25,
				ViewPoint->{1,-3.6,1.5}];

Strom in gleicher Richtung

Plot3D[Abs[h1-h2],{x,-0.1,0.1},{y,0,0.1},
				PlotRange->{0,500}, PlotPoints->25,
				ViewPoint->{1,-3.6,1.5}];

Probiere nun selbst!

Einige ParametricPlots
(Schläuche)

Probiere aus:

?ParametricPlot3D

ParametricPlot3D[{Cos[u],Sin[u],t},{u,0,2 Pi},{t,0,5}];

ParametricPlot3D[{(5+Sin[t]) Cos[u],(5+Sin[t]) Sin[u],
Cos[t]},
{u,0,2 Pi},{t,0,2 Pi}];

ParametricPlot3D[{(5+Sin[t]) Cos[u],(5+Sin[t]) Sin[u],
Cos[t]+u},
{u,0,4 Pi},{t,0,2 Pi}];

ParametricPlot3D[{(5+Sin[t]) Cos[u],(5+Sin[t]) Sin[u],
Cos[t]+t},
{u,0,2 Pi},{t,0,2 Pi}];

Zeichne den letzten Schlauch so, dass der zur Mittellinie senkrechte Schnitt ein Kreis ist:

Vektor

Remove[v];
v[t_]:={Cos[t],Sin[t],t};
ParametricPlot3D[Evaluate[v[t]],{t,0,2Pi}];

Ableitungen

Remove[v1];
v1[t_]:=D[v[t],t];
v1[t]

Remove[v2];
v2[t_]:=D[v[t],{t,2}];
v2[t]

Remove[langv1];
langv1[t_]:=Sqrt[v1[t].v1[t]];
langv1[t]

Krümmung

Remove[krumm];
krumm[t_]:=(v1[t].v1[t] v2[t].v2[t]-(v1[t].v2[t])^2)/
			langv1[t]^3;
krumm[t]

Begleitendes Dreibein

Remove[vT];
vT[t_]:=v1[t]/langv1[t];
vT[t]

Remove[vN];
vN[t_]:=(v2[t]/langv1[t]^2-v1[t] v2[t].v1[t] /
							langv1[t]^4)/krumm[t];
vN[t]

Remove[vB];
Needs["LinearAlgebra`CrossProduct`"];
vB[t_]:=Cross[vT[t],vN[t]];
vB[t]

Plots zum Probieren

ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t],v[t]+vN[t],
					v[t]+vB[t]}],{t,0,2Pi}];

ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t]}],{t,0,2Pi}];

Schlauch

Remove[param];
param[t_]:=v[t] + vN[t] Cos[u] + vB[t] Sin[u];
param[t]

ParametricPlot3D[Evaluate[param[t]],{t,0,3Pi},
{u,0,2Pi}];

ParametricPlot3D[Evaluate[param[t]],{t,0,4Pi},
{u,0,2Pi}];

Programm für einee beliebige Kurve

Gib Kurve v[t] sowie Parameter ein:

Remove[v];

v[t_]:={Cos[t],Sin[t],t};
g = 2 Pi;
h = 2 Pi;

ParametricPlot3D[Evaluate[v[t]],{t,0,2Pi}];
Remove[v1];
v1[t_]:=D[v[t],t];
Remove[v2];
v2[t_]:=D[v[t],{t,2}];
Remove[langv1];
langv1[t_]:=Sqrt[v1[t].v1[t]];
Remove[krumm];
krumm[t_]:=(v1[t].v1[t] v2[t].v2[t]-(v1[t].v2[t])^2)/
			langv1[t]^3;
Remove[vT];
vT[t_]:=v1[t]/langv1[t];
Remove[vN];
vN[t_]:=(v2[t]/langv1[t]^2-v1[t] v2[t].v1[t] /
							langv1[t]^4)/krumm[t];
Remove[vB];
Needs["LinearAlgebra`CrossProduct`"];
vB[t_]:=Cross[vT[t],vN[t]];
ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t],v[t]+vN[t],
					v[t]+vB[t]}],{t,0,2Pi}];
Remove[param];
param[t_]:=v[t] + vN[t] Cos[u] + vB[t] Sin[u];
ParametricPlot3D[Evaluate[param[t]],{t,0,g},
{u,0,h}];

Gib Kurve v[t] sowie Parameter ein:

Remove[v];

v[t_]:={3 E^(-(t^2)),5(1+Sin[t]),t^1.2};
g0 = 0.5;
h0 = 0;
g1 = 2.5;
h1 = 2 Pi;
r  = 0.8;

ParametricPlot3D[Evaluate[v[t]],{t,g0,g1}];
Remove[v1];
v1[t_]:=D[v[t],t];
Remove[v2];
v2[t_]:=D[v[t],{t,2}];
Remove[langv1];
langv1[t_]:=Sqrt[v1[t].v1[t]];
Remove[krumm];
krumm[t_]:=(v1[t].v1[t] v2[t].v2[t]-(v1[t].v2[t])^2)/
			langv1[t]^3;
Remove[vT];
vT[t_]:=v1[t]/langv1[t];
Remove[vN];
vN[t_]:=(v2[t]/langv1[t]^2-v1[t] v2[t].v1[t] /
							langv1[t]^4)/krumm[t];
Remove[vB];
Needs["LinearAlgebra`CrossProduct`"];
vB[t_]:=Cross[vT[t],vN[t]];
ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t],v[t]+vN[t],
					v[t]+vB[t]}],{t,g0,g1}];
Remove[param];
param[t_]:=v[t] + vN[t] r Cos[u] + vB[t] r Sin[u];
ParametricPlot3D[Evaluate[param[t]],{t,g0,g1},
{u,h0,h1}];

Gib Kurve v[t] sowie Parameter ein (verbessern...):

Remove[v];

v[t_]:={ Sin[3 t] Cos[t] , Sin[2 t], Cos[3 t]};
g0 = 0;
h0 = 0;
g1 = 2 Pi;
h1 = 2 Pi;
r  = 0.2;

ParametricPlot3D[Evaluate[v[t]],{t,g0,g1}];
Remove[v1];
v1[t_]:=D[v[t],t];
Remove[v2];
v2[t_]:=D[v[t],{t,2}];
Remove[langv1];
langv1[t_]:=Sqrt[v1[t].v1[t]];
Remove[krumm];
krumm[t_]:=(v1[t].v1[t] v2[t].v2[t]-(v1[t].v2[t])^2)/
			(langv1[t]^3 + 0.000000001);
Remove[vT];
vT[t_]:=v1[t]/(langv1[t]+ 0.000000001);
Remove[vN];
vN[t_]:=(v2[t]/(langv1[t]^2-v1[t] v2[t].v1[t]+ 0.000000001) /
						(langv1[t]^4+ 0.000000001))/krumm[t];
Remove[vB];
Needs["LinearAlgebra`CrossProduct`"];
vB[t_]:=Cross[vT[t],vN[t]];
ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t],v[t]+vN[t],
					v[t]+vB[t]}],{t,g0,g1}];
Remove[param];
param[t_]:=v[t] + vN[t] r Cos[u] + vB[t] r Sin[u];
ParametricPlot3D[Evaluate[param[t]],{t,g0,g1},
{u,h0,h1}];

Gib Kurve v[t] sowie Parameter ein (verbessern...):

Remove[v];

v[t_]:={2 Sin[3 t] Cos[t] ,4 Sin[t], Cos[3 t]};
g0 = 0;
h0 = 0;
g1 = 2 Pi;
h1 = 2 Pi;
r  = 0.8;

ParametricPlot3D[Evaluate[v[t]],{t,g0,g1}];
Remove[v1];
v1[t_]:=D[v[t],t];
Remove[v2];
v2[t_]:=D[v[t],{t,2}];
Remove[langv1];
langv1[t_]:=Sqrt[v1[t].v1[t]];
Remove[krumm];
krumm[t_]:=(v1[t].v1[t] v2[t].v2[t]-(v1[t].v2[t])^2)/
			langv1[t]^3;
Remove[vT];
vT[t_]:=v1[t]/langv1[t];
Remove[vN];
vN[t_]:=(v2[t]/langv1[t]^2-v1[t] v2[t].v1[t] /
							langv1[t]^4)/krumm[t];
Remove[vB];
Needs["LinearAlgebra`CrossProduct`"];
vB[t_]:=Cross[vT[t],vN[t]];
ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t],v[t]+vN[t],
					v[t]+vB[t]}],{t,g0,g1}];
Remove[param];
param[t_]:=v[t] + vN[t] r Cos[u] + vB[t] r Sin[u];
ParametricPlot3D[Evaluate[param[t]],{t,g0,g1},
{u,h0,h1}];

Gib Kurve v[t] sowie Parameter ein:

Remove[v];

v[t_]:={3 E^(-(t^2)) Cos[t] ,5(1+Sin[t]),Sin[3 t]};
g0 = 0.1;
h0 = 0;
g1 = 2.5;
h1 = 2 Pi;
r  = 0.8;

ParametricPlot3D[Evaluate[v[t]],{t,g0,g1}];
Remove[v1];
v1[t_]:=D[v[t],t];
Remove[v2];
v2[t_]:=D[v[t],{t,2}];
Remove[langv1];
langv1[t_]:=Sqrt[v1[t].v1[t]];
Remove[krumm];
krumm[t_]:=(v1[t].v1[t] v2[t].v2[t]-(v1[t].v2[t])^2)/
			langv1[t]^3;
Remove[vT];
vT[t_]:=v1[t]/langv1[t];
Remove[vN];
vN[t_]:=(v2[t]/langv1[t]^2-v1[t] v2[t].v1[t] /
							langv1[t]^4)/krumm[t];
Remove[vB];
Needs["LinearAlgebra`CrossProduct`"];
vB[t_]:=Cross[vT[t],vN[t]];
ParametricPlot3D[Evaluate[{v[t],v[t]+vT[t],v[t]+vN[t],
					v[t]+vB[t]}],{t,g0,g1}];
Remove[param];
param[t_]:=v[t] + vN[t] r Cos[u] + vB[t] r Sin[u];
ParametricPlot3D[Evaluate[param[t]],{t,g0,g1},
{u,h0,h1}];

WIR93

Studium einer Welle

(Das Beispiel stammt von U. Altmann)

Studiere das folgende Beispiel (Simulation einer Welle)

Erst rechnen, dann animieren

Table[Plot3D[
	Sin[3t+(x-3)^2+(y-2)^2]/(1+(x-3)^2+(y-2)^2)+
	3/5Sin[5/3(3t+(x+3)^2+(y-2)^2)]/(1+(x+3)^2+(y-2)^2)+
	3/4Sin[4/3(3t+x^2+(y+3)^2)]/(1+x^2+(y+3)^2),
	{x,-8,8},{y,-8,8},
PlotRange->{{-8,8},{-8,8},{-1,1}},
PlotPoints->30],
{t,0,2Pi,2Pi/60}];

Wähle die Zelle an, die alle ausgegebenen Bilder umschliesst.
Animiere mit "Animate Selected Graphics y" in Menu "Graph".
Beobachte, welche Zonensich bewegen und welcht Zonen  ruhig
bleiben. Gibt es Interferenzen?

Probiere eigene Beispiele aus!

Abbildung eines Parallelgitters in der
komplexen Ebene

"Putzmaschine" einsetzen

Remove["Global`@*"]

Vorübung

Funktion w definieren

Clear[w]

w[s_,t_]:=1/(1+s+I t)

Diverse Plots

ParametricPlot[{Re[w[s,0]],Im[w[s,0]]},{s,-10,10}];

ParametricPlot[{Re[w[0,s]],Im[w[0,s]]},{s,-10,10}];

ParametricPlot[{Re[w[Cos[t],Sin[t]]],Im[w[Cos[t],Sin[t]]]},
{t,-10,10}];

Alles in einem Schritt; das Gitter, das abgebildet wird

Remove["Global`@*"];
w[s_,t_]:=(s+I t);
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Alles in einem Schritt, 1., 1/(1+z)

Remove["Global`@*"];
w[s_,t_]:=1/(1+s+I t);
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-5,5}]],{s,-20,20}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-5,5}]],{s,-20,20}];
Show[a,b];

Alles in einem Schritt, 2., E^z

Remove["Global`@*"];
w[s_,t_]:=E^(s+I t);
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Show[{a,b},AspectRatio->GoldenRatio];

Alles in einem Schritt, 3., z^2

Remove["Global`@*"];
w[s_,t_]:=(s+I t)^2;
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Alles in einem Schritt, 4., z^3

Remove["Global`@*"];
w[s_,t_]:=(s+I t)^3;
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Alles in einem Schritt, 5., z^4

Remove["Global`@*"];
w[s_,t_]:=(s+I t)^4;
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Alles in einem Schritt, 6., z^5

Remove["Global`@*"];
w[s_,t_]:=(s+I t)^5;
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Alles in einem Schritt, 7., Sin[z]

Remove["Global`@*"];
w[s_,t_]:=Sin[s+I t];
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,1,10}]],{s,1,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,1,10}]],{s,1,10}];
Show[a,b];

Alles in einem Schritt, 8., 1/z

w[s_,t_]:=1/(s+I t);
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Alles in einem Schritt, 9., (z-1)/(z+1)

w[s_,t_]:=(s+I t-1)/(s+I t + 1);
a=ParametricPlot[Evaluate[Table[{Re[w[n,s]],Im[w[n,s]]}, 
{n,-10,10}]],{s,-10,10}];
b=ParametricPlot[Evaluate[Table[{Re[w[s,n]],Im[w[s,n]]}, 
{n,-10,10}]],{s,-10,10}];
Show[a,b];

Minimalflächen

Enneper's Surface

Created by S. Dickson.

ParametricPlot3D[
  {Re[E ^ (I * th) * r - (E ^ (3 * I * th) * r ^ 3) / 3],
  Re[I * E ^ (I * th) * r + I / 3 * E ^ (3 * I * th) * r ^ 3],
  Re[E ^ (2 * I * th) * r ^ 2]},
  {th, 0, 2 Pi}, {r, 0, 3},
  ViewPoint -> {2.8, -1.9, 0.1}, Boxed -> False,
  PlotPoints -> {100, 30}, Axes -> False, 
  PlotRange -> All, LightSources -> {
    {{0.7071, 0, 0.7071}, RGBColor[0.9481, 0, 0]},
    {{0.5773, 0.5773, 0.5773}, RGBColor[0, 0.8888, 0]},
    {{0, 0.7071, 0.7071}, RGBColor[0, 0, 1]}
  }
]

Local Structure of an Immersed Minimal Surface

Created by R. Maeder.

ParametricPlot3D[
  {Re[E^ (I * phi) * r - (E ^ (3 * I * phi) * r ^ 3) / 3], 
  Re[I * E ^ (I * phi) * r + I / 3 * E ^ (3 * I * phi) * r ^ 3],
  Re[E ^ (2 * I * phi) * r ^ 2]},
  {r, 0, 1}, {phi, -Pi, Pi}, PlotPoints -> {10, 40},
  PlotRange -> All, Axes -> False, Boxed -> False
]

Local Structure of an Immersed Minimal Surface

Created by R. Maeder.

ParametricPlot3D[
  {-(r ^ 2 Cos[2 phi]) / 2 + Log[r],
   -phi - (r ^ 2 Sin[2 phi]) / 2, 2 r Cos[phi]},
  {r, 0.1, 3.5}, {phi, -3.1, 1.5},
  PlotPoints -> {10, 40}, Axes -> False,
  Boxed -> False, PlotRange -> All,
  ViewPoint -> {0.5906, -1.9715, -2.6871},
  LightSources -> {
    {{0.5901, -0.1296, 0.7968}, RGBColor[1, 0, 0]}, 
    {{0.7299, 0.6626, 0.1677}, RGBColor[0, 1, 0.0081]}, 
    {{-0.2038, 0.724, 0.6589}, RGBColor[0, 0, 1]}
  }
]

Upright Conical Spiral

Created by S. Dickson.

X = 2 * (1 - E ^ (u / (6 * Pi))) * Cos[v / 2] ^ 2 * Cos[u];
Y = 2 * (-1 + E ^ (u / (6 * Pi))) * Cos[v / 2] ^ 2 * Sin[u];
Z = 1 - E ^ (u / (3 * Pi)) - Sin[v] + E ^ (u / (6 * Pi)) * Sin[v];

ParametricPlot3D[
  {X, Y, Z}, {u, 0, 6 Pi}, {v, 0, 2 Pi},
  PlotPoints -> {96, 20}, PlotRange -> All,
  ViewPoint -> {1.0992, 3.2001, 0.0255}, Axes -> False,
  Boxed -> False, RenderAll -> False,
  LightSources -> {
    {{0.7071, 0, 0.7071}, RGBColor[1, 0, 0]}, 
    {{0.5773, 0.5773, 0.5773}, RGBColor[0, 1, 0]}, 
    {{0, 0.7071, 0.7071}, RGBColor[0, 0, 1]}
  }
]

Upright Conical Spiral with Gradual Slope

Created by S. Dickson.

X = 2 * (-1 + E ^ (u / (6 *Pi))) * Cos[v / 2] ^ 2 * Cos[u];
Y = 2 * (-1 + E ^ (u / (6 *Pi))) * Cos[v / 2] ^ 2 * Sin[u];
Z = 1 / 2 - E^ (u / (3 * Pi)) / 2 - Sin[v] +
  E ^ (u / (6 * Pi)) * Sin[v];

ParametricPlot3D[
  {X, Y, Z}, {u, 0, 6 Pi}, {v, 0, 2 Pi},
  PlotPoints -> {100, 20}, PlotRange -> All,
  Axes -> False, Boxed -> False
]

X = 2 * (-1 + E ^ (u / (6 * Pi))) * Cos[v / 2] ^ 2 *
  (Cos[u] * Cos[Sin[2 * v] / 2] +
  Sin[u] * Sin[Sin[2 * v] / 2]);
Y =  2 * I * (-1 + E ^ (u / (6 * Pi))) * Cos[v / 2] ^ 2 *
   (-I * Cos[Sin[2 * v] / 2] * Sin[u] +
   I * Cos[u] * Sin[Sin[2 * v] / 2]);
Z = (-1 + E ^ (u / (6 * Pi))) * Sin[v];

ParametricPlot3D[
  {X, Y, Z}, {u, 0, 6 Pi}, {v, -Pi, 0},
  ViewPoint -> {3.6305, -0.2046, 1.8322},
  PlotPoints -> {108, 14}, PlotRange -> All,
  Axes -> False, Boxed -> False
]

Trinoid Minimal Surface

Created by S. Dickson.

ParametricPlot3D[
  {Re[(E ^ (I * th) * r) / (3 * (1 + E ^ (I * th) * r +
   E ^ (2 * I * th) * r ^ 2)) - (4 * Log[1 - E ^ (I * th) * r]) /
   9 + (2 * Log[1 + E ^ (I * th) * r + E ^ (2 * I * th) * r ^ 2]) / 9], 
  Re[((-2 * I) / 9 * Pi) / 3 ^ (1 / 2) - (I / 3 * (E ^ (I * th) * r +
   E ^ (2 * I * th) * r ^ 2)) / (-1 + E ^ (3 * I * th) * r ^ 3) +
   ((4 * I) / 3 * ArcTan[(1 + 2 * E ^ (I * th) * r) / 3 ^ (1 / 2)]) /
   3 ^ (1 / 2)], Re[-2 / 3 - 2 / (3 * (-1 + E ^ (3 * I * th) * r ^ 3))]},
  {th, 0, 2 Pi}, {r, 0, 2},
  ViewPoint -> {-0.081, -0.003, 1.618},
  PlotPoints -> {15, 20}, PlotRange -> All,
  Boxed -> False, Axes -> False
]

Trefoil Knot

Created by H. Cejtin after R. W. Gosper.

The Mathematica code below makes use of the tubes package, available on MathSource as part of the  Mathematica Graphics Gallery  notebook.

Show[
  tube[
    {-10 Cos[t] - 2 Cos[5 t] + 15 Sin[2 t], 
     -15 Cos[2 t] + 10 Sin[t] - 2 Sin[5 t],
     10 Cos[3 t]},
    {t, 0, 2 Pi, 75}, 9, 4.
  ],
  Boxed -> False
]

Figure-Eight Knot

Created by S. Dickson.

The Mathematica code below makes use of the tubes package, available on MathSource as part of the  Mathematica Graphics Gallery  notebook.

Show[
  tube[
    {10 (Cos[t] + Cos[3 t]) + Cos[2 t] + Cos[4 t],
     6 Sin[t] + 10 Sin[3 t],
     4 Sin[3 t] Sin[5 t/2] + 4 Sin[4 t] - 2 Sin[6 t]},
    {t, 0, 2 Pi, 90}, 8, 2.5
  ],
  Boxed -> False
]

Closed Three-Braid Knot

Created by S. Dickson.

The Mathematica code below makes use of the tubes package, available on MathSource as part of the  Mathematica Graphics Gallery  notebook.

Show[
  tube[
    {Cos[3 t] (25 + 5 Sin[10 t]),
     Sin[3 t] (25 + 5 Sin[10 t]),
     10 Sin[5 t]},
    {t, 0, 2 Pi, 200}, 10, 3.
  ],
  Boxed -> False
]

(3, 4)-Torus Knot

Created by S. Dickson.

The Mathematica code below makes use of the tubes package, available on MathSource as part of the  Mathematica Graphics Gallery  notebook.

Show[
  tube[
    {4 Cos[t + N[Pi]] / 3 + 2 Cos[3 t],
     4 Sin[t] / 3 + 2 Sin[3 t],
     Sin[4 t] + Sin[2 t] / 2}, 
    {t, 0, 2 Pi, 90}, 9, .25
  ],
  Boxed -> False
]

Klein Bottle

Created by S. Dickson.

bx = 6 Cos[u] (1 + Sin[u]);
by = 16 Sin[u];
rad = 4 (1 - Cos[u] / 2);
X = If[Pi < u <= 2 Pi, bx + rad Cos[v + Pi], bx + rad Cos[u] Cos[v]];
Y = If[Pi < u <= 2 Pi, by, by + rad Sin[u] Cos[v]];
Z = rad Sin[v];

ParametricPlot3D[
  {X, Y, Z}, {u, 0, 2 Pi}, {v, 0, 2 Pi}, 
  PlotPoints -> {48, 12},
  Axes -> False, Boxed -> False, 
  ViewPoint -> {1.4, -2.6, -1.7}
]

Cross-Section of a Flat Conical Spiral

Created by S. Dickson.

X = 2 * (-1 + E ^ (u / (6 * Pi))) * Cos[v / 2] ^ 2 *
  (Cos[u] * Cos[Sin[2 * v] / 2] +
  Sin[u] * Sin[Sin[2 * v] / 2]);
Y =  2 * I * (-1 + E ^ (u / (6 * Pi))) * Cos[v / 2] ^ 2 *
   (-I * Cos[Sin[2 * v] / 2] * Sin[u] +
   I * Cos[u] * Sin[Sin[2 * v] / 2]);
Z = (-1 + E ^ (u / (6 * Pi))) * Sin[v];

ParametricPlot3D[
  {X, Y, Z}, {u, 0, 6 Pi}, {v, -Pi, 0},
  ViewPoint -> {3.6305, -0.2046, 1.8322},
  PlotPoints -> {108, 14}, PlotRange -> All,
  Axes -> False, Boxed -> False
]

Density Plot of Sin[Re[Gamma[x + I y]]]

Created by P. Abbott.

DensityPlot[Sin[Re[Gamma[x + I y]]], {x, 3, 7}, {y, 0, 5},
  PlotPoints -> 200, Mesh -> False, ColorFunction ->
    (RGBColor[Sin[(# + 1) / 2],
    Sin[(1 - # ^ 2) / 2],
    Abs[(1 - (# + .2) / 1)]]&)
];

Recursively Constructed Tree

Created by T. Gray and J. Glynn.

<<Geometry`Rotations`;

BeautifulTree[nestingDepth_, rotation_, shrinkage_,
 thicknessShrinkage_, originalThickness, colors_, options___] :=
  Block[{twinLine, doTwins, rotateLine, colorCounter},
    rotateLine[Line[{start_, end_}], angle_] :=
      Line[{end, end + shrinkage *
      Rotate2D[end - start, angle Random[Real, {0,5, 1.5}]]}];
    twinLine[Line[points_]] := 
      {rotateLine[Line[points], rotation], 
      rotateLine[Line[points], -rotation]};
    twinLine[Thickness[th_]] :=
      Thickness[th thicknessShrinkage];
    twinLine[RGBColor[r_, g_, b_]] := {};

    colorCounter = 1;
    doTwins [lines_] :=
      Flatten[{colors[[1 + Mod[colorCounter++, Length[colors]]]],
    Map[twinLine, lines]}];

    Show[Graphics[NestList[doTwins,
     {colors[[1]], Thickness[originalThickness],
     Line[{{0, 0}, {0, 1}}]}, nestingDepth]], options,
     AspectRatio -> Automatic, PlotRange -> All]
  ];

BeautifulTree[10, 0.65, 0.75, 0.7, 0.07, {
 RGBColor[0.430, 0.185, 0.064], RGBColor[0.465, 0.201, 0.069],
 RGBColor[0.535, 0.231, 0.080], RGBColor[0.590, 0.255, 0.088],
 RGBColor[0.219, 0.465, 0.000], RGBColor[0.245, 0.520, 0.000],
 RGBColor[0.275, 0.582, 0.000], RGBColor[0.322, 0.684, 0.000],
 RGBColor[0.332, 0.714, 0.000], RGBColor[0.346, 0.734, 0.000],
 RGBColor[0.984, 0.731, 0.024]
}]

Part of Scherk's Second Minimal Surface

Created by S. Dickson.

ParametricPlot3D[
  {Re[2 * (-Log[1 - E ^ (I * th) * r] + Log[1 + E ^ (I * th) * r])], 
  Re[4 * I * ArcTan[E ^ (I * th) * r]], 
  Re[2 * I * (-Log[1 - E ^ (2 * I * th) * r ^ 2] +
  Log[1 + E ^ (2 * I * th) * r ^ 2])]},

  {th, 0, 2 Pi}, {r, .01, .98}, PlotPoints -> {60, 19},
  ViewPoint -> {1.3, -2.4, 2.0}, 
  Boxed -> False, Axes -> False, PlotRange -> All
]

Strange Attractor in the Equations for an Electric Circuit

Created by P. Boyland.

Hue[x_] := HSBColor[-1.2 x + .2, 1, 1];
hue[x_] := Hue[5 x, 1, 1];

goodscroll = Graphics3D[{
  {hue[0.028], Line[{{1.64339, 0.156082, -1.52554}, 
   {1.70846, 0.150505, -1.67952}}]}, 
  {hue[0.024], Line[{{1.70846, 0.150505, -1.67952}, 
   {1.75783, 0.139013, -1.82497}}]}, 
  {hue[0.019], Line[{{1.75783, 0.139013, -1.82497}, 
   {1.79065, 0.121695, -1.95594}}]}, 
  {hue[0.013], Line[{{1.79065, 0.121695, -1.95594}, 
   {1.80645, 0.0989796, -2.06684}}]}, 
  {hue[0.0081], Line[{{1.80645, 0.0989796, -2.06684}, 
   {1.80511, 0.07161055963021647, -2.15259}}]}, 
  {hue[0.0045], Line[{{1.80511, 0.07161055963021647, -2.15259}, 
   {1.78706, 0.0406063, -2.20902}}]}, 
  {hue[0.0028], Line[{{1.78706, 0.0406063, -2.20902}, 
   {1.7533, 0.007210519606065681, -2.2331}}]}, 
  {hue[0.0036], Line[{{1.7533, 0.007210519606065681, -2.2331}, 
   {1.70548, -0.0271716, -2.22312}}]}, 
  {hue[0.0067], Line[{{1.70548, -0.0271716, -2.22312}, 
   {1.64578, -0.0610319, -2.17885}}]}, 
  {hue[0.012], Line[{{1.64578, -0.0610319, -2.17885}, 
   {1.57691, -0.0928326, -2.10157}}]}, 
  {hue[0.018], Line[{{1.57691, -0.0928326, -2.10157}, 
   {1.50204, -0.121082, -1.99411}}]}, 
  {hue[0.024], Line[{{1.50204, -0.121082, -1.99411}, 
   {1.42461, -0.144406, -1.86073}}]}, 
  {hue[0.03], Line[{{1.42461, -0.144406, -1.86073}, 
   {1.34825, -0.161614, -1.70696}}]}, 
  {hue[0.033], Line[{{1.34825, -0.161614, -1.70696}, 
   {1.27662, -0.171765, -1.53944}}]}, 
  {hue[0.034], Line[{{1.27662, -0.171765, -1.53944}, 
   {1.2132, -0.174215, -1.36557}}]}, 
  {hue[0.032], Line[{{1.2132, -0.174215, -1.36557}, 
   {1.16121, -0.168657, -1.19325}}]}, 
  {hue[0.028], Line[{{1.16121, -0.168657, -1.19325}, 
   {1.12338, -0.155142, -1.03051}}]}, 
  {hue[0.022], Line[{{1.12338, -0.155142, -1.03051}, 
   {1.10186, -0.134091, -0.885119}}]}, 
  {hue[0.015], Line[{{1.10186, -0.134091, -0.885119}, 
   {1.09809, -0.10628, -0.764271}}]}, 
  {hue[0.0094], Line[{{1.09809, -0.10628, -0.764271}, 
   {1.11273, -0.0728202, -0.674198}}]}, 
  {hue[0.0055], Line[{{1.11273, -0.0728202, -0.674198}, 
   {1.14561, -0.0351099, -0.619882}}]}, 
  {hue[0.0044], Line[{{1.14561, -0.0351099, -0.619882}, 
   {1.1957, 0.005219149746375589, -0.60478}}]}, 
  {hue[0.0066], Line[{{1.1957, 0.005219149746375589, -0.60478}, 
   {1.26114, 0.0463729, -0.630627}}]}, 
  {hue[0.012], Line[{{1.26114, 0.0463729, -0.630627}, 
   {1.33931, 0.0864675, -0.697297}}]}, 
  {hue[0.02], Line[{{1.33931, 0.0864675, -0.697297}, 
   {1.42695, 0.123627, -0.802798}}]}, 
  {hue[0.029], Line[{{1.42695, 0.123627, -0.802798}, 
   {1.52027, 0.156054, -0.94328}}]}, 
  {hue[0.039], Line[{{1.52027, 0.156054, -0.94328}, 
   {1.61509, 0.182127, -1.11317}}]}, 
  {hue[0.046], Line[{{1.61509, 0.182127, -1.11317}, 
   {1.70709, 0.200473, -1.30541}}]}, 
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Show[goodscroll,
 Axes -> False, Lighting -> False, ViewPoint -> {4.1, 1.53, -0.3}]

Einfache Ostereierkonstuktion

ParametricPlot[{
{2  Cos[t] + 1, 
            2  Sin[t]}, 
{2  Cos[t] - 1, 2  Sin[t]}}, {t, 
          0, 2  Pi}, AspectRatio -> Automatic];

ParametricPlot[{
{2  Cos[t] + 1, 
            2  Sin[t]}, 
{2  Cos[t] - 1, 
            2  Sin[t]}, 
{Cos[t/2 + Pi], 
            Sin[t/2 + Pi]}, 
{(2 - Sqrt[2]) 
              Cos[Pi/4 + t/4], (2 - Sqrt[2]) Sin[Pi/4 + t/4] + 
              1}
}, {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

ParametricPlot[{
{2  Cos[t/8 + 3  Pi/4] + 1, 
            2  Sin[t/8 + 3  Pi/4]}, 
{2  Cos[t] - 1, 
            2  Sin[t]}, 
{Cos[t/2 + Pi], 
            Sin[t/2 + Pi]}, 
{(2 - Sqrt[2]) 
              Cos[Pi/4 + t/4], (2 - Sqrt[2]) Sin[Pi/4 + t/4] + 
              1}
}, {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

ParametricPlot[{
{2  Cos[t/8 + 3  Pi/4] + 1, 
            2  Sin[t/8 + 3  Pi/4]}, 
{2 
                 Cos[-t/8  + 1  Pi/4] - 1, 
            2  Sin[-t/8 + 1  Pi/4]}, 
{Cos[t/2 + Pi], 
            Sin[t/2 + Pi]}, 
{(2 - Sqrt[2]) 
              Cos[Pi/4 + t/4], (2 - Sqrt[2]) Sin[Pi/4 + t/4] + 
              1}
}, {t, 0, 2  Pi}, 
        AspectRatio -> Automatic];

ParametricPlot3D[{
{(2  Cos[t/8 + 3  Pi/4] + 
                  1) Cos[u], (2  Cos[t/8 + 3  Pi/4] + 1) Sin[u], 
            2  Sin[t/8 + 
                  3  Pi/4]}, 
{(2 
                     Cos[-t/8  + 1  Pi/4] - 1) Cos[
                u], (2  Cos[-t/8  + 1  Pi/4] - 1) Sin[u], 
            2  Sin[-t/8 + 1  Pi/4]}, 
{Cos[
                t/2 + Pi] Cos[u], Cos[t/2 + Pi] Sin[u], 
            Sin[t/2 + Pi]}, 
{(2 - Sqrt[2]) 
              Cos[Pi/4 + t/4] Cos[u], (2 - Sqrt[2]) Cos[Pi/4 + t/4] 
              Sin[u], (2 - Sqrt[2]) Sin[Pi/4 + t/4] + 
              1}
}, {t, 0, 2  Pi}, {u, 0, 2  Pi}, 
        AspectRatio -> Automatic];

ParametricPlot3D[{
{(2  Cos[t/8 + 3  Pi/4] + 
                  1) Cos[u], (2  Cos[t/8 + 3  Pi/4] + 1) Sin[u], 
            2  Sin[t/8 + 
                  3  Pi/4]}, 
{(2 
                     Cos[-t/8  + 1  Pi/4] - 1) Cos[
                u], (2  Cos[-t/8  + 1  Pi/4] - 1) Sin[u], 
            2  Sin[-t/8 + 1  Pi/4]}, 
{Cos[
                t/2 + Pi] Cos[u], Cos[t/2 + Pi] Sin[u], 
            Sin[t/2 + Pi]}, 
{(2 - Sqrt[2]) 
              Cos[Pi/4 + t/4] Cos[u], (2 - Sqrt[2]) Cos[Pi/4 + t/4] 
              Sin[u], (2 - Sqrt[2]) Sin[Pi/4 + t/4] + 
              1}
}, {t, 0, 2  Pi}, {u, 0, 2  Pi}, 
        AspectRatio -> 
          Automatic, 
ViewPoint -> {2.167,  -1.158,  
            1.708}];