Blumen - Fleurs

2D

Remove["Global`*"]

pP[n_,h_,a_,b_]:=ParametricPlot[f[n,x,a,b],{x,0,h},AspectRatio->Automatic,Axes->False];
p[1,q,r,s]

f[1,x_,a_,b_]:= {a,b}+{Cos[x],Sin[x]}(3+ Cos[5x])/3;
f[1,x,3,1]

pP[1,2Pi,1,4];

h=2Pi

f[2,x_,a_,b_]:= x/h {a,b}+{Cos[ArcTan[b/a]+Pi],Sin[ArcTan[b/a]+Pi]}(3+ Cos[5(ArcTan[b/a]+Pi)])/3+{Sin[x]/10,0};
pP[2,2Pi,1,4];

f[3,x_,a_,b_]:= {a,b}+{Cos[x],Sin[x]}/2;
f[3,x,3,1]

pP[3,2Pi,1,4];

Show[pP[3,2Pi,1,4],pP[1,2Pi,1,4],pP[2,2Pi,1,4],PlotRange->{{-1,4},{0,6}}];

Remove["Global`*"]

p[n_,h_,a_,b_,colo_,thick_]:=ParametricPlot[f[n,x,a,b],{x,0,h},AspectRatio->Automatic,Axes->False,PlotStyle -> {{Thickness[thick], Hue[colo]}}];
p[1,q,r,s]

f[1,x_,a_,b_]:= {a,b}+{Cos[x],Sin[x]}(3+ Cos[5x])/3;
f[1,x,3,1]

p[1,2Pi,1,4,1,0.02];

h=2Pi

f[2,x_,a_,b_]:= x/h {a,b}+{Cos[ArcTan[b/a]+Pi],Sin[ArcTan[b/a]+Pi]}(3+ Cos[5(ArcTan[b/a]+Pi)])/3+{Sin[x]/10,0};
p[2,2Pi,1,4,0.4,0.02];

f[3,x_,a_,b_]:= {a,b}+{Cos[x],Sin[x]}/2;
f[3,x,3,1]

p[3,2Pi,1,4,0.1,0.02];

Show[p[3,2Pi,1,4,0.1,0.02],p[1,2Pi,1,4,1,0.02],p[2,2Pi,1,4,0.4,0.02],PlotRange->{{-1,4},{0,6}}];

f[4,x_,a_,b_]:= {-a,b}+{-Cos[x],Sin[x]}(3+ Cos[5x])/3;

f[5,x_,a_,b_]:= x/h {-a,b}+{-Cos[ArcTan[b/a]+Pi],Sin[ArcTan[b/a]+Pi]}(3+ Cos[5(ArcTan[b/a]+Pi)])/3+{-Sin[x]/10,0};

f[6,x_,a_,b_]:= {-a,b}+{-Cos[x],Sin[x]}/2;

Show[p[3,2Pi,1,4,0.1,0.02],p[1,2Pi,1,4,1,0.02],p[2,2Pi,1,4,0.4,0.02],p[4,2Pi,2,4,0.9,0.02],p[5,2Pi,2,4,0.35,0.02],p[6,2Pi,2,4,0.05,0.02],PlotRange->{{-4,4},{0,6}}];

?*Hue*

f[4,x_,a_,b_]:= {-a,b}+{-Cos[x],Sin[x]}(3+ Cos[5x])/3;

f[5,x_,a_,b_]:= x/h {-a,b}+{-Cos[ArcTan[b/a]+Pi],Sin[ArcTan[b/a]+Pi]}(3+ Cos[5(ArcTan[b/a]+Pi)])/3+{-Sin[x]/10,0};

f[6,x_,a_,b_]:= {-a,b}+{-Cos[x],Sin[x]}/2;

Show[p[3,2Pi,1,4,0.1,0.02],p[1,2Pi,1,4,1,0.02],p[2,2Pi,1,4,0.4,0.02],p[4,2Pi,-1,4,0.9,0.02],p[5,2Pi,-1,4,0.35,0.02],p[6,2Pi,-1,4,0.05,0.02],PlotRange->{{-4,4},{0,6}}];

f[3, x_, a_, b_] := {a, b} + {Cos[x], Sin[x]}/4;
f[6, x_, a_, 
          b_] := {-a, b} + {-Cos[x], Sin[x]}/
            4;

dd = 0.1;

Show[p[2, 2  Pi, 1, 4, 0.4, dd], p[3, 2  Pi, 1, 4, 0.1, dd], 
        p[1, 2  Pi, 1, 4, 1, dd], p[4, 2  Pi, 2, 4, 0.9, dd], 
        p[5, 2  Pi, 2, 4, 0.35, dd], p[6, 2  Pi, 2, 4, 0.05, dd], 
        PlotRange -> {{-4, 4}, {0, 6}}];

3D

Remove["Global`*"]

h1 = 2  Pi; h2 = 2  Pi;

p3D[n_, h1_, h2_, a_, b_, c_] := 
        ParametricPlot3D[f3D[n, u, v, a, b, c], {u, 0, h1}, {v, 0, h2}, 
          AspectRatio -> Automatic, Boxed -> False];
p3D[1, q1, q2, r, s, t];

f3D[1, u_, v_, a_, b_, 
          c_] := {a, b, c} + {Cos[v] (Cos[5  v])/3, 
            v Cos[u] (3 + Cos[5  u])/3, 
            v Sin[u] (3 + Cos[5  u])/3};
f3D[1, u, v, 3, 1, 1]

p3D[1, h1, h2, 2, 4, 1];

p3D[n_, h1_, h2_, a_, b_, c_] := 
        ParametricPlot3D[f3D[n, u, v, a, b, c], {u, 0, h1}, {v, 0, h2}, 
          AspectRatio -> Automatic, Boxed -> False, 
          Axes -> False, PlotPoints -> 70, 
          DefaultColor -> RGBColor[0, 1, 0], 
          Background -> Hue[0.55], 
          ViewPoint -> {-2.465,  -3.752,  2.377}];
p3D[1, h1, h2, 2, 4, 1];

p3D[n_, h1_, h2_, a_, b_, c_] := 
        ParametricPlot3D[f3D[n, u, v, a, b, c], {u, 0, h1}, {v, 0, h2}, 
          AspectRatio -> Automatic, Boxed -> False, 
          Axes -> False, PlotPoints -> 50, 
          ViewPoint -> {-2.465,  -3.752,  2.377}];
p3D[1, h1, h2, 2, 4, 1];

p3D[n_, h1_, h2_, a_, b_, c_] := 
        ParametricPlot3D[f3D[n, u, v, a, b, c], {u, 0, h1}, {v, 0, h2}, 
          AspectRatio -> Automatic, Boxed -> False, 
          Axes -> False, PlotPoints -> 50, 
          DefaultColor -> Hue[1], Background -> Hue[0.55], 
          ViewPoint -> {-2.465,  -3.752,  2.377}];
p3D[1, h1, h2, 2, 4, 1];

p3D[n_, h1_, h2_, a_, b_, c_] := 
        ParametricPlot3D[f3D[n, u, v, a, b, c], {u, 0, h1}, {v, 0, h2}, 
          AspectRatio -> Automatic, Boxed -> False, 
          Axes -> False, PlotPoints -> 50, 
          DefaultColor -> RGBColor[0, 1, 0], 
          Background -> Hue[0.55], 
          ViewPoint -> {-2.465,  -3,  2}, 
          PlotRange -> {{-8, 8}, {-8, 8}, {-8, 
                8}}];
p3D[1, h1, h2, 2, 4, 1];

q3D[n_, h1_, h2_, a_, b_, c_, d_] := 
        ParametricPlot3D[k3D[n, u, v, a, b, c, d], {u, 0, h1}, {v, 0, h2}, 
          AspectRatio -> Automatic, Boxed -> False, 
          Axes -> False, PlotPoints -> 50, 
          DefaultColor -> RGBColor[0, 1, 0], 
          Background -> Hue[0.55], 
          ViewPoint -> {-2.465,  -3,  2}, 
          PlotRange -> {{-8, 8}, {-8, 8}, {-8, 
                8}}];

k3D[2, u_, v_, a_, b_, c_, d_] := 
        10  u/h1 {0, b, c} + 
          10 {0, Cos[ArcTan[c/b] + Pi], 
              Sin[ArcTan[c/b] + Pi]} (3 + Cos[5 (ArcTan[c/b] + Pi)])/
              3 + 
10 
             u/h2 {a, 0, 0} + {Cos[u] (Cos[5  u])/3, 0, 0} + 
          d {Cos[v], Sin[v], 0};

q3D[2, h1, h2, 2, 4, 1, 0.1];

Show[q3D[2, h1, h2, 2, 4, 1, 0.1], p3D[1, h1, h2, 2, 4, 1], 
        PlotRange -> {{-8, 8}, {-8, 8}, {-8, 
              8}}];

Options[ParametricPlot3D]

?PlotRange

?DefaultColor

?Background

?PlotStyle

?GrayLevel

Show[p3D[1, h1, h2, 2, 4, 1], 
        ViewPoint -> {-2.465,  -3.752,  2.377}, 
        Boxed -> False];

Generierung von Linien mit Punkten - 
Créer des lignes à l'aide de points

a 2D

Remove["Global`*"]

p={1,1};Show[Graphics[{PointSize[0.04],Point[p],Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}],AspectRatio->Automatic];

?Union

p = {5, 2} // MatrixForm

Union[{x22}, {x11, z11}]

p = {5, 2} // Union // MatrixForm

Join[{x22}, {x11, z11}]

p = {5, 2} // Join // MatrixForm

Prepend[{a, b, c}, x]

in1=Prepend[{PointSize[0.04],Point[p],Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]},PointSize[0.04]]

in1=Prepend[{PointSize[0.04],Point[p]},PointSize[0.04]]

Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}]

v[t_]:={Cos[t],Sin[3t]};
tab1=Table[v[t],{t,0,2Pi,0.3}]

pts=Map[Point,tab1]

in1=Prepend[{pts},PointSize[0.04]]

plpts=Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}]

Show[Graphics[plpts],AspectRatio->Automatic];

h=0.05;
v[t_]:={Cos[t],Sin[3t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
plpts=Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}];
Show[Graphics[plpts],AspectRatio->Automatic];

h=0.01;
v[t_]:={Cos[t],Sin[3t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
plpts=Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}];
Show[Graphics[plpts],AspectRatio->Automatic];

h=0.05;
v[t_]:={Cos[t],Sin[3t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
plpts=Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}];
Show[Graphics[plpts],AspectRatio->Automatic];

?*Dash*

Show[Graphics[{Dashing[{0.15,0.05}],Circle[{0,0},1]},AspectRatio->Automatic]];

Plot[{Sin[x],Cos[x]},{x,0,2Pi},PlotStyle->{Dashing[{0.15,0.05}],Dashing[{0.05,0.05}]}];

h=0.05;
v[t_]:={Cos[t],Sin[3t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
plpts=Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}];
Show[Graphics[plpts],AspectRatio->Automatic];

h=0.01;
v[t_]:={Cos[t],Sin[3t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
plpts=Join[in1,{Line[{{-1,0},{1,0}}],Line[{{0,-1},{0,1}}]}];
Show[Graphics[plpts],AspectRatio->Automatic];

b 3D

Remove["Global`*"]

p={1,2,4} // MatrixForm

p={-1,0.5,0.3};
Show[Graphics3D[{PointSize[0.04],Point[p],Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}],AspectRatio->Automatic];

p={-1,0.5,0.3};
Show[Graphics3D[{PointSize[0.04],Point[p],Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}],AspectRatio->Automatic,Boxed->False];

Options[Graphics]

?AxesStyle

?Union

Prepend[{a, b, c}, x]

in1=Prepend[{Point[p],Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]},PointSize[0.04]]

in1=Prepend[{Point[p]},PointSize[0.04]]

Join[in1,{Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}]

v[t_]:={Cos[t],Sin[3t],t^2/(t^2+1) Sin[2t]};
tab1=Table[v[t],{t,0,2Pi,0.3}]

pts=Map[Point,tab1]

in1=Prepend[{pts},PointSize[0.04]]

plpts=Join[in1,{Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}]

Show[Graphics3D[plpts],AspectRatio->Automatic];

Show[Graphics3D[plpts],AspectRatio->Automatic,Boxed->False];

h=0.05;
v[t_]:={Cos[t],Sin[3t],t^2/(t^2+1) Sin[2t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
plpts=Join[in1,{Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}];
Show[Graphics3D[plpts],AspectRatio->Automatic,Boxed->False];

?Hue

?#

farb1=0.3;
farb2=0.9;
h=0.01;
v[t_]:={Cos[t],Sin[3t],t^2/(t^2+1) Sin[2t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
in2=Prepend[{in1},Hue[farb1]];
plpts=Join[in2,{Hue[farb2],Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}];
Show[Graphics3D[plpts],AspectRatio->Automatic,Boxed->False];

Table[farb1=k;
farb2=0.9;
h=0.01;
v[t_]:={Cos[t],Sin[3t],t^2/(t^2+1) Sin[2t]};
tab1=Table[v[t],{t,0,2Pi,h}];
pts=Map[Point,tab1];
in1=Prepend[{pts},PointSize[0.04]];
in2=Prepend[{in1},Hue[farb1]];
plpts=Join[in2,{Hue[farb2],Line[{{-1,0,0},{1,0,0}}],Line[{{0,-1,0},{0,1,0}}],Line[{{0,0,-1},{0,0,1}}]}];
Show[Graphics3D[plpts],AspectRatio->Automatic,Boxed->False],{k,0,1,0.1}];

Anwendung - Application

Remove["Global`*"]

farb1 = 0.7;
farb2 = 0.9;
h = 0.05;  a = 2;  b = 4; c = 1; 
    vec[u_, v_, a_, b_, c_, 
        h_] := {a, b, 
          c} + {Cos[v] (Cos[5  v])/
              3, (a + (b - a)/2/Pi v) Cos[
              u] (3 + Cos[5  u])/
              3, (a + (b - a)/2/Pi v) Sin[
              u] (3 + Cos[5  u])/3};
vec[u, v, a, b, c, h]

?Flatten

farb1 = 0.7;
farb2 = 0.9;
h = 0.05;  a = 2;  b = 4; c = 1; 
    vec[u_, v_, a_, b_, c_, 
        h_] := {a, b, 
          c} + {Cos[v] (Cos[5  v])/
              3, (a + (b - a)/2/Pi v) Cos[
              u] (3 + Cos[5  u])/
              3, (a + (b - a)/2/Pi v) Sin[
              u] (3 + Cos[5  u])/3};

farb1 = 0.7;
farb2 = 0.9;
h = 0.05;  a = 2;  b = 4; c = 1; 
    vec[u_, v_, a_, b_, c_, 
        h_] := {a, b, 
          c} + {Cos[v] (Cos[5  v])/
              3, (a + (b - a)/2/Pi v) Cos[
              u] (3 + Cos[5  u])/
              3, (a + (b - a)/2/Pi v) Sin[
              u] (3 + Cos[5  u])/3}; 
    tab1 = Flatten[
        Table[N[vec[u, v, a, b, c, h]], {u, a, b, h}, {v, a, b, h}], 
        1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.04]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt1 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False];

farb1 = 0.7;
farb2 = 0.9;
h = 0.05;  a = 2;  b = 4; c = 1; 
    vec[u_, v_, a_, b_, c_, 
        h_] := {a, b, 
          c} + {Cos[v] (Cos[5  v])/
              3, (a + (b - a)/2/Pi v) Cos[
              u] (3 + Cos[5  u])/
              3, (a + (b - a)/2/Pi v) Sin[
              u] (3 + Cos[5  u])/3}; 
    tab1 = Flatten[
        Table[N[vec[u, v, a, b, c, h]], {u, a, b, h}, {v, a, b, h}], 
        1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.04]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt2 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False];

{a, b, c} + {Cos[v] (Cos[5  v])/3, 
          v Cos[u] (3 + Cos[5  u])/3, 
          v Sin[u] (3 + Cos[5  u])/3};

farb1 = 0.99;
farb2 = 0.9;
h = 0.1;  a = 2;  b = 4; c = 1; 
    vec[u_, v_, a_, b_, c_, 
        h_] := ({a, b, c} + {Cos[v] (Cos[5  v])/3, 
              v Cos[u] (3 + Cos[5  u])/3, 
              v Sin[u] (3 + Cos[5  u])/3}) 10; 
    tab1 = Flatten[
        Table[N[vec[u, v, a, b, c, h]], {u, 0, 2  Pi, h}, {v, 0, 2  Pi, h}], 
        1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.04]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt1 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False, 
          ViewPoint -> {3.050,  -3.859,  -1.272}];

farb1 = 0.4;
farb2 = 0.9;
h = 0.1;  a = 0.5;  b = 4; c = 1; d = 1; 
    vec1[u_, v_, a_, b_, c_, d_, 
        h_] := (10 (a + (b - a)/2/Pi u)/h {0, b, c} + 
            10 {0, Cos[ArcTan[c/b] + Pi], 
                Sin[ArcTan[c/b] + Pi]} (3 + 
                    Cos[5 (ArcTan[c/b] + Pi)])/
                3 + 
19 (a + (b - a)/2/Pi u)/
                h {a, 0, 
                0} + {Cos[(a + (b - a)/2/Pi u)] (Cos[
                      5 (a + (b - a)/2/Pi u)])/3, 0, 0} + 
            d {Cos[(a + (b - a)/2/Pi v)], 
                Sin[(a + (b - a)/2/Pi v)], 0}) 0.1; 
    tab1 = Flatten[
        Table[N[vec1[u, v, a, b, c, d, h]], {u, 0, 2  Pi, h}, {v, 0, 2  Pi, 
            h}], 1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.04]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt2 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False, 
          ViewPoint -> {3.050,  -3.859,  -1.272}];

Show[prnt1, prnt2];

farb1 = 0.4;
farb2 = 0.9;
h = 0.1;  a = 0.5;  b = 4; c = 1; d = 1; 
    vec1[u_, v_, a_, b_, c_, d_, 
        h_] := (10 {a, b, c} - 20 {Sin[u/4], Sin[u/4] Sin[v/10], v^2}); 
    tab1 = Flatten[
        Table[N[vec1[u, v, a, b, c, d, h]], {u, 0, 2  Pi, h}, {v, 0, 2  Pi, 
            h}], 1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.04]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt2 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False, 
          ViewPoint -> {3.050,  -3.859,  -1.272}];

Show[prnt1, prnt2];

farb1 = 0.4;
farb2 = 0.9;
h = 0.1;  a = 0.5;  b = 4; c = 1; d = 1; 
    vec1[u_, v_, a_, b_, c_, d_, 
        h_] := (10 {a, b, c} - {5  Sin[u], 5  Cos[u], 20  v} + {5  Sin[v], 
            5  Sin[v], 0}); 
    tab1 = Flatten[
        Table[N[vec1[u, v, a, b, c, d, h]], {u, 0, 2  Pi, h}, {v, 0, 2  Pi, 
            h}], 1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.1]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt2 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False, 
          ViewPoint -> {3.050,  -3.859,  -1.272}];

Show[prnt2, prnt1];

Achtung Runtime!!!  Attention Runtime!!!

farb1 = 0.99;
farb2 = 0.9;
h = 0.05;  a = 2;  b = 4; c = 1; 
    vec[u_, v_, a_, b_, c_, 
        h_] := ({a, b, c} + {Cos[v] (Cos[5  v])/3, 
              v Cos[u] (3 + Cos[5  u])/3, 
              v Sin[u] (3 + Cos[5  u])/3}) 10; 
    tab1 = Flatten[
        Table[N[vec[u, v, a, b, c, h]], {u, 0, 2  Pi, h}, {v, 0, 2  Pi, h}], 
        1];
pts = Map[Point, tab1];
in1 = Prepend[{pts}, PointSize[0.04]];
in2 = Prepend[{in1}, Hue[farb1]];
plpts = 
        Join[in2, {Hue[farb2], Line[{{-1, 0, 0}, {1, 0, 0}}], 
            Line[{{0, -1, 0}, {0, 1, 0}}], 
            Line[{{0, 0, -1}, {0, 0, 1}}]}];
prnt1 = 
        Show[Graphics3D[plpts], AspectRatio -> Automatic, 
          Boxed -> False, 
          ViewPoint -> {3.050,  -3.859,  -1.272}];

Show[prnt2, prnt1];

Beispiele von Studentenarbeiten
(kleine Projektaufgabe)

Drei Arbeiten aus dem Jahre 2005
Weitere unter http://www.hta-bi.bfh.ch/~wir/StudentenProjekteArbeiten/index.html
Blumen - Fleurs - von Studenten 04 / 05 (ZIP-Datei)

hofstetter_thomas.nb

Remove["Global`*"]

<< Graphics`SurfaceOfRevolution`

wave1 = 
        SurfaceOfRevolution[-Sin[x] +  .1, {x, 0, 
            2}, 
PlotPoints -> {20, 
              200}, 
ViewPoint -> {1.4, - .8, 
              1.3}, 
Boxed -> 
            True, 
Axes -> False]

wave2 = 
        SurfaceOfRevolution[-Sin[x] -  .4, {x, 0, 
            2.5}, 
PlotPoints -> {20, 
              200}, 
ViewPoint -> {1.4, - .8, 
              1.3}, 
Boxed -> 
            True, 
Axes -> False]

wave3 = 
        SurfaceOfRevolution[-Sin[x] -  .9, {x, 0, 
            3}, 
PlotPoints -> {20, 
              200}, 
ViewPoint -> {1.4, - .8, 
              1.3}, 
Boxed -> 
            True, 
Axes -> False]

q :=  (1.5) Exp[-theta/(8  Pi)];a := 1  +   .5 ((5/
                          4) (1  -  
                            Mod[2 theta,  2 Pi]/Pi)^3  -  1/8)^2;b := 2 x^2 (1.2 x  -  1)^2 Sin[q];c := a (x Sin[q]  +  b Cos[q]);
searose=ParametricPlot3D[{c Sin[theta],  c Cos[theta],  
              a (x Cos[q]  -  b Sin[q]), 
              EdgeForm[]},  
{x,  0,  
              1},  {theta,  - Pi,  
              10  Pi}, 
PlotPoints  ->  {20,  300},  
            ViewPoint  ->  {1.4,  - .8,  
                1.3}, 
Boxed  ->  
              True,  
Axes  ->  False]

Show[
      Graphics3D[EdgeForm[]], 
wave1, wave2, wave3, 
      searose, 
Boxed -> 
        False, 
LightSources ->  {{{1, 0, 1}, 
            RGBColor[ .6, 
              1,  .8]}}, 
ViewPoint -> {1.4, - .8, 
          1.3}, 
ImageSize -> {1000, 
          1000}]

Drei Arbeiten aus dem Jahre 2005
Weitere unter http://www.hta-bi.bfh.ch/~wir/StudentenProjekteArbeiten/index.html
Blumen - Fleurs - von Studenten 04 / 05 (ZIP-Datei)

Blumenstrauss F. Steck I1a

Einzelteile (Blatt, Blüte)

<< Graphics`Graphics3D`;
<< Graphics`Shapes`;

    
p = 
        ParametricPlot3D[{Sin[x y], y*3, x^2*y^2,  EdgeForm[]}, {x, -1, 
            1}, {y, -1, 1}, 
Axes -> 
            False, 
Boxed -> 
            False, 
PlotPoints -> 
            12,  
AmbientLight -> 
            RGBColor[1, 0, 0]
];

    
c  =  
        ParametricPlot3D[{(Cos[t] Cos[u])/4, (Cos[t] Sin[u])/4, 
            0.1 + (Sin[t]/4), EdgeForm[]}, {t,  0, 2  Pi}, {u,  0, 
            10 Pi}, 
PlotPoints -> 
            8, 
Ticks -> 
            Automatic,  
Boxed -> 
            False,  
FaceGrids -> 
            None, 
Lighting -> 
            True, 
Axes -> 
            False
];

Einzelne Blume

myFlower = 
      Show[RotateShape[p, 1/3*(Pi), 0, 0],  
        RotateShape[p, 2/3*(Pi), 0, 0],  p,  c]

Strauss

Show[RotateShape[myFlower, 2 Pi/3, 0, 0],  
      TranslateShape[myFlower, {5, -1, 0}],  
      RotateShape[TranslateShape[myFlower, {-5, 1, 0}], Pi/2, 0, 
        0]]

Drei Arbeiten aus dem Jahre 2005
Weitere unter http://www.hta-bi.bfh.ch/~wir/StudentenProjekteArbeiten/index.html
Blumen - Fleurs - von Studenten 04 / 05 (ZIP-Datei)

3D Plot - "Funktionelle Blume"  kohler_s_blume.nb

Remove["Global`*"]

Aufbau - Vase 1

h1  = ((100 - 7.1^2) (7.1^2 - 49))^(1/8)

Plot[((100 - r^2) (r^2 - 49))^(1/8), {r, 7.1, 
          9.6}];

Der Plot zweigt den Querschnitt welcher der Rand haben soll.
    h1: Höhendifferenz für das Zusammenfügen der Vase

VRx  := r Cos[φ];
VRy :=  r Sin[φ];
VRz  :=  
        17 - h1 + ((100 - r^2) (r^2 - 49))^(1/
                8);
p2  = 
        ParametricPlot3D[{VRx, VRy, 
            VRz, {EdgeForm[], SurfaceColor[RGBColor[1, 1, 1]]}}, {r, 7.1, 
            9.6}, {φ, 0, 2  Pi}, 
          PlotPoints -> 80];

Sinus / Cosinus zum Erzeugen der Kreisform mit Streckparameter r. Höhe (VRz)

Aufbau - Vase 2

Sx  :=  (4  Sin[x]) (0.4  Sin[y/3] + 
              2);
Sy  :=  (4  Cos[x]) (0.4  Sin[y/3] + 
              2);
Sz  :=   y;
p3  = 
        ParametricPlot3D[{Sx, Sy, 
            Sz, {EdgeForm[], SurfaceColor[RGBColor[1, 1, 1]]}}, {x, 0, 
            2 Pi}, {y, 0, 17}, PlotPoints  ->  80];

Sinus / Cosinus -> Kreisform, zusammen mit der Sz-Funktion ergiebt sich ein Zylinder.Der angehängte Teil bewirkt die Stauchung  und die Ausdehnung in xy-Richtung. Der Faktor vor dem Sinus beeinflusst den Amplitudenausschlag, jener im Sinus die Periodizität. Der Sinus selbst ist gut in der Aufsicht zu erkennen.

Aufbau - Erde

p5  = 
        ParametricPlot3D[{1.05  y Sin[x], 1.05  y Cos[x], 
            16 + 0.3  Sin[4  y], {EdgeForm[], 
              SurfaceColor[RGBColor[0.5, 0.3, 0]]}}, {x, 0, 2  Pi}, {y, 0, 
            2  Pi}, PlotPoints ->  80];

Einfache Kreisfunktion in xy-Ebene und eine Sinusschwankung  für Wellenform.

Aufbau - Stengel

Stx  :=  
        Sqrt[y] + (Sin[x]) (0.2 Sin[y/2] + 
                1);
Sty  :=  
        Sqrt[y] + (Cos[x]) (0.2 Sin[y/2] + 
                1);
Stz  :=  16 + y;
p1  = 
        ParametricPlot3D[{Stx, Sty, 
            Stz, {EdgeForm[], SurfaceColor[RGBColor[0, 1, 0]]}}, {x, 0, 
            2 Pi}, {y, 0, 20}, PlotPoints  ->  80];

Wurzel bewirkt Entfernung der Kreismittelpunkte von der Z-Achse. Eben erwähnte Kreise entstehen durch die Sinus und Cosinus-Funktion. Der Faktor der Winkelfunktionen 
ergibt die unterschiedlichen Radien (Sinus bewirkt die Schwankung.Faktor 0.5 Schränkt die Amplitude ein. y/2 beeinflusst die Periodizität und das Addieren von 2 garantiert positive Werte);

Aufbau - Blüte 1

aZ  :=  11;
Vx  :=  4.4;
Vy  :=  4.4;
Sxy  :=  0.8;
Et  :=  6;
p4  =  
        ParametricPlot3D[{Vx + y Cos[x] (Et + Cos[aZ x]), 
            Vy + y (Et + Cos[aZ x]) Sin[x], 
            36 +  y^7,  {EdgeForm[], SurfaceColor[RGBColor[0, 1, 0]]}}, {x, 
            0, 2 Pi}, {y, 0, 1.1}, PlotPoints  ->  80];

Eine mit Parametern modifizierte Polarfunktion in der xy-Ebene. Die Parameter verändern die Periode und Amplitude der Schwingung auf dem Kreis,ebenso die Streckung und Verschiebung in xy-Dimension.
Der nächste Plot ist gleich aufgebaut aber unterschiedlich modifiziert.

Aufbau - Blüte 2

aZ  :=  5;
Vx  :=  4.4;
Vy  :=  4.4;
Sxy  :=  1;
Et  :=  4;
p7  =  
        ParametricPlot3D[{Vx + y Cos[x] (Et + Cos[aZ x]), 
            Vy + y (Et + Cos[aZ x]) Sin[x], 
            36 +  y^7,  {EdgeForm[], SurfaceColor[RGBColor[1, 1, 0]]}}, {x, 
            0, 2 Pi}, {y, 1.1, 0}, PlotPoints  ->  80];

Aufbau - Blüte 3

FBSx  :=  (1/y) ( Sin[x]) + 
          4.4;
FBSy  :=  (1/y ) (Cos[x]) + 
          4.4;
p6  = 
        ParametricPlot3D[{FBSx, FBSy, 
            38 + 3  Sin[y], {EdgeForm[], 
              SurfaceColor[RGBColor[0.8, 0, 0]]}}, {x, 0, 2 Pi}, {y, 0.3, 
            15}, PlotPoints ->  100];

Der Kehrwert von y ist massgebend für die grosse Amplitude im Kreiszentrum. Zum äusseren Bereich hin, nimmt die Funktion Grenzwertverhalten an (-> 4,4).

Zusammenfügen der Graphikobjekte

Show[{p1, p2, p3, p4, p5, p6, p7}, AspectRatio  -> Automatic,  
        Boxed -> False, Axes -> False];

Show[{p1, p2, p3, p4, p5, p6, p7}, AspectRatio  -> Automatic,  
        ViewPoint -> {5, -10, 0}];

Show[{p1, p2, p3, p4, p5, p6, p7}, AspectRatio  -> Automatic,  
        ViewPoint -> {0, 0, 35}];