Algebraische Kurven Version 2001 Gleichungsübersicht

Mathematik mit Mathematica , Prof Dr. Dörte Haftendorn Dez 2001 
Ausfühliche Seiten zu den einzelnen Kurven

Klingelbrett:
Kegelschnitte,       Konchoiden,           Kissoiden,     Cassinische Kurven    
Kartesisches Blatt,    Rosettte,    Booth-Lemniskate,    Dreiblatt
Kappakurve,    Serpentine,     Versiera,     Kartesische Ovale   Ei-Kurven

Needs["Graphics`master`"];

Kegelschnitte    ⇑

Clear[a, b, xmin, xmax, ymin, ymax];

Clear[a, b]; ellipse = ((x^2)/(a^2)) + ((y^2)/(b^2)) == 1;

xmin = -3.5; xmax = 3.5; ymin = -2.5; 
    ymax = 2.5;
a = 3; b = 2; 
    ImplicitPlot[ellipse, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

xmin = -3.5; xmax = 3.5; ymin = -1.5; 
    ymax = 1.5;
a = 2; b = 1; 
    ImplicitPlot[ellipse, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

xmin = -2.5; xmax = 2.5; ymin = -3; 
    ymax = 3;
a = 2; b = 3; 
    ImplicitPlot[ellipse, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.03], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

Clear[a, b]; hyperbel = ((x^2)/(a^2)) - ((y^2)/(b^2)) == 1;

xmin = -3.5; xmax = 3.5; ymin = -2; 
    ymax = 2;
a = 2; b = 1; 
    ImplicitPlot[{hyperbel, a y == b x,  
        a y == -b x}, {x, xmin, xmax}, 
      PlotRange -> {ymin, ymax}, 
karopapier[xmin, xmax, 
          ymin, ymax] // Evaluate, 
      PlotStyle -> {{Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
{Thickness[0.01], 
            RGBColor[0, 1, 0]}, {Thickness[0.01], 
            RGBColor[0, 0, 1]}}, 
AspectRatio -> 
        Automatic];

Clear[a, b, p]; parabel = (y^2) == 2 p x;

xmin = 0; xmax = 4.5; ymin = -1; 
    ymax = 3;
p = 1; 
    ImplicitPlot[parabel, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {{Thickness[0.02], 
            RGBColor[1, 0, 0]}}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

xmin = 0; xmax = 4.5; ymin = -2; 
    ymax = 2;
p = (1/2); 
    ImplicitPlot[parabel, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {{Thickness[0.02], 
            RGBColor[1, 0, 0]}}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

Clear[a, b, p, ε]; 
    konics = (y^2) == 
        2 p x  -  ( 1 - (ε^2)) (x^2);

xmin = 0; xmax = 4; ymin = -2; ymax = 2;
p = (1/2); 
    ImplicitPlot[{konics /. ε -> 1.25, 
        konics /. ε -> 1, 
        konics /. ε -> 0.8}, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {{Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
{Thickness[0.02], 
            RGBColor[0, 1, 0]}, {Thickness[0.02], RGBColor[0, 0, 1]}}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

Konchoiden     ⇑

Clear[a, b, p, ε, k]; 
    hundekurven = ((x^2) + 
              (y^2)) (((y - a))^2) == (k^(2 )) 
          (y^2);

xmin = -2; xmax = 2; ymin = -0; 
    ymax = 2;
a = 1; k = 1; 
    ImplicitPlot[hundekurven, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

xmin = -2; xmax = 2; ymin = -0; 
    ymax = 2;
a = 1; k = (1/2); 
    ImplicitPlot[hundekurven, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

xmin = -4; xmax = 4; ymin = -1; 
    ymax = 3;
a = 1; k = 2; 
    ImplicitPlot[hundekurven, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

Clear[a, b, p, ε, k]; 
    pascalschnecke = ((((x^2) + (y^2) - a y))^2) == (k^(2 )) ((x^2) + (y^2));

xmin = -2;xmax = 2;ymin = -1;ymax = 2; (*Kardioide*)
a = 1;k = 1;ImplicitPlot[pascalschnecke, {x, xmin, xmax}, 
        PlotRange -> {ymin, 
            ymax}, 
PlotStyle -> {Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
        karopapier[xmin, xmax, ymin, ymax] // 
          Evaluate, 
AspectRatio -> Automatic];

xmin = -2; xmax = 2; ymin = -1; 
    ymax = 3;
a = 2; k = 1; 
    ImplicitPlot[pascalschnecke, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

xmin = -2; xmax = 2; ymin = -1; 
    ymax = 3;
a = 1; k = 1.5; 
    ImplicitPlot[pascalschnecke, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.01], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> Automatic,  
      PlotPoints -> 100];

Kissoiden     ⇑

Clear[a, b, p, ε, k, c ]; 
    kissoiden = 
      (x^(2 )) (k - 
              y)  == (y^(2 )) (k ((ε^2)  - 1) + 
              y );
 (*Speziell :   Efeukurve  und Strophoide und Trisektix*)

xmin = -2; xmax = 2; ymin = -0; ymax = 1.1; (* Efeukurve,  eigentliche Kissoide*)
a = 1; 
    k = 1; ε = 1; 
    ImplicitPlot[kissoiden, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
      AspectRatio -> Automatic];

xmin = -4; xmax = 4; ymin = -0; 
    ymax = 1; 
a = 1; k = 1; ε = 0.8; 
    ImplicitPlot[kissoiden, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.01], 
          RGBColor[1, 0, 0]}, karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
      AspectRatio -> Automatic];

xmin = -4;xmax = 4;ymin = -1;ymax = 1;       (*  Strophoide  *)
a = 1;k = 1;ε = (Sqrt[2]);ImplicitPlot[kissoiden, {x, xmin, xmax}, 
        PlotRange -> {ymin, 
            ymax}, 
PlotStyle -> {Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
        karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
        AspectRatio -> Automatic];

xmin = -4; xmax = 4; ymin = -3; ymax = 1;       (*  Trisektrix  *)
a = 1; 
    k = 1; ε = 2; 
    ImplicitPlot[kissoiden, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
      AspectRatio -> Automatic];

Cassinische Kurven    ⇑

Clear[a, b, p, ε, k, c ];  
    cassini = ((((x^2) + (y^2)))^2)   -  
          2 (k^2) ( (x^(2 )) - (y^2)) == 
        4 (a^(2 )) -  ((k )^2);
 (*Speziell :   Bernoullische Lemniskate  *)

xmin = -5; xmax = 5; ymin = -2; 
    ymax = 2;
a = 1; k = 3; 
    ImplicitPlot[cassini, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
      AspectRatio -> Automatic];

xmin = -4; xmax = 4; ymin = -2; ymax = 2; (*Bernoulilli Lemniskate *)
a = 1 ; k = 2 ; ImplicitPlot[
          cassini, {x, xmin, xmax}, 
          PlotRange -> {ymin, 
              ymax}, 
PlotStyle -> {Thickness[0.02], 
              RGBColor[1, 0, 0]}, 
          karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
          AspectRatio -> Automatic] ;

xmin = -3; xmax = 3; ymin = -1; 
    ymax = 1; 
a = 1 ; k = 1.8 ; ImplicitPlot[
        cassini, {x, xmin, xmax}, 
        PlotRange -> {ymin, 
            ymax}, 
PlotStyle -> {Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
        karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
        AspectRatio -> Automatic] ;

xmin = -3; xmax = 3; ymin = -1; 
    ymax = 1;
a = 1 ; k = 1 ; ImplicitPlot[
        cassini, {x, xmin, xmax}, 
        PlotRange -> {ymin, 
            ymax}, 
PlotStyle -> {Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
        karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
        AspectRatio -> Automatic] ;

Kartesisches Blatt     ⇑

Clear[a, b, p, ε, k, c ]; 
    karthesischesBlatt = ((x^3) + (y^3)) ==  
        3  k  x  y     ;

xmin = -2; xmax = 2; ymin = -2; 
    ymax = 2;
k = 1; 
    ImplicitPlot[karthesischesBlatt, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
      AspectRatio -> Automatic];

Rosette   ⇑

Clear[a, b, p, ε, k, c ]; 
    rosette  = ((((x^2) + (y^2)))^3) ==    (k^2)  (x^2)  (y^2)     ;

xmin = -1.7; xmax = 2; ymin = -1.7; 
    ymax = 1.7;
k = 4; 
    ImplicitPlot[rosette, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.03], 
          RGBColor[1, 0, 0]}, karopapier[xmin, xmax, ymin, ymax] // Evaluate, 
      AspectRatio -> Automatic];

Boothsche Lemniskaten          ⇑

Clear[a, b, p, ε, 
        k] ;boothPlus = ((((x^2) + (y^2)))^2) == (k^(4 )) (((x^2)/(a^2)) + 
                ((y^2)/(b^2))) ;boothMinus = ((((x^2) + (y^2)))^2) == (k^(4 )) (((x^2)/(a^2)) - ((y^2)/(b^2))) ;   (* 
      Speziell :  Bernoulische Lemniskate *)

xmin = -4; xmax = 4; ymin = -3; 
    ymax = 3;
a = 1 ; b = 4 ; k = 2 ; ImplicitPlot[
        boothPlus, {x, xmin, xmax}, 
        PlotRange -> {ymin, 
            ymax}, 
PlotStyle -> {Thickness[0.02], 
            RGBColor[1, 0, 0]}, 
        karopapier[xmin, xmax, ymin, ymax] // 
          Evaluate, 
AspectRatio -> 
          Automatic] ;

xmin = -((k^2)/a); xmax = ((k^2)/a); ymin = -((k^2)/a); 
    ymax = ((k^2)/a);
a = 1 ; b = 3 ; k = 
        2 ; ImplicitPlot[{boothPlus, (((x - ((k^2)/(2  a))))^2) + 
              (y^2) == 
            ((k^4)/(4  (a^2))), (((x + ((k^2)/(2  a))))^2) + (y^2) == 
            ((k^4)/(4  (a^2))), (x^2) + (y^2) == ((k^4)/(a^2))}, {x, xmin, 
          xmax}, PlotRange -> {ymin, 
            ymax}, 

PlotStyle -> {{Thickness[0.02], RGBColor[1, 0, 0]}, 
{Thickness[0.01], 
              RGBColor[0, 1, 0]}, {Thickness[0.01], 
              RGBColor[0, 1, 0]}, {Thickness[0.01], RGBColor[0, 0, 1]}}, 
        karopapier[xmin, xmax, ymin, ymax] // 
          Evaluate, 
AspectRatio -> 
          Automatic] ;

xmin = -((k^2)/a); xmax = ((k^2)/a); 
    ymin = -((k^2)/(2  a)); 
    ymax = ((k^2)/(2  a));
a = 1 ; b = 1 ; k = 
        2 ; ImplicitPlot[{boothMinus, (((x - ((k^2)/(2  a))))^2) + 
              (y^2) == 
            ((k^4)/(4  (a^2))), (((x + ((k^2)/(2  a))))^2) + (y^2) == 
            ((k^4)/(4  (a^2)))}, {x, xmin, xmax}, 
        PlotRange -> {ymin, 
            ymax}, 
PlotStyle -> {{Thickness[0.02], 
              RGBColor[1, 0, 0]}, 
{Thickness[0.01], 
              RGBColor[0, 1, 0]}, {Thickness[0.01], RGBColor[0, 0, 1]}}, 
        karopapier[xmin, xmax, ymin, ymax] // 
          Evaluate, 
AspectRatio -> 
          Automatic] ;

Weiteres in einer eigenen Seite über die Boothschen Lemniskaten

Dreiblatt    ⇑

xmin = -1; xmax = 2; ymin = -1; 
    ymax = 1;
c = 2; 
    ImplicitPlot[((((x^2) + (y^2)))^2) == c x ((x^2) - (y^2)), {x, 
        xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

Kappakurve    ⇑

xmin = -4; xmax = 4; ymin = -2; 
    ymax = 2;
k = 2; 
    ImplicitPlot[   (y^4) == (x^2) ((k^2) - (y^2)), {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> 
        Automatic];

Serpentine      ⇑

Clear[a, b, p, ε, k, r];   serpentine = 
      y ((a^2) + (x^2)) == 2 a r x;

xmin = -4; xmax = 4; ymin = -1; 
    ymax = 1.1;
a = 1; r = 1; 
    ImplicitPlot[serpentine, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> Automatic,  
      PlotPoints -> 60];

Maximumsuche in expliziter Form

f[x_] := ((2 a r x)/((a^2) + (x^2)));

f'[x]

Solve[f'[x] == 0, x]

f[x] /. %

Also ist im Punkt (a,r) das Maximum.

Do[r = 1; 
        ImplicitPlot[serpentine, {x, xmin, xmax}, 
          PlotRange -> {ymin, 
              ymax}, 
PlotStyle -> {Thickness[0.01], 
              RGBColor[1, 0, 0]}  , 
          Axes -> None, 
AspectRatio -> 
            Automatic,  PlotPoints -> 180], {a, 0.2, 3, 
          0.2}];

Versiera der Maria Angnesi   ⇑

Clear[a, b, p, ε, k, r];   versiera = 
      y ((a^2) + (x^2)) == (a^3) ;

xmin = -4; xmax = 4; ymin = 0; ymax = 2;
a = 2; 
    ImplicitPlot[versiera, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> Automatic,  
      PlotPoints -> 60, 
      Ticks -> {{-4, -2, 0, 2, 4}, {1, 2}}];

Kartesische Ovale    ⇑

Clear[a, b, p, ε, k, r, 
      c];   ovale =  ((((x^2) + (y^2) - (k^2)))^2) == (c - 
              x) a ;

xmin = -4; xmax = 4; ymin = -3; 
    ymax = 3;
a = 4; c = 1; k = 2; 
    ImplicitPlot[ovale, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.03], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> Automatic,  
      PlotPoints -> 60];

Ei-Kurven    ⇑

Clear[a, b, p, ε, k, 
      r];   doppelei =  ((((x^2) + (y^2)))^3) ==  
        (a^2)  (x^4) ;

xmin = -4; xmax = 4; ymin = -1; 
    ymax = 1;
a = 2; 
    ImplicitPlot[doppelei, {x, xmin, xmax}, 
      PlotRange -> {ymin, 
          ymax}, 
PlotStyle -> {Thickness[0.02], 
          RGBColor[1, 0, 0]}, 
      karopapier[xmin, xmax, ymin, ymax] // 
        Evaluate, 
AspectRatio -> Automatic,  
      PlotPoints -> 60];

allgemeine Definitionen

Farbig = 
        Table[{Thickness[0.01], farbig[[k]]}, {k, 1, 
            Length[farbig]}];

karopapier[xmin_, xmax_, ymin_, ymax_] := 
        GridLines -> {Table[i, {i, xmin, xmax, 0.5}], 
            Table[i, {i, ymin, ymax, 0.5}]};