Dieses Notebook soll zeigen, wie man komplexe Funktionen anhand ihrer Operationen auf frei wählbaren Gebilden darstellen kann. Insbesondere wird gezeigt, wie die Hooke- und Newton-Kegelschnitte aus einfachen Kreisen und Geraden erzeugt werden können mithilfe der Zhukovskii- und der Bohlin-Abbildung

 (vgl. V.I. Arnol'd, "Huygens & Barrow, Newton & Hooke", Birkhäuser 1990, Appendix 1 )

Zshu[z_] := z + (1/z)

Squ[z_] := (z^2)

Boh[z_] := Composition[Squ, Zshu][z]

Tr2[z_] := z + 2

Es gilt die folgende Identität aufgrund der Definitionen :
			
			Squ   ˚    Zshu    =    Boh    =    Tr2   ˚    Zshu   ˚   Squ

MyCircle[t_, r_] := r Cos[t] + I r Sin[t]

MyLine[t_, a_] := t a

MyFunc[t_] := Composition[Zshu, MyCircle][t, 2]

ParametricPlot[{Composition[Re, MyFunc][t], 
        Composition[Im, MyFunc][t]}, {t, 0, 2 π}]

MyFunc2[t_] := Composition[Zshu, MyLine][t, 1.2 + I]

ParametricPlot[{Composition[Re, MyFunc2][t], 
        Composition[Im, MyFunc2][t]}, {t, 0.04, 5}, 
      AspectRatio -> 1, PlotRange -> {{-2, 6}, {-4, 4}}]

Show[{%8, %10}, PlotRange -> {{-4, 8}, {-5, 5}}, 
      AspectRatio -> (10/12)]

a := 2 + 5 ISpiral[z_] := z a

MyFunc3[t_] := Composition[Spiral, Zshu, MyCircle][t, 2]

ParametricPlot[{Composition[Re, MyFunc3][t], 
        Composition[Im, MyFunc3][t]}, {t, 0, 2 π}, 
      PlotStyle -> {{RGBColor[1, 0, 0]}}, AspectRatio -> 1.5, 
      PlotRange -> {{-10, 10}, {-15, 15}}, 
      Epilog -> {RGBColor[0, 0, 1], 
          Line[{{Re[-3 a], Im[-3 a]}, {Re[3 a], Im[3 a]}}], 
          Line[{{-Im[2 a], Re[2 a]}, {Im[2 a], -Re[2 a]}}], 
          PointSize[0.025], RGBColor[1, 0, 0], 
          Point[{Re[-2 a], Im[-2 a]}], Point[{Re[2 a], Im[2 a]}]}]

MyFunc4[t_] := Composition[Squ, Spiral, Zshu, MyCircle][t, 1.4]

MyFunc5[t_] := 
      Composition[Spiral, Spiral, Squ, Zshu, MyCircle][t, 2]

ParametricPlot[{Composition[Re, MyFunc4][t], 
        Composition[Im, MyFunc4][t]}, {t, 0, 2 π}, 
      AspectRatio -> 1.0, 
      Epilog -> {RGBColor[0, 0, 1], 
          Line[{{Re[6 (a^2)], Im[6 (a^2)]}, {Re[-2 (a^2)], 
                Im[-2 (a^2)]}}], 
          Line[{{Re[(2 + 2 I) (a^2)], Im[(2 + 2 I) (a^2)]}, {
                Re[(2 - 2 I) (a^2)], Im[(2 - 2 I) (a^2)]}}], 
          PointSize[0.025], RGBColor[1, 0, 0], 
          Point[{Re[4 (a^2)], Im[4 (a^2)]}], Point[{0, 0}]}]

Newton[a_, r_] := 
      Block[{}, 
        ParametricPlot[{Re[Squ[a Zshu[MyCircle[t, r]]]], 
            Im[Squ[a Zshu[MyCircle[t, r]]]]}, {t, 0, 2 π}, 
          AspectRatio -> Automatic, 
          Epilog -> {RGBColor[0, 0, 1], 
              Line[{{Re[6 (a^2)], Im[6 (a^2)]}, {Re[-2 (a^2)], 
                    Im[-2 (a^2)]}}], 
              Line[{{Re[(2 + 4 I) (a^2)], Im[(2 + 4 I) (a^2)]}, {
                    Re[(2 - 4 I) (a^2)], Im[(2 - 4 I) (a^2)]}}], 
              PointSize[0.025], RGBColor[1, 0, 0], 
              Point[{Re[4 (a^2)], Im[4 (a^2)]}], Point[{0, 0}]}]]

Newton[2 + 5 I, 1.4]

Newton[2 + 0.5 I, 1.8]

Hooke[a_, r_] := 
      Block[{}, 
        ParametricPlot[{Re[a Zshu[MyCircle[t, r]]], 
            Im[a Zshu[MyCircle[t, r]]]}, {t, 0, 2 π}, 
          AspectRatio -> Automatic, 
          Epilog -> {RGBColor[0, 0, 1], 
              Line[{{Re[4 a], Im[4 a]}, {Re[-4 a], Im[-4 a]}}], 
              Line[{{-Im[4 a], Re[4 a]}, {+Im[4 a], 
                    -Re[4 a]}}], PointSize[0.025], RGBColor[1, 0, 0], 
              Point[{Re[2 a], Im[2 a]}], 
              Point[{-Re[2 a], -Im[2 a]}]}]]

Hooke[2 + 3 I, 1.8]