Cayleyjeva transformacija je konformna preslikava zgornje kompleksne polravnine na enotski krog:
$$w(z) = \frac{z-i}{z+i}.$$Konstruiramo lahko zvezno Cayleyjevo transformacijo:
$$w_t(z) = \frac{-\frac{(1+i) \sin \left(\sqrt{3} t\right)}{\sqrt{3}}+\frac{1}{3} z \left(3 \cos \left(\sqrt{3} t\right)-i \sqrt{3} \sin \left(\sqrt{3} t\right)\right)}{\frac{(1-i) z \sin \left(\sqrt{3} t\right)}{\sqrt{3}}+\frac{1}{3} \left(3 \cos \left(\sqrt{3} t\right)+i \sqrt{3} \sin \left(\sqrt{3} t\right)\right)}$$s parametrom \(t \in [0, \tfrac{\pi}{3\sqrt3} ]\).
w[t_, z_] = (-(((1 + I) Sin[Sqrt[3] t])/Sqrt[3]) +
1/3 z (3 Cos[Sqrt[3] t] - I Sqrt[3] Sin[Sqrt[3] t]))/
(((1 - I) z Sin[Sqrt[3] t])/Sqrt[3] + 1/3 (3 Cos[Sqrt[3] t] +
I Sqrt[3] Sin[Sqrt[3] t]));
L = 20;
Manipulate[
ParametricPlot[{Re[w[t, u + I v]], Im[w[t, u + I v]]}, {u, -L, L}, {v, 0, 2 L},
PlotRange -> 1.5, PlotPoints -> 50, Mesh -> 65,
PlotStyle -> Gray, BoundaryStyle -> Darker[Gray],
Axes -> None, FrameTicks -> None
MeshStyle -> {{CapForm["Butt"], Opacity[1], Thickness[.005], ColorData[97][4]},
{CapForm["Butt"], Opacity[1], Thickness[.005], ColorData[97][1]}}],
{t, 0, π/(3 Sqrt[3])}]
