I have the following mapping on the complex plane: $$ z \mapsto \tau \mu z-1, $$ where $\mu$ is complex, $\tau$ is real number. I want to draw the image of left unit semidisk and play with $\tau$.
μ = 0.16255558520216132` + 0.1849493244071408` I;
pic[τ_] := Block[{d, ds},
d = Disk[{0, 0}, 1, {π/2, 3 π/2}];
ds =
Fold[GeometricTransformation, d,
{RotationTransform[Arg@#, {0, 0}],
ScalingTransform[τ Abs@# {1, 1}],
TranslationTransform[{-1, 0}]
}] & @ μ;
Graphics@{Gray, d, Red, Opacity[.5], ds}
];
Manipulate[
Dynamic@Show[pic[τ], Frame -> True, Axes -> True,
AxesOrigin -> {0, 0}, PlotRange -> zz {{-100, 100}, {-100, 100}},
ImageSize -> 400
],
{τ, 0.01, 1000, 0.01},
{zz, 0.01, 10, 0.01}
]
Here zz is introduced to control zooming. The problem is that I get different (wrong) images for different zooms. It is clearly seen for big $\tau$'s.
It looks very like a bug, but I can not rule out that I'm making some silly mistake (Mathematica 8.0.4). So, what is the problem here and what is the best way of doing the job (I have several mappings for different $\mu$ actually)?
pic[]this way:pic[τ_] := Block[{d, ds}, d = Disk[{0, 0}, 1, {π/2, 3 π/2}]; ds = GeometricTransformation[d, Composition[TranslationTransform[{-1, 0}], ScalingTransform[τ Abs[μ] {1, 1}], RotationTransform[Arg[μ], {0, 0}]]]; Graphics @ {Gray, d, Red, Opacity[.5], ds}]– J. M.♦ Aug 17 '12 at 17:24ParametricPlot[Through[{Re, Im}[τ μ r Exp[I θ] - 1]], {r, 0, 1}, {θ, π/2, 3 π/2}]]– J. M.♦ Aug 17 '12 at 17:38Compositionproduces the same erroneous result.ParametricPlotis a viable alternative. Still the problem with compositeGeometricTransformations remains. – faleichik Aug 17 '12 at 20:18Graphics@GeometricTransformation[Disk[],Composition[ScalingTransform[{800,800}],RotationTransform[0.1,{0,0}]]]and resize the graphic with the mouse. – Simon Woods Aug 17 '12 at 21:41Disk[{0, 0}, 1, {α, β}]toDisk[{-1, 0}, τ Abs[μ], {α, β} + Arg[μ]]. This explicit form will work fine withGraphics. – Silvia Aug 18 '12 at 17:57