# Unexpected behavior of GeometricTransformation

Bug introduced in 8.0 or earlier and persisting through 11.2.0 or later

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)?

• I'd have implemented 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. will be back soon Aug 17 '12 at 17:24
• Here's another way to visualize your transformation of a disk: ParametricPlot[Through[{Re, Im}[τ μ r Exp[I θ] - 1]], {r, 0, 1}, {θ, π/2, 3 π/2}]] – J. M. will be back soon Aug 17 '12 at 17:38
• @J.M. thank you very much! Unfortunately Composition produces the same erroneous result. ParametricPlot is a viable alternative. Still the problem with composite GeometricTransformations remains. – faleichik Aug 17 '12 at 20:18
• Minimal example of the problem: Evaluate Graphics@GeometricTransformation[Disk[],Composition[ScalingTransform[{800,800}],RotationTransform[0.1,{0,0}]]] and resize the graphic with the mouse. – Simon Woods Aug 17 '12 at 21:41
• You can apply the transformations beforehand to the key-coordinates of the disk, which transforms Disk[{0, 0}, 1, {α, β}] to Disk[{-1, 0}, τ Abs[μ], {α, β} + Arg[μ]]. This explicit form will work fine with Graphics. – Silvia Aug 18 '12 at 17:57

This is surely a bug. The misbehavior certainly persists through V10.2. In fact, the two images below are of the same computation. The only difference is where they appear on the screen (as I scrolled the notebook, the transformed red disk jumped around).

μ = 0.16255558520216132 + 0.1849493244071408 I;

pic[τ_] := Block[{d, ds, arc},
d = Disk[{0, 0}, 1, {π/2, 3 π/2}];
ds = GeometricTransformation[d,
TranslationTransform[{-1, 0}] .
ScalingTransform[τ Abs@# {1, 1}] .
RotationTransform[Arg@#, {0, 0}]] &@ μ;
Graphics@ {Gray, d, Red, Opacity[.5], ds}];

Manipulate[
Dynamic@Show[pic[τ], Frame -> True, Axes -> True,
AxesOrigin -> {0, 0}, PlotRange -> 100 zz ,
ImageSize -> 222],
{τ, 0.01, 1000, 0.01}, {zz, 0.01, 10, 0.01}]  As others have observed, PlotRange and ImageSize can affect what is displayed.

Note that the graphics are sent to the front end with the disk inside GeometricTransformationBox (one with my code, three nested ones with the OP's code). They appear to be correct, so the issue is with the front end (or possibly the GPU, I suppose). The workarounds by Silvia and J. M. avoid the use of GeometricTransformationBox. This sort of misbehavior has been noted since, e.g. Weird behaviour when using Scale, Rotate in Graphics.