Why can't an annulus be mapped as a rectangle

Fiddling around...not an answer, other than that some obvious attempts don't work.

Try two half annuluses (annuli?)

S7 = ImplicitRegion[(1/10 <= x^2 + y^2 <= 1 && (x >= 0 )), {x, y}];
S8 = ImplicitRegion[(1/10 <= x^2 + y^2 <= 1 && (x <= 0 )), {x, y}];

GraphicsGrid[{{RegionPlot[S8 ], RegionPlot[S7]}}]

Use ToPolarCoordinates[{x, y}] to do the transformation. Now here is a weird one...I have to run ToPolarCoordinates[{x, y}] at least once outside of the transformation call or it fails. Very odd.

ToPolarCoordinates[{x, y}]
r7 = TransformedRegion[S7, Evaluate@ToPolarCoordinates];
r8 = TransformedRegion[S8, Evaluate@ToPolarCoordinates];

The r7 region plots exactly right. The r8 region plots wrong and jagged.

rp7 = RegionPlot[r7, PlotRange -> {{-0.2, 1.55}, 3/2 {-π, π}},
      PerformanceGoal -> "Quality", 
      GridLines -> {Automatic, π (Range[9] - 5)/4}, 
      FrameTicks -> {{π (Range[9] - 5)/4, None}, Automatic}
      ];

rp8 = RegionPlot[r8, PlotRange -> {{-0.2, 1.55}, 3/2 {-π, π}},
      PerformanceGoal -> "Quality", 
      GridLines -> {Automatic, π (Range[9] - 5)/4}, 
      FrameTicks -> {{π (Range[9] - 5)/4, None}, Automatic}
      ];

GraphicsGrid[{{rp7, rp8}}]

I could only get the r7 region to come out right using ToPolarCoordinates. It then gets the range of the angles right too. Didn't work with the hand-rolled transformation as in this question. The r8 region is wrong in the jagged edges and the angle range.

I tried this, but it crashes the kernel when you try to RegionPlot it.

S9 = ImplicitRegion[(1/10 <= x^2 + y^2 <= 1 && (x >= 0 || x <= 0)), {x, y}];

EDIT

Some more experimenting, things appear to work fine until you get x and y values in the lower left quadrant, both negative values. For example, this works fine.

S9 = ImplicitRegion[(1/10 <= x^2 + y^2 <= 1 && ((x >= 0) || (y >= 0))), {x, y}];
RegionPlot[S9]

ToPolarCoordinates[{x, y}]
r9 = TransformedRegion[S9, Evaluate@ToPolarCoordinates];  

rp9 = RegionPlot[r9, PlotRange -> {{-0.2, 1.55}, 3/2 {-π, π}},
                PerformanceGoal -> "Quality", 
                GridLines -> {Automatic, π (Range[9] - 5)/4}, 
                FrameTicks -> {{π (Range[9] - 5)/4, None},Automatic}
                ]

One possible workaround is to add a thin gap on the annulus:

r = TransformedRegion[
  Annulus[{0, 0}, {Sqrt[1/10], 1}, {-Pi + 0.001, Pi}],
                  {(#1^2 + #2^2)^0.5, ArcTan[#, #2]} &]; 

RegionPlot[r, AspectRatio -> Automatic]