# How to plot a complicated Region?

This works:

n = Infinity;
Region[ParametricRegion[{1 - (b1 (1 + b2))/(1 + b1^2 + b2^2),
1 - 2 ArcSin[
1/2 Sqrt[(1 + b1^2 + (-1 + b2) b2)/(
1 + b1^2 + b2^2 - b1 (1 + b2))]]/π}, {{b1, -n, n}, {b2, -n,
n}}], Frame -> True, AspectRatio -> 1/GoldenRatio]


but this doesn't:

n = Infinity;
Region[ParametricRegion[{1 - a1 + ((-1 + a1^2) b1)/(
1 + 2 a1 b1 + b1^2),
1 - 2 ArcSin[
1/2 Sqrt[((1 + a1) (1 + b1 (a1 + b1)))/(
1 + b1 (-1 + a1 + b1))]]/π}, {{a1, -1, 1}, {b1, -n, n}}],
Frame -> True, AspectRatio -> 1/GoldenRatio]


I.e., it runs for too long for me to hope it will give any output. It works for n=1, though. But theory says that $$-\infty, and the functions I'm trying to plot are well defined for all $$b_1\in\mathbb{R}$$. However, the first component is restricted to $$[0,1]$$, thus the contraints:

Reduce[0 < 1 - a1 + ((-1 + a1^2) b1)/(1 + 2 a1 b1 + b1^2) < 1 && -1 < a1 < 1, {a1, b1}]


(-1 < a1 < 0 && -a1 < b1 < -(1/a1)) || (a1 == 0 && b1 > 0) || (0 < a1 < 1 && (b1 < -(1/a1) || b1 > -a1))

which look like this:

On other hand,

RegionBounds@
ParametricRegion[{1 - a1 + ((-1 + a1^2) b1)/(1 + 2 a1 b1 + b1^2),
1 - 2 ArcSin[
1/2 Sqrt[((1 + a1) (1 + b1 (a1 + b1)))/(
1 + b1 (-1 + a1 + b1))]]/π}, {{a1, -1, 1}, {b1, -Infinity,
Infinity}}]


gives

{{0, 2}, {1/3, 1}}

What can be done to have it plotted in MMA v11.3?

We can split the domain into multiple finite pieces:

opts = {Mesh -> None, PlotStyle -> {Opacity[1], Hue[0.6, 0.3, 0.95]},
BoundaryStyle -> None, PlotPoints -> 50};

f[a1_, b1_] := {1 - a1 + ((-1 + a1^2) b1)/(1 + 2 a1 b1 + b1^2),
1 - 2 ArcSin[1/2 Sqrt[((1 + a1) (1 + b1 (a1 + b1)))/(1 + b1 (-1 + a1 + b1))]]/π}

Show[
ParametricPlot[f[a1, b1], {a1, -1, 1}, {b1, -1, 1}, Evaluate@opts],
ParametricPlot[f[a1, 1/b1], {a1, -1, 1}, {b1, 0, 1}, Evaluate@opts],
ParametricPlot[f[a1, 1/b1], {a1, -1, 1}, {b1, -1, 0}, Evaluate@opts],
AspectRatio -> 1/GoldenRatio,
PlotRange -> All
] // Quiet