If $$\left(x_{0},y_{0}\right)\in\left(0,\infty\right)\times\left(0,\infty\right)$$, then it easy to get $$\left(u_{0},v_{0}\right)\in\left(0,\infty\right)\times\left(0,1\right)$$ though the parametric equations $$u=x+y\quad\textrm{and}\quad v=\frac{x}{x+y}.$$

So I tried running the below code:

Region[ParametricRegion[{{x + y, x/(x + y)}, 0 < x && 0 < y}, {x, y}],Axes -> True, AxesOrigin -> {0, 0}]


but I got the below region:

Why the region does not coincide with $$\left\{\left(u,v\right):0? Could you please help me diagnose this issue?

• Since we can only plot a mapping from a finite region to another finite region. And by default, we map a finite rectangle to another region which the boundary may not be a rectangle.
ani = Manipulate[
GraphicsRow[{ParametricPlot[{x, y}, {x, 0, c}, {y, 0, c},
Mesh -> 10, PlotStyle -> Yellow, MeshStyle -> Cyan,
BoundaryStyle -> Red, PlotRange -> Automatic],
ParametricPlot[{x + y, x/(x + y)}, {x, 0, c}, {y, 0, c},
Mesh -> 10, PlotStyle -> Yellow, MeshStyle -> Cyan,
BoundaryStyle -> Red, PlotRange -> {{0, 6}, {0, 1}},
PerformanceGoal -> "Quality"]}, ImageSize -> Full], {c, 1, 8}]


• If we want the region be a rectangle,we need to limit the range domain be a triangle. ImplicitRegion[{x > 0, y > 0, x + y <= c}, {x, y}]
c = 10;
GraphicsRow[{ParametricPlot[{x, y}, {x, y} ∈
ImplicitRegion[{x > 0, y > 0, x + y <= c}, {x, y}], Mesh -> 10,
MeshFunctions -> {#3 + #4 &}, PlotStyle -> Yellow,
MeshStyle -> Cyan, BoundaryStyle -> Red, PlotRange -> Automatic],
ParametricPlot[{x + y, x/(x + y)}, {x, y} ∈
ImplicitRegion[{x > 0, y > 0, x + y <= c}, {x, y}], Mesh -> 10,
MeshFunctions -> {#3 + #4 &}, PlotStyle -> Yellow,
MeshStyle -> Cyan, BoundaryStyle -> Red, PlotRange -> All]},
ImageSize -> Full]


Appendix

FunctionBijective[{{x + y, x/(x + y)}, x > 0 && y > 0,
u > 0 && 0 < v < 1}, {x, y}, {u, v}]


True

FunctionRange[{{x + y, x/(x + y)}, x > 0, y > 0, x + y > 0}, {x,
y}, {u, v}, Reals] // Simplify


u > 0 && v > 0 && v < 1

• Thanks for your help.Yes,indeed,this transformation $\psi :\mathbb{R^2}\rightarrow\mathbb{R^2}$ $$\psi(x,y)=(x+y,\frac{x}{x+y})$$ is one-to-one on a unit square $\mathcal{R}=\{(x,y):0<x<1,0<y<1\}$ and maps the region $\mathcal{R}$ in the $xy$-plane onto the non-retangle $\mathcal{R^*}=\{(u,v):0<uv<1,0<u(1-v)<1\}$ in the $uv$-plane. Jan 29 at 7:36