Implicit Equality and RegionPlot

Question

I have the following problem:

I would like to do a RegionPlot like

RegionPlot[F[x,z,y]<1,{x,0,1},{y,0,1}]

The problem is that the variable z is an implicit (and not reducable via e.g. Solve, meaning too non-linear) but continuous function of x and y, for example

G[x,y,z]==0.

The functions F and G are well-defined. Would anyone know how to solve this issue in a general way?

Answer 1

You can define the implicit solution for z

bigG[x_?NumberQ, y_?NumberQ] :=  FindRoot[Sin[x + z] + Cos[y + z] == 0, {z, \[Pi]}][[1, 2]]

and then use it in the your F :

bigF[x_?NumberQ, y_?NumberQ] = x^2 + y^2 + 2 x y bigG[x, y]

RegionPlot[bigF[x, y] < 1, {x, 0, 1}, {y, 0, 1}]

Answer 2

For surface $G(x,y,z)=0$ which is bounded in $z$ direction on $(x,y)\in[0,1]\times[0,1]$, by specifying large enough $z$-interval, ContourPlot3D with the option RegionFunction can be used to generate the required region graphics.

Take $G(x,y,z)=x^3+y^3+z^3-20$ and $F(x,y,z)=30-(x-2)^4-(y-0.5)^4-0.8 (z-0.5)^4$ as an example.

graph = ContourPlot3D[
 x^3 + y^3 + z^3 - 20 == 0,
 {x, -3, 3}, {y, -3, 3}, {z, -5, 5},
 RegionFunction -> Function[{x, y, z},
   30 - (x - 2)^4 - (y - .5)^4 - .8 (z - .5)^4 < 1],
 BoundaryStyle -> None, Mesh -> None,
 PlotPoints -> 50]

Then convert the 3D graphics to 2D one:

Graphics[{##}[[1]],
   PlotRange -> All, Frame -> True, Axes -> False,
   Sequence @@ Rest[{##}]] & @@
 DeleteCases[graph /.
    GraphicsComplex[pts_, others__] :>
     GraphicsComplex[pts[[All, 1 ;; 2]], others] /.
   Polygon[pts_] :>
    Sequence[FaceForm[{Lighter[Blue, .8]}], EdgeForm[{Lighter[Blue, .8]}], Polygon[pts]],
  _?(MatchQ[#,
      (VertexNormals -> _) | (BoxRatios -> _) | (PlotRangePadding -> _) | (PlotRange -> _)
      ] &), \[Infinity]]

Note: for an unbounded $G(x,y,z)$, it will be much harder to determine whether a point $(x,y)$ satisfies the region condition, and I suppose, as Artes said, impossible to find a general way.