Plotting the image of a region in $\mathbb{R}^3$ defined by an inequality under a map

Question

I have two functions:

• $f: \mathbb{R}^3 \to \mathbb{R}^3$
• $g: \mathbb{R}^3 \to \mathbb{R}$

I want to plot the image of $\{ (x,y,z) | g(x,y,z)>0 \}$ under $f$. Is there a nice way to do this? The set $g(x,y,z)>0$ is nice, it is topologically a ball but I don't think that there is any nice way to parameterize the surface $g=0$.

EDIT: There are a few different f,g that I am interested in. All are fairly complicated but generally f is the gradient of some scalar function and I want to plot the image of the set where the Hessian of the function is positive semi-definite. (It is actually slightly more complicated than that, and it really involves a function from ${\mathbb R}^4$ to ${\mathbb R}^4$ but I only really care about a particular projection into ${\mathbb R}^3$. One example would be

$g(x,y,z) = \cos (x) \cos (x-y) \cos (x-z)+\cos (x) \cos (y) \cos (x-z)+\cos (y) \cos (x-y) \cos (x-z)+\cos (x) \cos (x-z) \cos (y-z)+\cos (y) \cos (x-z) \cos (y-z)+\cos (z) \cos (x-y) \cos (x-z)+\cos (y) \cos (z) \cos (x-z)+\cos (z) \cos (x-z) \cos (y-z)+\cos (x) \cos (x-y) \cos (y-z)+\cos (x) \cos (y) \cos (y-z)+\cos (y) \cos (x-y) \cos (y-z)+\cos (x) \cos (z) \cos (x-y)+\cos (x) \cos (y) \cos (z)+\cos (y) \cos (z) \cos (x-y)+\cos (x) \cos (z) \cos (y-z)+\cos (z) \cos (x-y) \cos (y-z)$

$f_1(x,y,z) = \frac{-2 \sin (x-y)-\sin (x-z)-\sin (x)+\sin (y-z)+\sin (y)}{\sqrt{2}}$

$f_2(x,y,z) = -\frac{3 \sin (x-z)+\sin (x)+3 \sin (y-z)+\sin (y)-2 \sin (z)}{\sqrt{6}}$

$f_3(x,y,z) = -\frac{2 (\sin (x)+\sin (y)+\sin (z))}{\sqrt{3}}$

I'd be interested in the image of the region containing the origin where g is positive. There is a second piece of the set $g>0$ that does not contain the origin but I am not interested in that piece.

Answer 1

Perhaps ContourPlot3D and RegionFunction would be useful, for example:

Show[ContourPlot3D[x  y z, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}, 
  Contours -> 10, Mesh -> None, ContourStyle -> Opacity[0.6], 
  PlotLegends -> Automatic, 
  RegionFunction -> Function[{x, y, z}, x^2 + y^2 + z^2 > 1]], 
 Graphics3D[Sphere[]]]

Answer 2

Here is a very awkward approach, but one that works.

First get a set of points that lie in $g$:

inGpoints = Select[Flatten[Table[If[x^2 + 2 y^2 + 3 z^2 < 1, {x, y, z}], 
     {x, -2, 2, .1}, {y, -2, 2, .1}, {z, -2, 2, .1}], 2], 
   Length[#] == 3 &];

Now define your $f$ function:

f[q_List] := {2 q[[1]]^2 + q[[2]], 3 q[[3]]^2 - q[[1]], q[[3]] + q[[2]]};

Now apply $f$ to the points that satisfy your $g$ constraint:

finalpoints = f /@ inGpoints;

Now plot the outer boundary of these final points:

ConvexHullMesh[finalpoints]