Plot 3D image of function

Question

I'm (very!) new to Mathematica, and trying to use it plot a set in $\mathbb{R}^3$. In particular, I want to plot the set

$$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{u^2+v^2+w^2}(vw,uw,uv) \text{ for some } (u,v,w) \in \mathbb{R}^3\setminus\{0\} \big\}. $$ This is just the image of function from $\mathbb{R}^3\setminus\{0\} \to \mathbb{R}^3$. I haven't been able to make any of the plot functions display this. Any advice would be greatly appreciated.

If it makes it easier, the set above is also $$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{u^2+v^2+w^2}(vw,uw,uv) \text{ for some } (u,v,w) \in \mathbb{S}^2 \big\}. $$ Alternatively, one could plot the three sets $$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{1+v^2+w^2}(vw,w,v) \text{ for some } (v,w) \in \mathbb{R}^2 \big\}, $$ $$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{1 +u^2+w^2}(w,uw,u) \text{ for some } (u,w) \in \mathbb{R}^2 \big\}, $$ $$ \big\{ (x,y,z) : (x,y,z) = \frac{1}{1+u^2+v^2}(v,u,uv) \text{ for some } (u,v) \in \mathbb{R}^2 \big\}. $$

Answer 1

Here are a couple more ways.

1. Parametrize $\mathbb S^2$ with latitude and longitude:

   With[{u = Cos[lat] Cos[lon], v = Cos[lat] Sin[lon], w = Sin[lat]}, 
    ParametricPlot3D[{v w, u w, u v}/(u^2 + v^2 + w^2), {lat, -Pi/2, Pi/2}, {lon, -Pi, Pi}]]
   

   

2. Parametrize $\mathbb R^2$ compactly using the $\tan$ function:

   With[{u = Tan[a], v = Tan[b]}, 
    ParametricPlot3D[{v, u, u v}/(1 + u^2 + v^2), {a, -Pi/2, Pi/2}, {b, -Pi/2, Pi/2}]]
   

   

Answer 2

With[{eqn = Eliminate[{x, y, z} == 1/(u^2 + v^2 + w^2) {v w, u w, u v}, {u, v, w}]},
 ContourPlot3D[eqn, {x, -0.6, 0.6}, {y, -0.6, 0.6}, {z, -0.6, 0.6}]
 ]

Answer 3

We can choose a sample and and visualize it. First we create a grid of values in the 3D-room:

sample = Select[
   Tuples[Range[-10, 10, 0.8], 3],
   Norm[#] > 0.5 &];

Select is used to remove points that are too close to the singularity. Now the plotting is easy using ListPointPlot3D:

f[{u_, v_, w_}] := {v w, u w, u v}/Norm[{u, v, w}]^2
ListPointPlot3D[f /@ sample]