Plotting image a set under perspective transformation

Question

I want to plot the following set in Mathematica:

$$\{ (x_1, x_2)/x_3 \mid (x_1-3)^2 + (x_2-3)^2 + (x_3 - 3)^2 \leq 1\}.$$

In other words, this set is the output of applying the perspective function $f(x_1, x_2,x_3) = (x_1,x_2)/x_3$ to each point of the unit sphere centered at $(3,3,3)$ (note that on this sphere $x_3 >0$ so division by zero is not an issue). Is there a way of plot this set in Mathematica? Any help will be much appreciated.

Answer 1

Method-1

ParametricRegion[{{x1, x2}/x3, (x1 - 3)^2 + (x2 - 3)^2 + (x3 - 3)^2 <=
     1}, {x1, x2, x3}] // Region

RegionPlot[
 DiscretizeRegion[
  ParametricRegion[{{x1, x2}/
     x3, (x1 - 3)^2 + (x2 - 3)^2 + (x3 - 3)^2 <= 1}, {x1, x2, x3}], 
  AccuracyGoal -> 3], BoundaryStyle -> Red]

Method-2

plot = RegionPlot3D[
   ImplicitRegion[(x1 - 3)^2 + (x2 - 3)^2 + (x3 - 3)^2 <= 1, {x1, x2, 
     x3}], PlotRange -> All, PlotPoints -> 80, 
   ViewProjection -> "Orthographic"];
Graphics[plot[[1, 1]] /. {x_Real, y_Real, z_Real} -> {x, y}/z]