# Plotting image a set under perspective transformation

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.

## 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] • That is great, thanks so much!
– TDH
Mar 17 at 8:00
• Quick question @cvgmt : is there a way to show the boundary along with the region?
– TDH
Mar 17 at 8:22
• Great, thank you very much!
– TDH
Mar 17 at 8:42