Bloch ball for a given transformation: Applying transformations to regions

How can we apply transformations to Regions in the Wolfram language?

For example, given the transformation $$r_x \rightarrow r_x\sqrt{1-g}, r_y \rightarrow r_y \sqrt{1-g}, r_z \rightarrow g + r_z (1-g)$$, how can one apply such transformation to a unit sphere regions representing Bloch sphere/ball ?

r = Region[Sphere[]]

Edit: What I originally meant by "show" is to take a unit sphere and then apply this transformation and see how it changes its shape/position (given that it may shrink and at the same time its center can be displaced).

• I am little bit confused. You want a graphic representation of the Bloch same as given in Wikipedia page? Sep 9 '21 at 14:27
• It is unclear what you are asking, or if it is even related to Mathematica. "how can one show this" <- What do you mean by "this"? What do you mean by "show? (Prove?) Please rewrite the question to prevent it from getting closed. Sep 9 '21 at 14:29
• The question was clear enough for me to give an acceptable answer. I have edited the question to clarify a bit and suggest we re-open it now. Sep 10 '21 at 10:42

Region[Sphere[]] tr = With[{g=0.7},
Composition[
TranslationTransform[{0, 0, g}],
ScalingTransform[{Sqrt[1-g],Sqrt[1-g],(1-g)}]
]
]; Region[
TransformedRegion[Sphere[],tr]
,PlotTheme->"Detailed"
,PlotRange->{{-1,1},{-1,1},{-1,1}}
] • Thanks, @rhermans. How can one use some mesh function so that the curvature is visible clearly? Sep 9 '21 at 17:36
• For that please ask a new question. Sep 9 '21 at 17:37
• It would also be nice if we could put axes with the origin at x=y=z=0. Sep 9 '21 at 17:37
• @Zubin No, while clarification and improvements in questions are desirable, here we normally consider it a bad idea to move the goalpost and add new requests to old questions. You need to ask a new question. Sep 10 '21 at 10:43
• Okay, just asked a separate question here: mathematica.stackexchange.com/questions/255499/… Sep 10 '21 at 18:08