Stereographic projection combined with rotation is a Möbius transformation

So for my Complex Analysis class, I need to prove the following question:

Consider the function that maps a point from $$\mathbb{C} \cup \{\infty\}$$ to the sphere via inverse stereographic projection, applies a rotation to the sphere and then maps it back to the complex plane by inverse stereographic projection. Show that this function is a Möbius transformation.

Now, we use the complex analysis book of Stein and Shakarchi, where this sphere has center $$(0,0,1/2)$$ and radius $$1/2$$. The stereographic projection is defined as

$$x = \frac{X}{1-Z}$$ and $$y = \frac{Y}{1-Z}$$

and its inverse as

$$X = \frac{x}{x^2+y^2+1}$$, $$Y = \frac{y}{x^2+y^2+1}$$ and $$Z = \frac{x^2+y^2}{x^2+y^2+1}$$.

I managed to prove that this is indeed a Möbius transformation when the $$Z$$-axis is the rotation axis but I have no clue how I could prove this for every rotation of the sphere.

• Are you mapping it back to the plane or the plane union infinity? The latter can be seen as a special case of "isometries of the Riemann Sphere are mobius transforms" which is easily proved by noting mobius transforms are conformal; its easy to show any injective conformal maps on the (open) unit disk are mobius transforms, e.g. by Schwartz, and then compose on both sides to get all conformal maps. The former case (mapping back to the plane minus infinity) is a little more nuanced – Brevan Ellefsen May 2 at 17:31
• Alternatively, if you want a direct proof in this case, can you think of a function $f$ that rotates the sphere? There are a few ways to do this; you could do it directly on the sphere in Euclidian coordinates, use spherical coordinates, or use polar coordinates on the plane and consider how the projection works. – Brevan Ellefsen May 2 at 17:43
• We map back to the plane union infinity – Mee98 May 2 at 17:46