I read the article Characterizations of some domains about Carathéodory Extremals.

To let $ Omega $$ subset mathbb {C} ^ n $ be an openly connected sentence (domain). To let $ T ( Omega) $ denote the tangent bundle of $ Omega $, To let $ mathbb {D} $ be the open unit disk in the complex plane $ mathbb {C} $ , To let $ f: Omega longrightarrow mathbb {D} $ is to be called holomorphic with $ f in H ( Omega, mathbb {D}) $,

The Carathéodory Reiffen Pseudometric is for every point $ (z; xi) $ in the $ T ( Omega) $ is defined as

$ c_ Omega (z, xi) = sup {| f ^ * (z) xi |; f: Omega longrightarrow mathbb {D}, ; f in H ( Omega, mathbb {D}) }

= sup {| sum_ {i = 1} ^ {n} { frac {{ partial f}} {{ partial z_i}} (z) xi_i} |; f: Omega longrightarrow mathbb {D}, ; f in H ( Omega, mathbb {D}) } $

A function $ f_0 in H ( Omega, mathbb {D}) $ is said to be a universal caratheodory extremal function if $ | f_0 ^ * (z) xi | = c_ Omega (z, xi) $ for each $ (z; xi) $ in the $ T ( Omega) $,

I can find the universal set in the case of $ Omega = mathbb {D} $,

However, in the case of the Polydisc and the Euclidean sphere unit, the author of the above article (Example 2.2, i and ii) claimed / mentioned that they are the coordinate functions and compositions of projections onto the planes through the center Unity Euclidean sphere. Can anyone tell how these were calculated or provide a reference for them?