conformal mapping from complex plane → Riemann sphere → horn torus, Z → P' → P or |z| → α → φ,
      when |z| and α are zero at south pole 0  and angle φ is measured as shown in the sketch on this page,
      the simpler and more consistent version (in the first considerations I had chosen α = 0 at north pole N)
       
      stereographic projection from complex plane to Riemann sphere, |z| → α:
       
       α = 2·arctan(|z|)
       
      mapping from Riemann sphere to horn torus, α → φ  (C any real number*):
       
       φ = 2·arccot(-ln(|tan(α/2)|) - C)
       
      mapping from complex plane to horn torus, |z| → φ:
       
       φ = 2·arccot(-ln(|z|) - C)
       
      inverse mappings P → P' → Z, equivalent φ → α → |z|:
       
       α = 2·arctan(e^(-cot(φ/2) + C))
       |z| = tan(α/2)
       |z| = e^(-cot(φ/2) + C)

 ⇐   preceding page

      derivation of formulas (analytically by the method of infinitesimal circles)  ← 
      excel chart for numerical computation of angles and |z|  (15 decimal places)
      a graphical method (stereographic projection) Z → P  probably does not exist
      Vyacheslav Puha spotlights that e lies on 'top',  1/e on 'bottom' of horn torus
      but note: that's only valid for C = 0, one of infinite many conformal mappings
      it's not sure whether conformality has any relevance in the horn torus model
      example for a mapping that preserves right angles between coordinates only

        *)  C seems to be a kind of 'zoom factor' for the mappings, diminishing
         projected figures and, for C > 0, moving α and φ towards π resp. 2π,
         for C < 0 towards 0 both.  Likewise, but then for C > 1 resp. 0 < C < 1
         (C = 1 for the stereographic projection), there exists the well known
         generalised conformal mapping between complex plane and sphere
        α = 2·arctan(C·|z|)  resp.  |z| = tan(α/2) / C

 
 


Riemann sphere:  0N = 1 ,  ∠0NP' = α/2 ,  ∠0ZN = π/2 - α/2 ,  |z| = 0Z
 
 
note: conformal mapping can be applied to any sphere and any horn torus!
solutions are independent from radii,  see numerical check  (Excel-file .xlsx)