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
gif
conformal mapping sphere ↔ horn torus
