conformal mapping of point P' on a sphere to point P on a horn torus, equivalent to
α → φ, when choosing α = 0 at the north pole (later
we change to version with α = 0 at south pole,
which is symmetric, but more logical and consistent, because z is zero here simultaneous)
φ = 2·arccot(ln(tan(α/2))  C)
we observe a small circle around P' on sphere with radius dα (α is latitude, 0 at north pole)
the radius perpendicular to screen in P' with same length is dω = dα / sin(α) (ω is longitude)
the projected small circle around P on horn torus has same radius dω (as angle, not length)
and for corresponding perpendicular radius dφ we respectively have: dω = dφ / (1  cos(φ))
ω and dω, as angles, are identical on both figures, therefore: dα / sin(α) =
dφ / (1  cos(φ)) by integration
and solving for φ we get above result
values for α: 0 to π, resulting values for φ: 0 to 2·π
approximations to both limits are highly asymptotic
and absolute symmetrical from both sides of center *
to get φ < 4° on horn torus, α has to be < 10^(10)°,
to get φ < 3° on horn torus, α has to be < 10^(14)°,
to get closer than 2°  my computer's capacity fails! (α < 10^(26))
an extremly slow approach to the horn torus center
reminds me of conditions at the edge of black holes
and of the different views ahead and backwards, when riding a photon  with speed of light: []
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• even slower 'approach' when mapping direct from the complex plane (α = 2·arccot(z))
• latter inserted into above formula we eventually get: φ = 2·arccot(lntan(arccot(z)))
• inverse mapping from horn torus to complex plane: z = cot(π/2  arccot(e^(cot(φ/2))))
• inverse mapping from horn torus back to Riemann sphere: α = π  2·arccot(e^(cot(φ/2)))
• most likely no stereographic projection possible, but mathematicians surely will look for!
• nonconformality creates pattern in the
dynamic model, but conformality has a
meaning
* we obtain much simpler formulas when we determine angle α to be
zero at south pole 0
my rough crosschecks with various angles α and differentials
Δω indicate exact validity, but
analytical evidence and search for a projective geometry
I gladly surrender to professionals ...
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Riemann sphere: 0N = 1 , z = 0Z ,
∠0ZN = ∠N0P' = α/2
