conformal mapping from (Riemann) sphere to horn torus surface
endeavour:
transfer point P' on sphere to a point P on torus, equivalent angle α → angle φ (see sketch),
without changing shape and angles of projected or mapped figures, e.g. circles
denotation of points:
(all points lie on the same plane that contains the main symmetry axis of the horn torus)
P' : any point on (any!) sphere, here we particularly cover the Riemann sphere
P : mapped point on nested horn torus, but compilation will also be valid for any horn torus!
Z : position of related complex number (x,y) on complex plane in case of Riemann sphere mapping
0 : center of complex plane (Z = (0,0)) and south pole of Riemann sphere (plane is tangent to sphere)
N : north pole of sphere, origin of Riemannian stereographic projection Z ↔ P'
S : common center of sphere and horn torus (not the south pole of sphere, notabene! I insist
on this denotation for the horn torus, because S stands for for Symmetry and Singularity!)
M : center of torus bulge crosssection circle
Q : intersection of perpendicular from point P on line ON
Q' : intersection of perpendicular from point P' on line ON
L : auxiliary point on line through S and M, intersection of perpendicular from P
R : auxiliary point on line QP,
intersection of perpendicular from M
angles:
α = ∠0SP' (0 < α < π, α = 0 for P' = point 0, α = π for P' = N)
φ = ∠SMP (0 < φ < 2π, φ = 0 for P = S and/or
φ = 2π for P = S)
ω : rotation angle around symmetry axis (0N) =
position of longitude (meridian) on sphere resp. torus
procedure:
we don't declare any appropriate stereographic projection to proof afterwards the conformality but
use instead the conditions of conformality to compile and establish the wanted mapping analytically:
for this we construct small circles on the surfaces of sphere resp. horn torus around points P' resp. P
by equalling their radii that lie on longitudes and on latitudes of sphere and of horn torus respectively.
in the sketch the longitudes are the crosssection circles on wich points P' and P are positioned and
the latitudes are circles perpendicular to the drawing plane through P' and P with radius Q'P' resp. QP,
they lie, half each, in front and behind the drawing plane, Q' and Q being centers of the latitude circles
compilation:
for Riemann sphere is valid: length of longitude (meridian) m = 2π·r = 2π·½ = π
and length of latitude l = 2π·Q'P' =
2π·½·sinα = π·sinα
for horn torus we have as length of longitude m = 2π·¼ = π/2, the latitude is slightly trickier:
its radius is QP =
QR + RP =
SM + ML =
¼·(1 + cos(π  φ)), so latitude l = 2π·QP = ½·π·(1  cosφ)
now we consider infinitesimal circles around P' and P on the surfaces of sphere resp. horn torus
and equal their radii dm on longitude and dl on latitude, noting that dω is identic for both figures
on sphere is dm = ½·dα, dl = ½·sinα·dω, and so for dm = dl we get
dω = dα / sinα
on horn torus is dm = ¼·dφ, dl = ¼·(1  cosφ)·dω, for dm = dl we get
dω = dφ / (1  cosφ)
finally we have to integrate a relative simple differential equation:
dα / sinα = dφ / (1  cosφ)
 home
Riemann sphere: 0N = 1 , ∠0NP' = α/2 ,
∠0ZN = π/2  α/2 , z = 0Z
gif
note: the solution is valid for any sphere and
any horn torus ! ( → as pdf )
see : numerical computation and check for conformality
(Excelfiles .xlsx)
