We don't declare any appropriate stereographic projection and try to proof afterwards the conformality but we use instead the conditions of conformality to compile and establish the wanted mapping analytically:

We consider the small circle around any point P and state the condition that the radii dm (in direction of meridians) and dl (parallel to latitudes) have to be equal. Lengths m of longitudes (meridians) on both figures, sphere and horn torus, are m = 2π·r Lengths l of latitudes are computed differently (* see supplement for derivation):l = 2π·r·sinα on the sphere and dm = r·dα on the sphere and dl = dω·r·sinα on the sphere and dα = dω·sinα for the sphere and dα/sinα = dφ/(1 - cosφ) and finally by integration we obtain the condition for conformal mapping
(0 < α < π, 0 < φ < 2π, C any real number) C is a kind of 'zoom/diminishing factor' for the mapped figures and shifts them:case α → φ: φ moves towards 2π with increasing C > 0, towards 0 with C < 0, case φ → α: α moves towards π with increasing C > 0, towards 0 with C < 0, conformality is given for C ≠ 0 as well, i. e. there is an infinite set of solutions - but mappings are not bijective, when C ≠ 0 is the same in the inverse mapping (likewise: Riemann stereographic projection is a special case amongst others) © 2018 Wolfgang W. Daeumler, Perouse |

QP = QR + RP = SM + ML = r + r·cos(π - φ) = r − r·cosφ = r·(1 − cosφ) l = 2π·QP
l = 2π·r·(1 − cosφ) |

the horn torus is an excellent graphic representation of complex numbers, a compactification with considerable more properties than the Riemann sphere has, it connects zero and infinity in an amazing way, can be dynamised by two independent turns, rotation around the axis and revolution around the torus bulge, what creates an incredible complexity and forms a coherent comprehensive entity, also explaining the self-interacting capability and fractal order of complex numbers, and finally - most important - horn tori can be interlaced into one another and so symbolise interactions between entities, easily interpretable as physical objects: 'space', 'time', 'particles', 'forces' ... home