Homeomorphism by Radial Projection f: S² → ∂C
u ∈ S²  ·  f(u) ∈ ∂C  ·  rays

📐 Theorem: f is a homeomorphism

Definition: For u = (x,y,z) ∈ S², let t(u) = 1/max{|x|,|y|,|z|} be the first contact with ∂C.

f(u) = t(u)·u = u / max{|x|,|y|,|z|} [projection S² → ∂C]
f⁻¹(p) = p / ‖p‖ [normalization ∂C → S²]

λ = 0: Shows the cube C = f(S²). λ = 1: Shows the sphere S². The slider interpolates (1−λ)f(u) + λu, showing the continuous deformation.

Material prepared by Professor Sergio Gevatschnaider