Conformal compactification of Euclidean space (${\mathbb{R}}^4\cup \{\infty \}\cong S^4$)

Inverse stereographic projection identifies Euclidean four-space with a sphere minus one point. Multiplying its flat metric by $4/(1+|x{|}^2{)}^2$ gives the round metric, which extends smoothly over the added point. In the Klein quadric description, the compact space is the projectivized real null cone of signature $(1,5)$. It differs topologically from Minkowski conformal compactification.