Stereographic projection σ : H R → R and its inverse σ : R → H R are given by σ σ ( τ τ , x ) = u = R x R + τ τ , σ σ − − 1 ( u ) = ( τ τ , x ) = ( R R 2 + | u | 2 R 2 − − | u | 2 , 2 R 2 u R 2 − − | u | 2 ) , where, for simplicity, τ ≡ ct. The (τ, x) are coordinates on M and the u are coordinates on R.