Summary: Let \(M=M_{n,m}\) be the Euclidean space \(\mathbb{R}^p\) equipped with a symmetric bilinear form \(B_M\) of rank \(p=n+m\) and signature \(n-m\). We compactify \(M\) so that \(M_c\) is homogeneous and has as its group of isometries a Lie group whose dimension is the dimension of \(M\) plus \(2p +1\). We observe that \(M_c\) is in two ways the total space of a non-trivial sphere bundle with base space the real projective space. The compactification is well understood in the classical case when \(M\) is Minkowski space. The contribution here is to observe that the construction works generally and that it admits a natural bundle description.