We will obtain the warped product decompositions of spaces of constant
curvature (with arbitrary signature) in their natural models as subsets of
pseudo-Euclidean space. This generalizes the corresponding result by S. Nolker
to arbitrary signatures, and has a similar level of detail. Although our
derivation is complete in some sense, none is proven. Motivated by
applications, we will give more information for the spaces with Euclidean and
Lorentzian signatures. This is an expository article which is intended to be
used as a reference. So we also give a review of the theory of circles and
spheres in pseudo-Riemannian manifolds.