  This paper points out that, in a four-dimensional spacetime manifold, the
four coordinates of any point may be viewed as arising from five homogeneous
coordinates according to the well-established recipes of projective geometry.
Moreover, if the homogeneous coordinates change by a linear transformation,
ruled by the coefficients of a matrix of the group GL(5,R), this induces a
fractional linear transformation of spherically symmetric spacetime coordinates
which can be exploited to bring infinity down to a finite distance. The
boundary of spherically symmetric spacetimes here studied is the disjoint union
of three points: future timelike infinity, past timelike infinity, spacelike
infinity, and the three-dimensional products of half-lines with a 2-sphere. For
the single points, the underlying structure is a projective map which destroys
the 2-sphere, whereas the two products of a half-line with the 2-sphere arise
from a projective map which preserves the 2-sphere. Geodesics are then studied
in the projectively transformed (t',r',theta',phi') coordinates for
Schwarzschild spacetime, with special interest in their way of approaching our
points at infinity. Next, Nariai and de Sitter spacetimes are studied with our
projective method. Since the kinds of infinity here defined depend only on the
symmetry of interest in a spacetime manifold, they have a broad range of
applications, which motivate the innovative analysis of Schwarzschild, Nariai
and de Sitter spacetime.
