Affine space
can be embedded isomorphically in
by the
standard injection
.
Affine points can be recovered from projective ones with
by the mapping

A projective point with

However, these mappings and definitions are affine rather than
projective concepts. They are only meaningful if we are told in
advance that
represents ``normal'' affine space and
*x*_{n+1} is a special homogenizing coordinate. In a general
projective space any coordinate (or linear combination) can act as the
homogenizing coordinate and all hyperplanes are equivalent -- none is
especially singled out as the ``hyperplane at infinity''. These
issues will be discussed more fully in chapter 4.