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