The projective plane
is the set of all pairs
of antipodal points in
.
Two alternative definitions of
, equivalent to the
preceding one are
The set of all lines through the origin in
.
The set of all equivalence classes of ordered triples
of numbers (i.e., vectors in
) not all zero,
where two vectors are equivalent if they are proportional.
The space
can be thought of as the infinite plane tangent
to the space
and passing through the point
.
Let
be the mapping that
sends
to
. The
is a two-to-one
map of
onto
.
A line of
is a set of the form
, where
is a line of
. Clearly,
lies
on
if and only if
.
Homogeneous coordinates: In general, points of real
-dimensional projective space,
,
are represented by
component column vectors
such that at least one
is non-zero and
and
represent the same point of
for all
.
is the homogeneous representation of
a projective point.