*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.