Editing Projective plane (section)

===The extended Euclidean plane===

To turn the ordinary Euclidean plane into a projective plane, proceed as follows:
# To each parallel class of lines (a maximum set of mutually parallel lines) associate a single new point. That point is to be considered incident with each line in its class. The new points added are distinct from each other. These new points are called ''[[point at infinity|points at infinity]]''.
# Add a new line, which is considered incident with all the points at infinity (and no other points). This line is called ''the'' ''[[line at infinity]]''.

The extended structure is a projective plane and is called the '''extended Euclidean plane''' or the [[real projective plane]]. The process outlined above, used to obtain it, is called "projective completion" or ''projectivization''. This plane can also be constructed by starting from '''R'''<sup>3</sup> viewed as a vector space, see ''{{section link||Vector space construction}}'' below.