Editing Projective plane (section)

==Examples==

===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.

===Projective Moulton plane===
[[File:Moulton plane2.svg|thumb|upright=1.25|The ''Moulton plane''. Lines sloping down and to the right are bent where they cross the ''y''-axis.]]
The points of the [[Moulton plane]] are the points of the Euclidean plane, with coordinates in the usual way. To create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, but other lines will remain unchanged. Redefine all the lines with negative slopes so that they look like "bent" lines, meaning that these lines keep their points with negative ''x''-coordinates, but the rest of their points are replaced with the points of the line with the same ''y''-intercept but twice the slope wherever their ''x''-coordinate is positive.

The Moulton plane has parallel classes of lines and is an [[affine plane (incidence geometry)|affine plane]]. It can be projectivized, as in the previous example, to obtain the '''projective Moulton plane'''. [[Desargues' theorem]] is not a valid theorem in either the Moulton plane or the projective Moulton plane.

===A finite example===

This example has just thirteen points and thirteen lines. We label the points P<sub>1</sub>, ..., P<sub>13</sub> and the lines m<sub>1</sub>, ..., m<sub>13</sub>. The [[incidence relation]] (which points are on which lines) can be given by the following [[incidence matrix]]. The rows are labelled by the points and the columns are labelled by the lines. A 1 in row ''i'' and column ''j'' means that the point P<sub>''i''</sub> is on the line m<sub>''j''</sub>, while a 0 (which we represent here by a blank cell for ease of reading) means that they are not incident. The matrix is in Paige–Wexler normal form.
:::{| class="wikitable" style="text-align:center;"
|-
! {{diagonal split header|Points|Lines}}
! m<sub>1</sub>
! m<sub>2</sub> !! m<sub>3</sub> !! m<sub>4</sub>
! m<sub>5</sub> !! m<sub>6</sub> !! m<sub>7</sub>
! m<sub>8</sub> !! m<sub>9</sub> !! m<sub>10</sub>
! m<sub>11</sub>!! m<sub>12</sub>!! m<sub>13</sub>
|- style="border-bottom:2px solid #999;"
! P<sub>1</sub>
| bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp;
|-
! P<sub>2</sub>
| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp; || &nbsp;
|-
! P<sub>3</sub>
| bgcolor="#9cf"|1 || &nbsp; ||&nbsp;|| &nbsp; || &nbsp; || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp;
|- style="border-bottom:2px solid #999;"
! P<sub>4</sub>
| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || &nbsp; ||&nbsp;|| &nbsp;||&nbsp;||&nbsp; || &nbsp; || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || bgcolor="#9cf"|1
|-
! P<sub>5</sub>
| &nbsp; || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || bgcolor="#9cf"|1 ||&nbsp;|| &nbsp;|| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || &nbsp; || &nbsp;
|-
! P<sub>6</sub>
| &nbsp; || bgcolor="#9cf"|1 || &nbsp; || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || &nbsp;||&nbsp;|| bgcolor="#9cf"|1 || &nbsp; || &nbsp; || bgcolor="#9cf"|1 || &nbsp;
|- style="border-bottom:2px solid #999;"
! P<sub>7</sub>
|   || bgcolor="#9cf"|1 ||   ||   ||   || || bgcolor="#9cf"|1 || ||   || bgcolor="#9cf"|1 ||   ||   || bgcolor="#9cf"|1
|-
! P<sub>8</sub>
|   ||   || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 || ||  || || bgcolor="#9cf"|1 ||   ||   ||   || bgcolor="#9cf"|1
|-
! P<sub>9</sub>
|   ||   || bgcolor="#9cf"|1 ||   ||   || bgcolor="#9cf"|1 ||  || ||   || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 ||   ||
|- style="border-bottom:2px solid #999;"
! P<sub>10</sub>
|   ||   || bgcolor="#9cf"|1 ||   ||   || || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 ||   ||   ||   || bgcolor="#9cf"|1 ||
|-
! P<sub>11</sub>
|   ||   ||   || bgcolor="#9cf"|1 || bgcolor="#9cf"|1 || ||  || ||   || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 ||
|-
! P<sub>12</sub>
|   ||   ||   || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 ||  || bgcolor="#9cf"|1 ||   ||   ||   ||   || bgcolor="#9cf"|1
|-
! P<sub>13</sub>
|   ||   ||   || bgcolor="#9cf"|1 ||   || || bgcolor="#9cf"|1 || || bgcolor="#9cf"|1 ||   || bgcolor="#9cf"|1 ||   ||
|}

To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1s appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1s appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P<sub>1</sub>, P<sub>4</sub>, P<sub>5</sub>, and P<sub>8</sub>, for example, will satisfy the third condition. This example is known as the '''projective plane of order three'''.