Editing Projective plane (section)

===Construction of projective planes from affine planes===

The affine plane ''K''<sup>2</sup> over ''K'' embeds into ''K'''''P'''<sup>2</sup> via the map which sends affine (non-homogeneous) coordinates to [[homogeneous coordinates]],
: <math>(x_1, x_2) \mapsto (1, x_1, x_2).</math>

The complement of the image is the set of points of the form {{nowrap|(0, ''x''<sub>1</sub>, ''x''<sub>2</sub>)}}. From the point of view of the embedding just given, these points are the [[point at infinity|points at infinity]]. They constitute a line in ''K'''''P'''<sup>2</sup>&mdash;namely, the line arising from the plane
:<math>\{k (0, 0, 1) + m (0, 1, 0) : k, m \in K\}</math>

in ''K''<sup>3</sup>&mdash;called the [[line at infinity]]. The points at infinity are the "extra" points where parallel lines intersect in the construction of the extended real plane; the point (0, ''x''<sub>1</sub>, ''x''<sub>2</sub>) is where all lines of slope ''x''<sub>2</sub> / ''x''<sub>1</sub> intersect. Consider for example the two lines
: <math>u = \{(x, 0) : x \in K\}</math>
: <math>y = \{(x, 1) : x \in K\}</math>

in the affine plane ''K''<sup>2</sup>. These lines have slope 0 and do not intersect. They can be regarded as subsets of ''K'''''P'''<sup>2</sup> via the embedding above, but these subsets are not lines in ''K'''''P'''<sup>2</sup>. Add the point {{nowrap|(0, 1, 0)}} to each subset; that is, let
: <math>\bar{u} = \{(1, x, 0) : x \in K\} \cup \{(0, 1, 0)\}</math>
: <math>\bar{y} = \{(1, x, 1) : x \in K\} \cup \{(0, 1, 0)\}</math>

These are lines in ''K'''''P'''<sup>2</sup>; ū arises from the plane
: <math>\{k (1, 0, 0) + m (0, 1, 0) : k, m \in K\}</math>

in ''K''<sup>3</sup>, while ȳ arises from the plane
: <math>{k (1, 0, 1) + m (0, 1, 0) : k, m \in K}.</math>
The projective lines ū and ȳ intersect at {{nowrap|(0, 1, 0)}}. In fact, all lines in ''K''<sup>2</sup> of slope 0, when projectivized in this manner, intersect at {{nowrap|(0, 1, 0)}} in ''K'''''P'''<sup>2</sup>.

The embedding of ''K''<sup>2</sup> into ''K'''''P'''<sup>2</sup> given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding
: <math>(x_1, x_2) \to (x_2, 1, x_1),</math>
has as its complement those points of the form {{nowrap|(''x''<sub>0</sub>, 0, ''x''<sub>2</sub>)}}, which are then regarded as points at infinity.

When an affine plane does not have the form of ''K''<sup>2</sup> with ''K'' a division ring, it can still be embedded in a projective plane, but the construction used above does not work. A commonly used method for carrying out the embedding in this case involves expanding the set of affine coordinates and working in a more general "algebra".