Editing Projective plane (section)

==Affine planes==

Projectivization of the Euclidean plane produced the real projective plane. The inverse operation&mdash;starting with a projective plane, remove one line and all the points incident with that line&mdash;produces an '''affine plane'''.

===Definition===
More formally an '''[[Affine plane (incidence geometry)|affine plane]]''' consists of a set of '''lines''' and a set of '''points''', and a relation between points and lines called '''incidence''', having the following properties:
<div id="axioms-of-affine-planes">
#Given any two distinct points, there is exactly one line incident with both of them.
#Given any line ''l'' and any point ''P'' not incident with ''l'', there is exactly one line incident with ''P'' that does not meet ''l''.
#There are four points such that no line is incident with more than two of them.
</div>
The second condition means that there are [[Parallel (geometry)|parallel lines]] and is known as [[John Playfair|Playfair's]] axiom. The expression "does not meet" in this condition is shorthand for "there does not exist a point incident with both lines".

The Euclidean plane and the Moulton plane are examples of infinite affine planes. A finite projective plane will produce a finite affine plane when one of its lines and the points on it are removed. The '''order''' of a finite affine plane is the number of points on any of its lines (this will be the same number as the order of the projective plane from which it comes). The affine planes which arise from the projective planes PG(2,&nbsp;''q'') are denoted by AG(2,&nbsp;''q'').

There is a projective plane of order ''N'' if and only if there is an [[affine plane (incidence geometry)|affine plane]] of order ''N''.  When there is only one affine plane of order ''N'' there is only one projective plane of order ''N'', but the converse is not true. The affine planes formed by the removal of different lines of the projective plane will be isomorphic if and only if the removed lines are in the same orbit of the collineation group of the projective plane. These statements hold for infinite projective planes as well.

===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".

===Generalized coordinates===
{{main|Planar ternary ring}}

One can construct a coordinate "ring"&mdash;a so-called [[planar ternary ring]] (not a genuine ring)&mdash;corresponding to any projective plane.  A planar ternary ring need not be a field or division ring, and there are many projective planes that are not constructed from a division ring.  They are called [[non-Desarguesian projective plane]]s and are an active area of research.  The [[Cayley plane]] ('''OP'''<sup>2</sup>), a projective plane over the [[octonion]]s, is one of these because the octonions do not form a division ring.{{sfnp|Baez|2002|p=167}}

Conversely, given a planar ternary ring (''R'',&nbsp;''T''), a projective plane can be constructed (see below). The relationship is not one to one. A projective plane may be associated with several non-isomorphic planar ternary rings. The ternary operator ''T'' can be used to produce two binary operators on the set ''R'', by:
: ''a'' + ''b'' = ''T''(''a'', 1, ''b''), and
: ''a'' ⋅ ''b'' = ''T''(''a'', ''b'', 0).
The ternary operator is '''''linear''''' if {{nowrap|1=''T''(''x'', ''m'', ''k'') = ''x''⋅''m'' + ''k''}}. When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring.

Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane. For example, [[Desargues' theorem]] corresponds to the coordinate ring being obtained from a [[division ring]], while [[Pappus's hexagon theorem|Pappus's theorem]] corresponds to this ring being obtained from a [[commutative]] field. A projective plane satisfying Pappus's theorem universally is called a ''Pappian plane''. [[Alternative algebra|Alternative]], not necessarily [[associative]], division algebras like the octonions correspond to [[Moufang plane]]s.

There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian).  (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.)  The most common proof uses coordinates in a division ring and [[Wedderburn's little theorem|Wedderburn's theorem]] that finite division rings must be commutative; {{harvtxt|Bamberg|Penttila|2015}} give a proof that uses only more "elementary" algebraic facts about division rings.

To describe a finite projective plane of order ''N''(≥ 2) using non-homogeneous coordinates and a planar ternary ring:
:Let one point be labelled (''∞'').
:Label ''N'' points, (''r'') where ''r'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1).
:Label ''N''<sup>2</sup> points, (''r'', ''c'') where ''r'', ''c'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1).
On these points, construct the following lines:
:One line <nowiki>[</nowiki>''∞''<nowiki>]</nowiki> = { (''∞''), (0), ..., (''N''&nbsp;&minus;&nbsp;1)}
:''N'' lines <nowiki>[</nowiki>''c''<nowiki>]</nowiki> = {(''∞''), (''c'', 0), ..., (''c'', ''N''&nbsp;&minus;&nbsp;1)}, where ''c'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1)
:''N''<sup>2</sup> lines <nowiki>[</nowiki>''r'', ''c''<nowiki>]</nowiki> = {(''r'') and the points (''x'', '''T'''(''x'', ''r'', ''c'')) }, where ''x'', ''r'', ''c'' = 0, ..., (''N''&nbsp;&minus;&nbsp;1) and '''T''' is the ternary operator of the planar ternary ring.

For example, for {{nowrap|1=''N'' = 2}} we can use the symbols {0,&nbsp;1} associated with the finite field of order 2. The ternary operation defined by {{nowrap|1=''T''(''x'', ''m'', ''k'') = ''xm'' + ''k''}} with the operations on the right being the multiplication and addition in the field yields the following:
:One line <nowiki>[</nowiki>''∞''<nowiki>]</nowiki> = { (''∞''), (0), (1)},
:2 lines <nowiki>[</nowiki>''c''<nowiki>]</nowiki> = {(''∞''), (''c'',0), (''c'',1) : ''c'' = 0, 1},
::<nowiki>[</nowiki>0<nowiki>]</nowiki> = {(''∞''), (0,0), (0,1) }
::<nowiki>[</nowiki>1<nowiki>]</nowiki> = {(''∞''), (1,0), (1,1) }
:4 lines <nowiki>[</nowiki>''r'', ''c''<nowiki>]</nowiki>: (''r'') and the points  (''i'', ''ir'' + ''c''), where ''i'' = 0, 1 : ''r'', ''c'' = 0, 1.
::<nowiki>[</nowiki>0,0<nowiki>]</nowiki>: {(0), (0,0), (1,0) }
::<nowiki>[</nowiki>0,1<nowiki>]</nowiki>: {(0), (0,1), (1,1) }
::<nowiki>[</nowiki>1,0<nowiki>]</nowiki>: {(1), (0,0), (1,1) }
::<nowiki>[</nowiki>1,1<nowiki>]</nowiki>: {(1), (0,1), (1,0) }