Projective plane

To turn the ordinary Euclidean plane into a projective plane proceed as follows:

1. 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 points at infinity.
2. 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 R³ viewed as a vector space, see § Vector space construction below.

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. 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.

This example has just thirteen points and thirteen lines. We label the points P₁,...,P₁₃ and the lines m₁,...,m₁₃. 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_i is on the line m_j, 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.

Lines Points	m1	m2	m3	m4	m5	m6	m7	m8	m9	m10	m11	m12	m13
P1	1	1	1	1
P2	1				1	1	1
P3	1							1	1	1
P4	1										1	1	1
P5		1			1			1			1
P6		1				1			1			1
P7		1					1			1			1
P8			1		1				1				1
P9			1			1				1	1
P10			1				1	1				1
P11				1	1					1		1
P12				1		1		1					1
P13				1			1		1		1

To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1's appear (every pair of distinct points are on exactly one common line) and that every two columns have exactly one common row in which 1's appear (every pair of distinct lines meet at exactly one point). Among many possibilities, the points P₁,P₄,P₅,and P₈, for example, will satisfy the third condition. This example is known as the projective plane of order three.

The real projective plane RP², arises when K is taken to be the real numbers, R. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.[4]

In this construction consider the unit sphere centered at the origin in R³. Each of the R³ lines in this construction intersects the sphere at two antipodal points. Since the R³ line represents a point of RP², we will obtain the same model of RP² by identifying the antipodal points of the sphere. The lines of RP² will be the great circles of the sphere after this identification of antipodal points. This description gives the standard model of elliptic geometry.

The complex projective plane CP², arises when K is taken to be the complex numbers, C. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields (known as pappian planes) serve as fundamental examples in algebraic geometry.[5]

The quaternionic projective plane HP² is also of independent interest.

By Wedderburn's Theorem, a finite division ring must be commutative and so be a field. Thus, the finite examples of this construction are known as "field planes". Taking K to be the finite field of q = pⁿ elements with prime p produces a projective plane of q² + q + 1 points. The field planes are usually denoted by PG(2,q) where PG stands for projective geometry, the "2" is the dimension and q is called the order of the plane (it is one less than the number of points on any line). The Fano plane, discussed below, is denoted by PG(2,2). The third example above is the projective plane PG(2,3).

The Fano plane. Points are shown as dots; lines are shown as lines or circles.

The Fano plane is the projective plane arising from the field of two elements. It is the smallest projective plane, with only seven points and seven lines. In the figure at right, the seven points are shown as small balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" – this is an example of duality in the projective plane: if the lines and points are interchanged, the result is still a projective plane (see below). A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3,2) = PGL(3,2)) has 168 elements.

The theorem of Desargues is universally valid in a projective plane if and only if the plane can be constructed from a three-dimensional vector space over a skewfield as above.[6] These planes are called Desarguesian planes, named after Girard Desargues. The real (or complex) projective plane and the projective plane of order 3 given above are examples of Desarguesian projective planes. The projective planes that can not be constructed in this manner are called non-Desarguesian planes, and the Moulton plane given above is an example of one. The PG(2,K) notation is reserved for the Desarguesian planes. When K is a field, a very common case, they are also known as field planes and if the field is a finite field they can be called Galois planes.

A Fano subplane is a subplane isomorphic to PG(2,2), the unique projective plane of order 2.

If you consider a quadrangle (a set of 4 points no three collinear) in this plane, the points determine six of the lines of the plane. The remaining three points (called the diagonal points of the quadrangle) are the points where the lines that do not intersect at a point of the quadrangle meet. The seventh line consists of all the diagonal points (usually drawn as a circle or semicircle).

In finite desarguesian planes, PG(2,q), Fano subplanes exist if and only if q is even (that is, a power of 2). The situation in non-desarguesian planes is unsettled. They could exist in any non-desarguesian plane of order greater than 6, and indeed, they have been found in all non-desarguesian planes in which they have been looked for (in both odd and even orders).

An open question, apparently due to Hanna Neumann though not published by her, is: Does every non-desarguesian plane contain a Fano subplane?

A theorem concerning Fano subplanes due to (Gleason 1956) is:

If every quadrangle in a finite projective plane has collinear diagonal points, then the plane is desarguesian (of even order).

More formally an 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:

The second condition means that there are parallel lines and is known as 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,q) are denoted by AG(2,q).

There is a projective plane of order N if and only if there is an 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.

The affine plane K² over K embeds into KP² via the map which sends affine (non-homogeneous) coordinates to homogeneous coordinates,

${\displaystyle (x_{1},x_{2})\mapsto (1,x_{1},x_{2}).}$

The complement of the image is the set of points of the form (0, x₁, x₂). From the point of view of the embedding just given, these points are the points at infinity. They constitute a line in KP²—namely, the line arising from the plane

${\displaystyle \{k(0,0,1)+m(0,1,0):k,m\in K\}}$

in K³—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, x₁, x₂) is where all lines of slope x₂ / x₁ intersect. Consider for example the two lines

${\displaystyle u=\{(x,0):x\in K\}}$

${\displaystyle y=\{(x,1):x\in K\}}$

in the affine plane K². These lines have slope 0 and do not intersect. They can be regarded as subsets of KP² via the embedding above, but these subsets are not lines in KP². Add the point (0, 1, 0) to each subset; that is, let

${\displaystyle {\bar {u}}=\{(1,x,0):x\in K\}\cup \{(0,1,0)\}}$

${\displaystyle {\bar {y}}=\{(1,x,1):x\in K\}\cup \{(0,1,0)\}}$

These are lines in KP²; ū arises from the plane

${\displaystyle \{k(1,0,0)+m(0,1,0):k,m\in K\}}$

in K³, while ȳ arises from the plane

${\displaystyle {k(1,0,1)+m(0,1,0):k,m\in K}.}$

The projective lines ū and ȳ intersect at (0, 1, 0). In fact, all lines in K² of slope 0, when projectivized in this manner, intersect at (0, 1, 0) in KP².

The embedding of K² into KP² given above is not unique. Each embedding produces its own notion of points at infinity. For example, the embedding

${\displaystyle (x_{1},x_{2})\to (x_{2},1,x_{1}),}$

has as its complement those points of the form (x₀, 0, x₂), which are then regarded as points at infinity.

When an affine plane does not have the form of K² 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".

One can construct a coordinate "ring"—a so-called planar ternary ring (not a genuine ring)—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 planes and are an active area of research. The Cayley plane (OP²), a projective plane over the octonions, is one of these because the octonions do not form a division ring.[3]

Conversely, given a planar ternary ring (R,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 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 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, not necessarily associative, division algebras like the octonions correspond to Moufang planes.

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 theorem that finite division rings must be commutative; 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 − 1).

Label N² points, (r, c) where r, c = 0, ..., (N − 1).

On these points, construct the following lines:

One line [∞] = { (∞), (0), ..., (N − 1)}

N lines [c] = {(∞), (c,0), ..., (c, N − 1)}, where c = 0, ..., (N − 1)

N² lines [r, c] = {(r) and the points (x, T(x,r,c)) }, where x, r, c = 0, ..., (N − 1) and T is the ternary operator of the planar ternary ring.

For example, for N=2 we can use the symbols {0,1} associated with the finite field of order 2. The ternary operation defined by 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 [∞] = { (∞), (0), (1)},

2 lines [c] = {(∞), (c,0), (c,1) : c = 0, 1},

[0] = {(∞), (0,0), (0,1) }

[1] = {(∞), (1,0), (1,1) }

4 lines [r, c]: (r) and the points (i,ir + c), where i = 0, 1 : r, c = 0, 1.

[0,0]: {(0), (0,0), (1,0) }

[0,1]: {(0), (0,1), (1,1) }

[1,0]: {(1), (0,0), (1,1) }

[1,1]: {(1), (0,1), (1,0) }

A homography (or projective transformation) of PG(2,K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space. Using homogeneous coordinates they can be represented by invertible 3 × 3 matrices over K which act on the points of PG(2,K) by y = M x^T, where x and y are points in K³ (vectors) and M is an invertible 3 × 3 matrix over K.[8] Two matrices represent the same projective transformation if one is a constant multiple of the other. Thus the group of projective transformations is the quotient of the general linear group by the scalar matrices called the projective linear group.

Another type of collineation of PG(2,K) is induced by any automorphism of K, these are called automorphic collineations. If α is an automorphism of K, then the collineation given by (x₀,x₁,x₂) → (x₀^α,x₁^α,x₂^α) is an automorphic collineation. The fundamental theorem of projective geometry says that all the collineations of PG(2,K) are compositions of homographies and automorphic collineations. Automorphic collineations are planar collineations.