Projective plane


In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.
Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by, RP2, or P2, among other notations. There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane.
A projective plane is a 2-dimensional projective space. Not all projective planes can be embedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.

Definition

A projective plane is a rank 2 incidence structure consisting of a set of points, a set of lines, and a symmetric relation on the set called incidence, having the following properties:

  1. Given any two distinct points, there is exactly one line incident with both of them.
  2. Given any two distinct lines, there is exactly one point incident with both of them.
  3. There are four points such that no line is incident with more than two of them.

The second condition means that there are no parallel lines. The last condition excludes the so-called degenerate cases. The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "point P is incident with line ℓ" is used instead of either "P is on ℓ" or "ℓ passes through P".
It follows from the definition that the number of points incident with any given line in a projective plane is the same as the number of lines incident with any given point. The cardinal number is called order of the plane.

Examples

The extended Euclidean plane

To turn the ordinary Euclidean plane into a projective plane, proceed as follows:
  1. To each parallel class of 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. 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 '' [|below].

Projective Moulton plane

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's 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 P1,..., P13 and the lines m1,..., m13. The incidence relation 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 Pi is on the line mj, while a 0 means that they are not incident. The matrix is in Paige–Wexler normal form.
To verify the conditions that make this a projective plane, observe that every two rows have exactly one common column in which 1s appear and that every two columns have exactly one common row in which 1s appear. 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.

Vector space construction

Though the line at infinity of the extended real plane may appear to have a different nature than the other lines of that projective plane, this is not the case. Another construction of the same projective plane shows that no line can be distinguished from any other. In this construction, each "point" of the real projective plane is the one-dimensional subspace through the origin in a 3-dimensional vector space, and a "line" in the projective plane arises from a plane through the origin in the 3-space. This idea can be generalized and made more precise as follows.
Let K be any division ring. Let K3 denote the set of all triples x = of elements of K. For any nonzero x in K3, the minimal subspace of K3 containing x is the subset
of K3. Similarly, let x and y be linearly independent elements of K3, meaning that implies that. The minimal subspace of K3 containing x and y is the subset
of K3. This 2-dimensional subspace contains various 1-dimensional subspaces through the origin that may be obtained by fixing k and m and taking the multiples of the resulting vector. Different choices of k and m that are in the same ratio will give the same line.
The projective plane over K, denoted PG or KP2, has a set of points consisting of all the 1-dimensional subspaces in K''3. A subset L of the points of PG is a line in PG if there exists a 2-dimensional subspace of K3 whose set of 1-dimensional subspaces is exactly L.
Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.
An alternate view of this construction is as follows. The points of this projective plane are the equivalence classes of the set modulo the equivalence relation
Lines in the projective plane are defined exactly as above.
The coordinates of a point in PG are called homogeneous coordinates. Each triple represents a well-defined point in PG, except for the triple, which represents no point. Each point in PG, however, is represented by many triples.
If K is a topological space, then ''KP2 inherits a topology via the product, subspace, and quotient topologies.

Classical examples

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.
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 serve as fundamental examples in algebraic geometry.
The quaternionic projective plane HP2 is also of independent interest.

Finite field planes

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 elements with prime p produces a projective plane of points. The field planes are usually denoted by PG where PG stands for projective geometry, the "2" is the dimension and q is called the order of the plane. The Fano plane, discussed below, is denoted by PG. The [|third example above] is the projective plane PG.
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. A permutation of the seven points that carries collinear points 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 has 168 elements.