Projective Plane - Vector Space Construction

Vector Space Construction

The line at infinity of the extended real plane appears to have a different nature than the other lines of that projective plane. This, however, is not true. Another construction of the same projective plane shows that no line can be distinguished (on geometrical grounds) from any other. In this construction, the "points" of the real projective plane are the lines through the origin in 3-dimensional Euclidean 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 (skewfield). Let K3 denote the set of all triples x = (x₀, x₁, x₂) of elements of K (a Cartesian product viewed as a Vector space). For any nonzero x in K3, the line in K3 through the origin and x is the subset

of K3. Similarly, let x and y be linearly independent elements of K3, meaning that if k x + l y = 0 then k = l = 0. The plane through the origin, x, and y in K3 is the subset

of K3. This plane contains various lines through the origin which are obtained by fixing either k or l.

The projective plane over K, denoted PG(2,K) or KP2, has a point set consisting of all the lines in K3 through the origin (each is a vector subspace of dimension 1). A subset L of PG(2,K) is a line in PG(2,K) if there exists a plane in K3 whose set of lines is exactly L (a vector subspace of dimension 2).

Verifying that this construction produces a projective plane is usually left as a linear algebra exercise.

An alternate (algebraic) view of this construction is as follows. The points of this projective plane are the equivalence classes of the set K3 - {(0, 0, 0)} modulo the equivalence relation

x ~ k x, for all k in .

Lines in the projective plane are defined exactly as above.

The coordinates (x₀, x₁, x₂) of a point in PG(2,K) are called homogeneous coordinates. Each triple (x₀, x₁, x₂) represents a well-defined point in PG(2,K), except for the triple (0, 0, 0), which represents no point. Each point in PG(2,K), however, is represented by many triples.

If K is a topological space, then KP2, inherits a topology via the product, subspace, and quotient topologies.