Projective line

In mathematics, in particular projective geometry is the projective line a one - dimensional projective space.

Definition

There is a body, for example the body of the real or complex numbers, or a finite field. It is the (unique up to isomorphism ) of 2- dimensional vector space. The projective line is the set of 1-dimensional subspaces of.

In other words, the projective line is the quotient space

With respect to the equivalence relation

This equivalence relation identifies two points if and only if they are straight line through the origin in the same one -dimensional subspace, ie on the same.

Homogeneous coordinates

Each point of the projective line can as in homogeneous coordinates

Be presented with, which applies to all.

Number line expanded to include the point at infinity

The projective line can be, the extended a " point at infinity " can be precisely identified. One can namely by straight line with the in homogeneous coordinates

Given subset of identify. This subset will contain all the points except one, the so-called " point at infinity ":

Examples

• The real projective line is homeomorphic to the circle.
• The complex projective line is called the Riemann sphere, it is homeomorphic to the 2-sphere.
• The projective line over the finite field has elements.

Automorphisms

The general linear group acts on by linear mappings. The projective linear group is the factor group, the normal ( even central ) subgroup of scalar multiples of the identity is made ​​. The effect of on induces a well-defined action of on. The automorphisms of the illustrations are by definition described by elements of.

In homogeneous coordinates, the matrices act as a broken - linear transformations:

After the identification.

The automorphism group acts transitively on triples pairwise different points.

A fundamental invariant of projective geometry is the cross ratio of 4- tuples of pairwise different points. Two such 4- tuples can be accurately then by an automorphism into one another if their cross ratio matches.

In the case is referred to the automorphisms of the Möbius transformations.

Projective line in the projective plane

The projective given line through two points of the projective plane determined one by one as a straight line in conceives ( and by their linear equation describes ) the two points, the calculated (see plane equation ) and this level then projective plane containing them to a straight line in projected.

Analogously one determines projective line through two given points in a higher-dimensional projective space.