COLLINEATION

A 'collineation' is a one-to-one map from one projective space to another, or from a projective plane onto itself, such that the images of collinear points are themselves collinear. All automorphisms induce a collineation.

Contents

Definition

Fundamental theorem of projective geometry

See also

Definition

Let ''V'' be a vector space (of dimension at least three) over a field ''K'' and ''W'' a vector space over a field ''L''. Consider the projective spaces ''PG(V)'' and ''PG(W)''.
Call ''D(V)'' and ''D(W)'' the set of subspaces of ''V'' and ''W'' respectively. A collineation from ''PG(V)'' to ''PG(W)'' is a map
ightarrow D(W), such that :

★ is a bijection.

★
When ''V'' has dimension one, a collineation from ''PG(V)'' to ''PG(W)'' is a map
ightarrow D(W), such that :

★ $\{0\}$ is mapped onto the trivial subspace of ''W''.

★ ''V'' is mapped onto ''W''.

★ There is a nonsingular semilinear map from ''V'' to W'' such that :
The reason for the seemingly completely different definition when ''V'' has geometric dimension one will become clearer further on in this article.
When $V=W$ the collineations are also called automorphisms.

Fundamental theorem of projective geometry

Briefly, every collineation is the product of a homography and an automorphic collineation. In particular, the collineations of PG(2,R) are exactly the homographies.
Suppose $phi$ is a semilinear nonsingular map from ''V'' to ''W'', with the dimension of ''V'' at least three.
Define
ightarrow D(W) in this way:

As $phi$ is semilinear, one easily checks that this map is properly defined, and further more, as $phi$ is not singular, it is bijective. It is obvious now that is a collineation. We say is induced by $phi$.
The fundamental theorem of projective geometry states the converse:
Suppose ''V'' is a vector space over a field ''K'' with dimension at least three, ''W'' is a vector space over a field ''L'', and is a collineation from ''PG(V)'' to ''PG(W)''. This implies ''K'' and ''L'' are isomorphic fields, ''V'' and ''W'' have the same dimension, and there is a semilinear map $phi$ such that $phi$ induces .
The fundamental theorem explains the different definition for projective lines. Otherwise, every bijection between the points would be a collineation, and then there would be no nice algebraic relationship.

See also

★ Homography

★ Correlation