Collineation

projective

A projective transformation (projective isomorphism) of a projective space that is representable as the product of a finite number of perspectivities (cf. Perspective); if is a projective collineation, then for any subspace there exists a product of not more than perspectivities such that for any . For example, a projective transformation that leaves each point of some straight line fixed is a collineation, this is, a homology (in the narrow sense).

Let be interpreted as the collection of subspaces of the linear space over a skew-field . Then in order that a projective transformation be a projective collineation, it is necessary and sufficient that it be induced by a linear transformation of . The collection of all projective collineations forms a subgroup of the group of projective transformations which is a normal subgroup of .

The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field is inner. A field possesses this property if and only if any of its automorphisms is the identity, such as, for example, the field of real numbers . The complex field does not possess this property, whereas every automorphism of the skew-field of quaternions is inner.

If is a non-commutative skew-field, then there exists a non-trivial projective collineation that leaves every point of a given simplex fixed. Each simplex can be mapped onto any other simplex by precisely one projective collineation, if and only if is a field (the second fundamental theorem of projective geometry).

Comments

There is a large amount of confusion in the literature on projective geometry about the terminology for the different kinds of transformations. The transformation defined above, a projective collineation, sometimes also called a projectivity, is usually defined as transformation that is the product of a finite number of perspectivities. Some authors use the term projectivity, however, only for mappings between lines that are the product of a finite number of perspectivities. A projective transformation, usually called a projective isomorphism, or in older literature simply a collineation, is a bijection between projective spaces that preserves incidence.

References

[a1]	M. Berger, "Geometry" , I , Springer (1987)