Collineation

projective

A projective transformation (projective isomorphism) of a projective space $ \Pi _ {n} $ that is representable as the product of a finite number of perspectivities (cf. Perspective); if $ \nu $ is a projective collineation, then for any subspace $ S _ {q} $ there exists a product $ \pi $ of not more than $ q - 1 $ perspectivities such that $ \nu ( S _ {p} ) = \pi ( S _ {p} ) $ for any $ S _ {p} \subset S _ {q} $. 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 $ \Pi _ {n} $ be interpreted as the collection of subspaces of the linear space $ A _ {n + 1 } ^ {e} ( K) $ over a skew-field $ K $. 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 $ A _ {n + 1 } ^ {e} ( K) $. The collection of all projective collineations forms a subgroup $ G _ {0} $ of the group of projective transformations $ G $ which is a normal subgroup of $ G $.

The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field $ K $ 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 $ \mathbf R $. The complex field $ \mathbf C $ does not possess this property, whereas every automorphism of the skew-field of quaternions $ \mathbf H $ is inner.

If $ K $ 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 $ K $ 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 $ any $ 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)