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) |
Collineation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Collineation&oldid=17653