Collineation

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.

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 α : D(V) → D(W), such that:

  • α is a bijection.
  • ADAαBα for all A, B in D(V).

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

  • 0 is mapped onto the trivial subspace of W.
  • V is mapped onto W.
  • There is a nonsingular semilinear map β from V to W such that, for all v in V,

(langle vrangle)^{alpha}=langle v^{beta}rangle

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 φ is a semilinear nonsingular map from V to W, with the dimension of V at least three. Define α : D(V) → D(W) by saying that Zα = { φ(z) | zZ } for all Z in D(V). As φ is semilinear, one easily checks that this map is properly defined, and further more, as φ is not singular, it is bijective. It is obvious now that α is a collineation. We say α is induced by φ.

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 α 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 φ such that φ induces α.

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

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