**Collineations**

A Collineation of a projective plane is a bijective map of the plane to itself which maps points to points and lines to lines that preserves incidence, meaning that if σ is a bijection and point P is on line m, then Pσ is on mσ.

If σ is a collineation of a projective plane, a point P with P = Pσ is called a * fixed point* of σ, and a line m with m = mσ is called a

*of σ. The points on a fixed line need not be fixed points, their images under σ are just constrained to lie on this line. The collection of fixed points and fixed lines of a collineation form a*

**fixed line***, which is a system of points and lines that satisfy the first two but not necessarily the third condition in the definition of a projective plane. Thus, the fixed point and fixed line structure for any collineation either form a projective plane by themselves, or a degenerate plane. Collineations whose fixed structure forms a plane are called*

**closed configuration***.*

**planar collineations**Read more about this topic: Projective Plane

### Other articles related to "collineations, collineation":

**Collineations**- Homography

... A homography (or projective transformation) of PG(2,K) is a

**collineation**of this type of projective plane which is a linear transformation of the underlying vector space ... Another type of

**collineation**of PG(2,K) is induced by any automorphism of K, these are called automorphic

**collineations**... If α is an automorphism of K, then the

**collineation**given by (x0,x1,x2) → (x0α,x1α,x2α) is an automorphic

**collineation**...

... The main examples of

**collineations**are projective linear transformations (also known as homographies) and automorphic

**collineations**... fundamental theorem of projective geometry states that all

**collineations**are a combination of these, as described below ... A duality is a

**collineation**from a projective space onto its dual space, taking points to hyperplanes (and vice versa) and preserving incidence ...

**Collineations**

... A related group is the

**collineation**group, which is defined axiomatically ... A

**collineation**is an invertible (or more generally one-to-one) map which sends collinear points to collinear points ... the incidence relation, which is exactly a

**collineation**of a space to itself ...