Projective Plane - 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 fixed line 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 closed configuration, 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 planar collineations.

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Other articles related to "collineations, collineation":

Projective Plane - 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 ...
Collineation - Types
... 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 ...
Projective Linear Group - Name - 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 ...