Homography
Central collineations

In above sections, homographies have been defined through linear algebra. In synthetic geometry, they are traditionally defined as the composition of one or several special homographies called central collineations. It is a part of the fundamental theorem of projective geometry that the two definitions are equivalent.

In a projective space, P, of dimension n ≥ 2, a collineation of P is a bijection from P onto P that maps lines onto lines. A central collineation (traditionally these were called perspectivities, but this term may be confusing, having another meaning; see Perspectivity) is a bijection α from P to P, such that there exists a hyperplane H (called the axis of α), which is fixed pointwise by α (that is, α (X) = X for all points X in H) and a point O (called the center of α), which is fixed linewise by α (any line through O is mapped to itself by α, but not necessarily pointwise). There are two types of central collineations. Elations are the central collineations in which the center is incident with the axis and homologies are those in which the center is not incident with the axis. A central collineation is uniquely defined by its center, its axis, and the image α (P) of any given point P that differs from the center O and does not belong to the axis. (The image α (Q) of any other point Q is the intersection of the line defined by O and Q and the line passing through α (P) and the intersection with the axis of the line defined by P and Q.)

A central collineation is a homography defined by a (n +1) × (n +1) matrix that has an eigenspace of dimension n. It is a homology, if the matrix has another eigenvalue and is therefore diagonalizable. It is an elation, if all the eigenvalues are equal and the matrix is not diagonalizable.

The geometric view of a central collineation is easiest to see in a projective plane. Given a central collineation α, consider a line ℓ that does not pass through the center O, and its image under α, ℓ ′ = α (ℓ). Setting R = ℓ ∩ ℓ ′, the axis of α is some line M through R. The image of any point A of ℓ under α is the intersection of OA with ℓ ′. The image B ′ of a point B that does not belong to ℓ may be constructed in the following way: let S = AB ∩ M, then B ′ = SA ′ ∩ OB.

The composition of two central collineations, while still a homography in general, is not a central collineation. In fact, every homography is the composition of a finite number of central collineations. In synthetic geometry, this property, which is a part of the fundamental theory of projective geometry is taken as the definition of homographies.
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