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In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a
A mapping of a geometry to itself which sends lines to lines is called a collineation; it preserves the collinearity property. The linear maps (or linear functions) of vector spaces , viewed as geometric maps, map lines to lines; that is, they map collinear point sets to collinear point sets and so, are collineations.
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.
By the Fundamental theorem of projective geometry, the full collineation group (or automorphism group, or symmetry group) is the projective linear group PGL(3, 2), [a] Hirschfeld 1979, p. 131 [3] This is a well-known group of order 168 = 2 3 ·3·7, the next non-abelian simple group after A 5 of order 60 (ordered by size).
The archetypical example is the real projective plane, also known as the extended Euclidean plane. [1] This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP 2, or P 2 (R), among other notations.
Let x, y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by ,,, the coordinates of the image point of P on the sensor plane by x and y and the coordinates of the projection (optical) centre by ,,.
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As in the above example, matrices can be used to represent dualities. Let π be a duality of PG( n , K ) for n > 1 and let φ be the associated sesquilinear form (with companion antiautomorphism σ ) on the underlying ( n + 1 )-dimensional vector space V .
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