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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
In geometry, collinearity of a set of points is the property of their lying on a single line. [1] A set of points with this property is said to be collinear (sometimes spelled as colinear [ 2 ] ). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
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.
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. [1] It is a bijection that maps lines to lines, and thus a collineation.
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 term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of homogeneous coordinates, and in which Euclidean geometry may be embedded (hence its name ...
A Hughes plane H: [1]. is a non-Desarguesian projective plane of odd square prime power order of Lenz-Barlotti type I.1, has a Desarguesian Baer subplane H 0,; is a self-dual plane in which every orthogonal polarity of H 0 can be extended to a polarity of H,
In finite geometry, PG(3, 2) is the smallest three-dimensional projective space. It can be thought of as an extension of the Fano plane. It has 15 points, 35 lines, and 15 planes. [1] It also has the following properties: [2] Each point is contained in 7 lines and 7 planes. Each line is contained in 3 planes and contains 3 points.