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The affine planes which arise from the projective planes PG(2, q) are denoted by AG(2, q). There is a projective plane of order N if and only if there is an affine plane of order N. When there is only one affine plane of order N there is only one projective plane of order N, but the converse is not true. The affine planes formed by the removal ...
In contexts where there is no ambiguity, it is simply called the projective plane; the qualifier "real" is added to distinguish it from other projective planes such as the complex projective plane and finite projective planes. One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass ...
The Fano plane is the projective plane with the fewest points and lines. The smallest 2-dimensional projective geometry (that with the fewest points) is the Fano plane, which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
The projective -space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the -sphere, a simply connected space. It is a double cover.
As the intersection of two planes passing through O is a line passing through O, the intersection of two distinct projective lines consists of a single projective point. The plane P 1 defines a projective line which is called the line at infinity of P 2. By identifying each point of P 2 with the corresponding projective point, one can thus say ...
In the Cayley plane, lines and points may be defined in a natural way so that it becomes a 2-dimensional projective space, that is, a projective plane. It is a non-Desarguesian plane, where Desargues' theorem does not hold. More precisely, as of 2005, there are two objects called Cayley planes, namely the real and the complex Cayley plane.
Many hyperbolic lines through point P not intersecting line a in the Beltrami Klein model A hyperbolic triheptagonal tiling in a Beltrami–Klein model projection. In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit ...
A two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has two complex dimensions, which can alternately be represented using four real dimensions. A two-dimensional lattice is an infinite grid of points which can be represented using integer coordinates.
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