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In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the setting for planar projective geometry, in which the relationships between objects are not ...
The projective plane over K, denoted PG(2, K) or KP 2, has a set of points consisting of all the 1-dimensional subspaces in K 3. A subset L of the points of PG(2, K ) is a line in PG(2, K ) if there exists a 2-dimensional subspace of K 3 whose set of 1-dimensional subspaces is exactly L .
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:
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.
There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space . Not all projective planes can be embedded in 3-dimensional projective spaces; such embeddability is a consequence of a property known as Desargues' theorem ...
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.
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions and theorems of projective planes. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special ...
Projective space. In graphical perspective, parallel (horizontal) lines in the plane intersect at a vanishing point (on the horizon). In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus be viewed as the extension of a ...