projective plane


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projective plane

[prə′jek·tiv ′plān]
(mathematics)
The topological space obtained from the two-dimensional sphere by identifying antipodal points; the space of all lines through the origin in Euclidean space.
More generally, a plane (in the sense of projective geometry) such that (1) every two points lie on exactly one line, (2) every two lines pass through exactly one point, and (3) there exists a four-point.

Projective Plane

 

in its original meaning, the Euclidean plane with the addition of the points and line at infinity. From the topological standpoint, the projective plane is a closed, non-orientable surface with Euler characteristic 1.

projective plane

(mathematics)
The space of equivalence classes of vectors under non-zero scalar multiplication. Elements are sets of the form

kv: k != 0, k scalar, v != O, v a vector

where O is the origin. v is a representative member of this equivalence class.

The projective plane of a vector space is the collection of its 1-dimensional subspaces. The properties of the vector space induce a topology and notions of smoothness on the projective plane.

A projective plane is in no meaningful sense a plane and would therefore be (but isn't) better described as a "projective space".
References in periodicals archive ?
We shall consider the following kind of families of subvarieties: families of curves with prescribed invariants and singularities in surfaces (with special attention to the two cases of the projective plane, and of K3 surfaces), families of hyperplane sections with prescribed singularities of hypersurfaces in projective spaces, families of curves with a given genus in Calabi-Yau threefolds, and families of surfaces in the projective 3-space containing curves with unexpected singularities.
The main objective of the current study was to enhance channel utilization by applying a novel finite projective plane (FPP)-based scheme involving Markov chain modeling of N channels and N users, particularly when channel resources are often idle because of inefficient use.
As all the investigations in this work are in the Euclidean model of the projective plane we will denote its infinite line with cg.
Already in [16], [section] 20 and [17] it was established that in the 3-dimensional betweenness space every bundle of lines through a fixed point has the structure of a Desarguesian projective plane (see also [34], Sec.
For the nonquadratic case, it is convenient to explicitly use the concept of the projective plane over a field F.
n], isometric either to a complex projective line of maximal curvature, to a flat torus of dimension 2 or to a real projective plane.
and Canadian mathematicians recently submitted a proof problem--Could a finite projective plane of Order 10 exist?
Such a completed table, in which filled spaces can be represented by ones and empty spaces by zeros, is one way of expressing what is known as a "finite projective plane.
Coverage of the Real Projective Plane in chapter 11 has been re-written to be more clear to students.
3] [subset] PV (4,3) is a projective plane of kind [P.
i])i [member of] z of points in the projective plane that is periodic modulo some projective transformation [phi], i.