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 ?
Anghel, Coanda, and Manolache provide a complete classification of globally generated vector bundles with first Chern class of five or less on the projective plane and four or less on the projective n-space for n equaling three or more.
[4] extended partially Kronk's result by proving that la(S) = 3 if S is the projective plane or the torus, and la(S) [less than or equal to] 4 if S is the Klein bottle.
The Severi varieties are a typical example: they parametrize curves of given degree and geometric genus in the projective plane; the general such curve has a prescribed number of ordinary double points and no further singularity.
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.
To have Bezout's theorem we need to consider i) curves over the complex projective plane and ii) that each intersection point must be counted with its multiplicity.
Let C be the complex numbers field and let [P.sup.2](C) be the projective plane over C.
So there are only four possible surfaces on which fullerenes ((4,6)-fullerenes) can be embedded, namely, the sphere, the projective plane, the torus, and the Klein bottle.
As all the investigations in this work are in the Euclidean model of the projective plane we will denote its infinite line with cg.
Then we deal with functions on the Riemann sphere [S.sup.2], which, as is well known, can be identified to the complex projective plane [P.sup.2](C); at least this was known to me at the time [9].
Topics of the papers include finite field experiments, K3 surfaces of Picard Rank One (which are double covers of the projective plane), Beilinson conjectures in the non-commutative setting, rational curves on cubic hypersurfaces, Abelian varieties over finite fields, global information from computations over finite fields, the geometry of Shimura varieties of the Hodge type over finite fields, Zeta functions over finite fields, de Rham cohomology of varieties over fields of positive characteristics, and homomorphisms of Abelian varieties over finite fields.
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.
Professor Saniga's paper "Conics, (q+1)-Arcs, Pencil Concept of Time and Psychopathology," informs us that it is demonstrated in the (projective plane over) Galois fields GF (q) with q = 2" and n [greater than or equal to] 3 (n being a positive integer) we can define, in addition to the temporal dimensions generated by pencils of conics, also time coordinates represented by aggregates of (q+1)-arcs that are not conics.