|
Ovoid (polar Space)
In mathematics, an ovoid ''O'' of a (finite) polar space of rank ''r'' is a set of points, such that every subspace of rank r-1 intersects ''O'' in exactly one point.. Cases Symplectic polar space An ovoid of W_(q) (a symplectic polar space of rank ''n'') would contain q^n+1 points. However it only has an ovoid if and only n=2 and ''q'' is even. In that case, when the polar space is embedded into PG(3,q) the classical way, it is also an ovoid in the projective geometry sense. Hermitian polar space Ovoids of H(2n,q^2)(n\geq 2) and H(2n+1,q^2)(n\geq 1) would contain q^+1 points. Hyperbolic quadrics An ovoid of a hyperbolic quadric Q^(2n-1,q)(n\geq 2)would contain q^+1 points. Parabolic quadrics An ovoid of a parabolic quadric Q(2 n,q)(n\geq 2) would contain q^n+1 points. For n=2, it is easy to see to obtain an ovoid by cutting the parabolic quadric with a hyperplane, such that the intersection is an elliptic quadric. The intersection is an ovoid. If ''q'' is even, Q(2n,q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Space
In mathematics, in the field of geometry, a polar space of rank ''n'' (), or ''projective index'' , consists of a set ''P'', conventionally called the set of points, together with certain subsets of ''P'', called ''subspaces'', that satisfy these axioms: * Every subspace is isomorphic to a projective geometry with and ''K'' a division ring. By definition, for each subspace the corresponding ''d'' is its dimension. * The intersection of two subspaces is always a subspace. * For each point ''p'' not in a subspace ''A'' of dimension of , there is a unique subspace ''B'' of dimension containing ''p'' and such that is -dimensional. The points in are exactly the points of ''A'' that are in a common subspace of dimension 1 with ''p''. * There are at least two disjoint subspaces of dimension . It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (''P'',''L''), so that for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ovoid (projective Geometry)
In projective geometry an ovoid is a sphere like pointset (surface) in a projective space of dimension . Simple examples in a real projective space are hyperspheres (quadrics). The essential geometric properties of an ovoid \mathcal O are: # Any line intersects \mathcal O in at most 2 points, # The tangents at a point cover a hyperplane (and nothing more), and # \mathcal O contains no lines. Property 2) excludes degenerated cases (cones,...). Property 3) excludes ruled surfaces (hyperboloids of one sheet, ...). An ovoid is the spatial analog of an oval in a projective plane. An ovoid is a special type of a ''quadratic set.'' Ovoids play an essential role in constructing examples of Möbius planes and higher dimensional Möbius geometries. Definition of an ovoid * In a projective space of dimension a set \mathcal O of points is called an ovoid, if : (1) Any line meets \mathcal O in at most 2 points. In the case of , g\cap\mathcal O, =0, the line is called a ''passing'' (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Polar Space")
