Projective space

In Mathematical, a projective space is a fundamental construction starting from any vector Space.

Details

Projective space generalizes the projective Plan which can be built starting from a vector space with three dimensions, on any body.

Whereas the theory of the projective plans has an aspect Combinatoire, which is absent in the general case, projective space is fundamental in algebraic Géométrie, through the rich person projective Géométrie developed at the 19th century but also in constructions of the modern theory (based on the graduated Algèbre). Projective spaces and their generalization with overlapping Sous-espaces play also large a part in Topologie, in the theory of the groups of Dregs and the algebraic groups, and their theory of the representations.

Basic construction, being given a vector space V on a body K , is to form the whole of the classes of equivalence of the not-null vectors of V under the scalar relation of proportionality: v is proportional to W if v = CW with C not-no one in K . The idea is to regard each vectorial line of V as a point.

This idea goes back to mathematical descriptions of the Perspective. If the body K is that of the real numbers, and V has as a dimension N , then projective space P ( V ) - that one can see as the space of the right-hand sides passing by 0 in V - carries a natural structure of compact variety smooth of dimension N − 1. It is also very symmetrical, because any linear automorphism of V gives also a symmetry of P ( V ). In the traditional examples, these transformations are changes of prospect or projective transformations. The group of these symmetries is the quotient of the linear general Groupe of V by under group of the not-null multiples of the identity.

Favors for the consideration of the infinite ones

The use of projective spaces makes rigorous the concept of Droite ad infinitum (where the parallel straight lines meet), or Plan ad infinitum for three dimensions. One can ad infinitum choose arbitrarily a plan in projective space like plane . In this manner, the geometrical ideas introduced by Poncelet and others become part of the theory based on the Linear algebra. The part of a projective space which is not ad infinitum is called Espace refines; but symmetries of P ( V ) do not respect this division. The use of a base of V allows, if need be, the introduction of homogeneous Coordonnées for the execution of concrete calculations.

The use of vector spaces on the body of the complex numbers reveals different varieties, also used by the geometricians. Their use makes it possible to obtain a good theory of the intersection for the algebraic varieties.

Construction and uses

One obtains a projective space by adding an additional coordinate to that of an ordinary space; example: three for a space with two dimensions; four for a space with three dimensions; etc Thus the point of coordinates (X, there, Z) in 3D will have of projective representation the coordinates (X, there, Z, 1). The Not ad infinitum of the x axis the coordinates (1,0,0,0).
This provision makes it possible ad infinitum to avoid particular treatments for the points (which are those whose fourth coordinate is 0).
the systems of graphic treatment GL and OpenGL, of SGI, use projective spaces to represent space information out of computer.
All Variety kählérienne with null Curve and scalar Courbure positive is isomorphous with the projective complex, according to a communication made public by the Academy of Science in the Années 1970.

See too

Right projective
Plane projective, Plane projective reality, Plan projective complex
projective Espace complexes
projective Transformation
projective linear Groupe
projective Espace Hilbert
projective Représentation

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