Projective right-hand side

In Geometry, a projective right is a projective Espace of Dimension 1. A vector Space of dimension 2 is associated there.

More formally, a right projective on a body $K$ , indicated $\ mathbb P ^1 (K)$ can be defined like the whole of the Sous-espace S with a dimension of the vector space with 2 dimensions $K^2$ . One can conceive the projective line like the line of the $K$ to which one adds a Point ad infinitum.

The concept projective line spreads for the ring X associative.

Homogeneous coordinates

In Coordinated homogeneous, a point on the projective line $\ mathbb P ^1 (K)$ is a pair of the form:

$: x_2$

where

x_1, x_2 \ in K

is not both zeros. Two such pairs are known as equal if they differ only by one factor not-no one λ:

: x_2 = x_1: \lambda x_2.

The line of the

K

is identified with the subset of

\ mathbb P ^1 (K)

given by:

\{ : 1 \ in \ mathbb P^1 (K) \ mid X \ in K \}.

This subset covers all the points in

K

, except the Point ad infinitum:

\ infty =: 0.

Examples

Real numbers

If the body

K

is the whole of the real numbers, then the real projective right is obtained by projecting the point of

\ mathbb R^2

on the unit Cercle and while posing like equal the points diametrically opposite. In terms of Theory of the groups, this is equivalent taking the quotient with the Sous-groupe

\ {1,-1 \}

In terms of Topology, it is a Cercle. One can conceive it by imagining them + ∞ and - ∞ real numbers stuck together to form one Point ad infinitum, ∞, said Point ad infinitum in the real line direction . The résulat is thus a circle.

It should be noted that the real projective line is not equivalent to the completed real Droite, where a distinction is made between + ∞ and - ∞.

Complex numbers

If the body

K

is the whole of the complex numbers, then the addition of a Point ad infinitum in the plan complexes

\ mathbb C^2

results in a space which, topologically, is a Sphère. The projective right complexes is also known under the name of Sphère of Riemann or Sphère of Gauss.

It is the simplest example of Surface of Riemann. This explains that the projective line complexes is of common use in Analyze complexes, in algebraic Géométrie and theory of the complex Variétés.

Finished bodies

If the body

K

is finished and has

q

elements, then the projective line has

q+1

elements. One can write all his subfields, except one, in the form:

y = ax

where

a \ in K

. The case remaining is that of the right-hand side

x=0

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