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Given two projective frames of a projective space P, there is exactly one homography of P that maps the first frame onto the second one. If the dimension of a projective space P is at least two, every collineation of P is the composition of an automorphic collineation and a homography. In particular, over the reals, every collineation of a ...
If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. The complex projective line is also called the Riemann sphere. All these definitions extend naturally to the case where K is a division ring; see, for example, Quaternionic projective space.
A homography (or projective transformation) of PG(2, K) is a collineation of this type of projective plane which is a linear transformation of the underlying vector space. Using homogeneous coordinates they can be represented by invertible 3 × 3 matrices over K which act on the points of PG(2, K ) by y = M x T , where x and y are points in K 3 ...
In mathematics, real projective space, denoted or (), is the topological space of lines passing through the origin 0 in the real space +. It is a compact , smooth manifold of dimension n , and is a special case G r ( 1 , R n + 1 ) {\displaystyle \mathbf {Gr} (1,\mathbb {R} ^{n+1})} of a Grassmannian space.
These examples of topological rings have the projective line as their one-point compactifications. The case of the complex number field C has the Möbius group as its homography group. The projective line over the dual numbers was described by Josef Grünwald in 1906. [4] This ring includes a nonzero nilpotent n satisfying nn = 0.
The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent form as follows. A projective space is of:
The use of real numbers gives homogeneous coordinates of points in the classical case of the real projective spaces, however any field may be used, in particular, the complex numbers may be used for complex projective space. For example, the complex projective line uses two homogeneous complex coordinates and is known as the Riemann sphere.
This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = x or y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R 3.