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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.
For example, it maps the positive real numbers to the interval [−1, 1]. Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is
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 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.
Frequently cross ratio is introduced as a function of four values. Here three define a homography and the fourth is the argument of the homography. The distance of this fourth point from 0 is the logarithm of the evaluated homography. In a projective space containing P(R), suppose a conic K is given, with p and q on K.