Search results
Results from the WOW.Com Content Network
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 ...
A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies.
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 ...
Most significantly, representation of P 1 (R) in a projective space over a division ring K is accomplished with a (K, R)-bimodule U that is a left K-vector space and a right R-module. The points of P 1 ( R ) are subspaces of P 1 ( K , U × U ) isomorphic to their complements.
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.
Geometrical setup for homography: stereo cameras O 1 and O 2 both pointed at X in epipolar geometry. Drawing from Neue Konstruktionen der Perspektive und Photogrammetrie by Hermann Guido Hauck (1845 — 1905) In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole ...
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.