होमोग्राफ इसके अंग्रेजी अर्थ का उदाहरण

Military units in Queensland In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive.

In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two.

Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term homography, which, etymologically, roughly means "similar drawing", dates from this time.

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.

Historically, the concept of homography had been introduced to understand, explain and study visual perspective, and, specifically, the difference in appearance of two plane objects viewed from different points of view.

If f is a perspectivity from P to Q, and g a perspectivity from Q to P, with a different center, then is a homography from P to itself, which is called a central collineation, when the dimension of P is at least two.

Originally, a homography was defined as the composition of a finite number of perspectivities.

Given two projective spaces P(V) and P(W) of the same dimension, an homography is a mapping from P(V) to P(W), which is induced by an isomorphism of vector spaces f:V\rightarrow W.

Two such isomorphisms, f and g, define the same homography if and only if there is a nonzero element a of K such that .

This may be written in terms of homogeneous coordinates in the following way: A homography φ may be defined by a nonsingular matrix [ai,j], called the matrix of the homography.

which generalizes the expression of the homographic function of the next section.

With this representation of the projective line, the homographies are the mappings.

which are called homographic functions or linear fractional transformations.