Homography

In <a href="/facts/Projective_geometry/rHKSaEbx">projective geometry</a>, a homography is an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> of <a href="/facts/Projective_space/7MSpA9cM">projective spaces</a>, induced by an isomorphism of the <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> from which the projective spaces derive. It is a <a href="/facts/Bijection/j4gTuTmW">bijection</a> that maps <a href="/facts/Line_(geometry)/gJqsr4Gj">lines</a> to lines, and thus a <a href="/facts/Collineation/bNlLV50U">collineation</a>. 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. Synonyms include projectivity, projective transformation, and projective collineation.
Historically, homographies (and projective spaces) have been introduced to study <a href="/facts/Perspectivity/RMwmQSmV">perspective</a> and <a href="/facts/Projection_(mathematics)/QxxC9XTp">projections</a> in <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a>, and the term homography, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which extended <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean</a> and <a href="/facts/Affine_space/tlvI3oX1">affine spaces</a> by the addition of new points called <a href="/facts/Points_at_infinity/GAuS0wHQ">points at infinity</a>. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over a given <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> (the above definition is based on this version); this construction facilitates the definition of <a href="/facts/Projective_coordinates/TG9ebKN4">projective coordinates</a> and allows using the tools of <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (<a href="/facts/Incidence_geometry/F1yPfCGu">incidence geometry</a>, see also <a href="/facts/Synthetic_geometry/IRL0B4K7">synthetic geometry</a>); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative) <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. Equivalently <a href="/facts/Pappus%27s_hexagon_theorem/t0Hb33dG">Pappus's hexagon theorem</a> and <a href="/facts/Desargues%27s_theorem/vJzx0uPh">Desargues's theorem</a> are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.

Homography open-in-new

Homography