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- Homography abstract "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. It is a bijection that maps lines to lines, and thus a collineation. 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 perspective and projections in Euclidean geometry, and the term homography, which, etymologically, roughly means \"similar drawing\" date from this time. At the end of 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. 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 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. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); 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) field. Equivalently Pappus's hexagon theorem and Desargues' theorem 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 thumbnail Projection_geometry.svg?width=300.
- Homography wikiPageID "3051962".
- Homography wikiPageLength "29317".
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- Homography wikiPageRevisionID "706427107".
- Homography wikiPageWikiLink Affine_space.
- Homography wikiPageWikiLink Alternating_group.
- Homography wikiPageWikiLink Arthur_Cayley.
- Homography wikiPageWikiLink Automorphism.
- Homography wikiPageWikiLink Bijection.
- Homography wikiPageWikiLink Biquaternion.
- Homography wikiPageWikiLink Block_design.
- Homography wikiPageWikiLink Category:Projective_geometry.
- Homography wikiPageWikiLink Category:Transformation_(function).
- Homography wikiPageWikiLink Cengage_Learning.
- Homography wikiPageWikiLink Collineation.
- Homography wikiPageWikiLink Commutative_ring.
- Homography wikiPageWikiLink Conformal_symmetry.
- Homography wikiPageWikiLink Cross-ratio.
- Homography wikiPageWikiLink Desarguess_theorem.
- Homography wikiPageWikiLink Diagonalizable_matrix.
- Homography wikiPageWikiLink Division_algebra.
- Homography wikiPageWikiLink Dual_quaternion.
- Homography wikiPageWikiLink Eigenvalues_and_eigenvectors.
- Homography wikiPageWikiLink Electromagnetic_field.
- Homography wikiPageWikiLink Euclidean_geometry.
- Homography wikiPageWikiLink Euclidean_space.
- Homography wikiPageWikiLink Field_(mathematics).
- Homography wikiPageWikiLink Finite_field.
- Homography wikiPageWikiLink Function_composition.
- Homography wikiPageWikiLink General_linear_group.
- Homography wikiPageWikiLink Group_(mathematics).
- Homography wikiPageWikiLink Harold_Scott_MacDonald_Coxeter.
- Homography wikiPageWikiLink Homogeneous_coordinates.
- Homography wikiPageWikiLink Homography.
- Homography wikiPageWikiLink Hyperplane.
- Homography wikiPageWikiLink Hyperplane_at_infinity.
- Homography wikiPageWikiLink Identity_function.
- Homography wikiPageWikiLink Incidence_geometry.
- Homography wikiPageWikiLink Integer.
- Homography wikiPageWikiLink Inverse_function.
- Homography wikiPageWikiLink Invertible_matrix.
- Homography wikiPageWikiLink Isomorphism.
- Homography wikiPageWikiLink Line_(geometry).
- Homography wikiPageWikiLink Linear_algebra.
- Homography wikiPageWikiLink Modular_arithmetic.
- Homography wikiPageWikiLink Modular_group.
- Homography wikiPageWikiLink Möbius_transformation.
- Homography wikiPageWikiLink Oxford.
- Homography wikiPageWikiLink Oxford_University_Press.
- Homography wikiPageWikiLink Pappuss_hexagon_theorem.
- Homography wikiPageWikiLink Partial_function.
- Homography wikiPageWikiLink Patrick_du_Val.
- Homography wikiPageWikiLink Periodic_function.
- Homography wikiPageWikiLink Permutation.
- Homography wikiPageWikiLink Perspective_(visual).
- Homography wikiPageWikiLink Perspectivity.
- Homography wikiPageWikiLink Projection_(mathematics).
- Homography wikiPageWikiLink Projective_frame.
- Homography wikiPageWikiLink Projective_geometry.
- Homography wikiPageWikiLink Projective_line.
- Homography wikiPageWikiLink Projective_linear_group.
- Homography wikiPageWikiLink Projective_space.
- Homography wikiPageWikiLink Quaternionic_analysis.
- Homography wikiPageWikiLink Quotient_group.
- Homography wikiPageWikiLink Restriction_(mathematics).
- Homography wikiPageWikiLink Riemann_sphere.
- Homography wikiPageWikiLink Ring_(mathematics).
- Homography wikiPageWikiLink Screw_theory.
- Homography wikiPageWikiLink Simplex.
- Homography wikiPageWikiLink Synthetic_geometry.
- Homography wikiPageWikiLink Trace_(linear_algebra).
- Homography wikiPageWikiLink Unit_(ring_theory).
- Homography wikiPageWikiLink Up_to.
- Homography wikiPageWikiLink Vector_space.
- Homography wikiPageWikiLink W-curve.
- Homography wikiPageWikiLink File:Faisceaux_orthogonaux.gif.
- Homography wikiPageWikiLink File:Projection_geometry.svg.
- Homography wikiPageWikiLinkText "Homographies".
- Homography wikiPageWikiLinkText "Homography".
- Homography wikiPageWikiLinkText "Homography#Anti-homography".
- Homography wikiPageWikiLinkText "Homography#Fundamental theorem of projective geometry".
- Homography wikiPageWikiLinkText "Homography#Homographic function".
- Homography wikiPageWikiLinkText "fundamental theorem of projective geometry".
- Homography wikiPageWikiLinkText "group of homographies".
- Homography wikiPageWikiLinkText "homographic".
- Homography wikiPageWikiLinkText "homographies".
- Homography wikiPageWikiLinkText "homography".
- Homography wikiPageUsesTemplate Template:About.
- Homography wikiPageUsesTemplate Template:Anchor.
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- Homography wikiPageUsesTemplate Template:Commons_category.
- Homography wikiPageUsesTemplate Template:Main.
- Homography wikiPageUsesTemplate Template:Math.
- Homography wikiPageUsesTemplate Template:MathSciNet.
- Homography wikiPageUsesTemplate Template:Nobreak.
- Homography wikiPageUsesTemplate Template:Reflist.
- Homography wikiPageUsesTemplate Template:Section_link.
- Homography wikiPageUsesTemplate Template:See_also.