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{{General geometry}}
In [[mathematics]], '''affine geometry''' is what remains of [[Euclidean geometry]] when not using (mathematicians often say "when forgetting") the [[metric space|metric]] notions of distance and angle.
As the notion of ''[[parallel lines]]'' is one of the main properties that is independent of any metric, affine geometry is often considered as the study of parallel lines. Therefore, [[Playfair's axiom]] (''given a line L and a point P not on L, there is exactly one line parallel to L that passes through P'') is fundamental in affine geometry. Comparisons of figures in affine geometry are made with [[affine transformation]]s, which are mappings that preserve alignment of points and parallelism of lines.
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: The interest of these five axioms is enhanced by the fact that they can be developed into a vast body of propositions, holding not only in [[Euclidean geometry]] but also in [[Minkowski geometry|Minkowski’s geometry]] of time and space (in the simple case of 1 + 1 dimensions, whereas the special theory of relativity needs 1 + 3). The extension to either Euclidean or Minkowskian geometry is achieved by adding various further axioms of orthogonality, etc<ref>Coxeter 1955, ''Affine plane'', p. 8</ref>
The various types of affine geometry correspond to what interpretation is taken for ''rotation''. Euclidean geometry corresponds to the [[rotation (mathematics)|ordinary idea of rotation]], while
===Ordered structure===
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===Ternary rings===
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The first [[non-Desarguesian plane]] was noted by [[David Hilbert]] in his ''Foundations of Geometry''.<ref>[[David Hilbert]], 1980 (1899). ''[http://www.gutenberg.org/files/17384/17384-pdf.pdf The Foundations of Geometry]'', 2nd ed., Chicago: Open Court, weblink from [[Project Gutenberg]], p. 74.</ref> The [[Moulton plane]] is a standard illustration. In order to provide a context for such geometry as well as those where [[Desargues theorem]] is valid, the concept of a ternary ring has been developed.
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==Affine transformations==
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Geometrically, affine transformations (affinities) preserve collinearity: so they transform parallel lines into parallel lines and preserve ratios of distances along parallel lines.
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==Affine space==
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Affine geometry can be viewed as the geometry of an [[affine space]] of a given dimension ''n'', coordinatized over a [[Field (mathematics)|field]] ''K''. There is also (in two dimensions) a combinatorial generalization of coordinatized affine space, as developed in [[synthetic geometry|synthetic]] [[finite geometry]]. In projective geometry, ''affine space'' means the complement of a [[hyperplane at infinity]] in a [[projective space]]. ''Affine space'' can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2''x'' − ''y'', ''x'' − ''y'' + ''z'', (''x'' + ''y'' + ''z'')/3, '''i'''''x'' + (1 − '''i''')''y'', etc.
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