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In mathematics, affine geometry is what remains of Euclidean geometry, when not using (mathematicians often say "when forgetting") the 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 transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent.〔 ''(Reprint of the 1957 original; A Wiley-Interscience Publication)''〕 In synthetic geometry, an affine space is a set of ''points'' to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of ''points'' equipped with a set of ''transformations'' (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations. In more concrete terms, this amounts to having an operation that associates to any ordered pair of points a vector and another operation that allows translation of a point by a vector to give another point; these operations are required to satisfy a number of axioms (notably that two successive translations have the effect of translation by the sum vector). By choosing any point as "origin", the points are in one-to-one correspondence with the vectors, but there is no preferred choice for the origin; thus an affine space may be viewed as obtained from its associated vector space by "forgetting" the origin (zero vector). ==History== In 1748, Euler introduced the term ''affine''〔(【引用サイトリンク】 last=Miller )〕 (Latin ''affinis'', "related") in his book ''Introductio in analysin infinitorum'' (volume 2, chapter XVIII). In 1827, August Möbius wrote on affine geometry in his ''Der barycentrische Calcul'' (chapter 3). After Felix Klein's Erlangen program, affine geometry was recognized as a generalization of Euclidean geometry. In 1912, Edwin B. Wilson and Gilbert N. Lewis developed an affine geometry〔Edwin B. Wilson & Gilbert N. Lewis (1912). "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics", Proceedings of the American Academy of Arts and Sciences 48:387–507〕〔(Synthetic Spacetime ), a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite〕 to express the special theory of relativity. In 1918, Hermann Weyl referred to affine geometry for his text ''Space, Time, Matter''. He used affine geometry to introduce vector addition and subtraction〔Hermann Weyl (1918)''(Raum, Zeit, Materie )''. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 ''(Space Time Matter )'', Methuen, rept. 1952 Dover. ISBN 0-486-60267-2 . See Chapter 1 §2 Foundations of Affine Geometry, pp 16–27〕 at the earliest stages of his development of mathematical physics. Later, E. T. Whittaker wrote:〔E. T. Whittaker (1958). ''From Euclid to Eddington: a study of conceptions of the external world'', Dover Publications, p. 130.〕 : Weyl's geometry is interesting historically as having been the first of the affine geometries to be worked out in detail: it is based on a special type of parallel transport () worldlines of light-signals in four-dimensional space-time. A short element of one of these world-lines may be called a ''null-vector''; then the parallel transport in question is such that it carries any null-vector at one point into the position of a null-vector at a neighboring point. In 1984, "the affine plane associated to the Lorentzian vector space ''L''2 " was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry".〔Graciela S. Birman & Katsumi Nomizu (1984). "Trigonometry in Lorentzian geometry", American Mathematical Monthly 91(9):543–9, Lorentzian affine plane: p. 544〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Affine geometry」の詳細全文を読む スポンサード リンク
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