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A geometric transformation is any bijection of a set having some geometric structure to itself or another such set. Specifically, "A geometric transformation is a function whose domain and range are sets of points. Most often the domain and range of a geometric transformation are both R2 or both R3. Often geometric transformations are required to be 1-1 functions, so that they have inverses." 〔Zalman Usiskin, Anthony L. Peressini, Elena Marchisotto – ''Mathematics for High School Teachers: An Advanced Perspective'', page 84.〕 The study of geometry may be approached via the study of these transformations. Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between planar transformations and those of space, for example). They can also be classified according to the properties they preserve: * displacements preserve distances and oriented angles; * isometries preserve distances and angles; * similarities preserve the ratios between distances; * affine transformations preserve parallelism;〔 * projective transformations preserve collinearity;〔Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – ''〕 Each of these classes contains the previous one.〔 *inversions preserve the set of all lines and circles in the planar case (but may interchange lines and circles), and Möbius transformations conserve all planes and spheres in dimension 3. France identique.gif | original image France par rotation.gif | isometry France par similitude.gif | similarity France affine (1).gif | affine transformation France homographie.gif | projective transformation France circ.gif | inversion * Diffeomorphisms (bidifferentiable transformations) are the transformations that are affine in the first order; they contain the preceding ones as special cases, and can be further refined. * Conformal transformations, preserving angles, are, in the first order, similarities. * equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case.〔 Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.]〕 and are, in the first order, affine transformations of determinant 1. *Homeomorphisms (bicontinuous transformations), preserve the neighborhoods of points. Fconf.gif | Conformal transformation France aire.gif | Equiareal transformation France diff.gif | Diffeomorphism France homothetie.gif | Homeomorphism Transformations of the same type form groups that may be sub-groups of other transformation groups. == See also == * Erlangen program * Topology * Rigid transformation 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「geometric transformation」の詳細全文を読む スポンサード リンク
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