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geometric transformation wikipedia 88 captures 27 oct 2014 01 sep 2026 aug sep oct 18 2021 2022 2023 success fail about this capture timestamps the wayback machine http web archive org web 20220918070645 https en wikipedia org wiki geometric_transformation geometric transformation from wikipedia the free encyclopedia jump to navigation jump to search bijection of a set using properties of shapes in space not to be confused with transformation geometry for broader coverage of this topic see transformation mathematics in mathematics a geometric transformation is any bijection of a set to itself or to another such set with some salient geometrical underpinning more specifically it is a function whose domain and range are sets of points most often both r 2 displaystyle mathbb r 2 or both r 3 displaystyle mathbb r 3 such that the function is injective so that its inverse exists 1 the study of geometry may be approached via the study of these transformations 2 contents 1 classifications 2 opposite group actions 3 see also 4 references 5 further reading classifications edit geometric transformations can be classified by the dimension of their operand sets thus distinguishing between say planar transformations and spatial transformations they can also be classified according to the properties they preserve displacements preserve distances and oriented angles e g translations 3 isometries preserve angles and distances e g euclidean transformations 4 5 similarities preserve angles and ratios between distances e g resizing 6 affine transformations preserve parallelism e g scaling shear 5 7 projective transformations preserve collinearity 8 each of these classes contains the previous one 8 möbius transformations using complex coordinates on the plane as well as circle inversion preserve the set of all lines and circles but may interchange lines and circles original image based on the map of france isometry similarity affine transformation projective transformation inversion conformal transformations preserve angles and are in the first order similarities equiareal transformations preserve areas in the planar case or volumes in the three dimensional case 9 and are in the first order affine transformations of determinant 1 homeomorphisms bicontinuous transformations preserve the neighborhoods of points 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 10 conformal transformation equiareal transformation homeomorphism diffeomorphism transformations of the same type form groups that may be sub groups of other transformation groups opposite group actions edit main articles group action and opposite group many geometric transformations are expressed with linear algebra the bijective linear transformations are elements of a general linear group the linear transformation a is non singular for a row vector v the matrix product va gives another row vector w va the transpose of a row vector v is a column vector v t and the transpose of the above equality is w t v a t a t v t displaystyle w t va t a t v t here a t provides a left action on column vectors in transformation geometry there are compositions ab starting with a row vector v the right action of the composed transformation is w vab after transposition w t v a b t a b t v t b t a t v t displaystyle w t vab t ab t v t b t a t v t thus for ab the associated left group action is b t a t displaystyle b t a t in the study of opposite groups the distinction is made between opposite group actions for the only groups for which these opposites are equal are commutative groups see also edit coordinate transformation erlangen program symmetry geometry reflection rigid transformation rotation topology transformation matrix references edit zalman usiskin anthony l peressini elena marchisotto mathematics for high school teachers an advanced perspective page 84 venema gerard a 2006 foundations of geometry pearson prentice hall p 285 isbn 9780131437005 geometry translation www mathsisfun com retrieved 2020 05 02 geometric transformations euclidean transformations pages mtu edu retrieved 2020 05 02 a b geometric transformation p 131 at google books transformations www mathsisfun com retrieved 2020 05 02 geometric transformations affine transformations pages mtu edu retrieved 2020 05 02 a b leland wilkinson d wills d rope a norton r dubbs geometric transformation p 182 at google books geometric transformation p 191 at google books bruce e meserve fundamental concepts of geometry page 191 stevecheng 2013 03 13 first fundamental form pdf planetmath org retrieved 2014 10 01 further reading edit wikimedia commons has media related to transformations geometry adler irving 2012 1966 a new look at geometry dover isbn 978 0 486 49851 5 dienes z p golding e w 1967 geometry through transformations 3 vols geometry of distortion geometry of congruence and groups and coordinates new york herder and herder david gans transformations and geometries hilbert david cohn vossen stephan 1952 geometry and the imagination 2nd ed chelsea isbn 0 8284 1087 9 john mccleary geometry from a differentiable viewpoint modenov p s parkhomenko a s 1965 geometric transformations 2 vols euclidean and affine transformations and projective transformations new york academic press a n pressley elementary differential geometry yaglom i m 1962 1968 1973 2009 geometric transformations 4 vols random house i ii iii maa i ii iii iv retrieved from https en wikipedia org w index php title geometric_transformation oldid 1101590875 categories geometry functions and mappings symmetry transformation function hidden categories articles with short description short description is different from wikidata commons category link is on wikidata navigation menu personal tools not logged in talk contributions create account log in namespaces article talk english views read edit view history more search navigation main page contents current events random article about wikipedia contact us donate contribute help learn to edit community portal recent changes upload file tools what links here related changes upload file special pages permanent link page information cite this page wikidata item print export download as pdf printable version in other projects wikimedia commons languages العربية azərbaycanca bosanski català чӑвашла čeština cymraeg deutsch ελληνικά español esperanto français bahasa indonesia עברית latviešu magyar nederlands 日本語 polski português română русский simple english svenska தமிழ் ไทย українська 粵語 中文 edit links this page was last edited on 31 july 2022 at 20 59 utc text is available under the creative commons attribution sharealike license 3 0 additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about 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