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Geometric transformation
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<h2 id="classifications">Classifications</h2> <p>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: </p> <ul><li><a href="/facts/Displacement_(vector)/jkmEfS3C">Displacements</a> preserve <a href="/facts/Distance/hbXlaEFk">distances</a> and <a href="/facts/Angle/gFUTEQYq">oriented angles</a> (e.g., <a href="/facts/Translation_(geometry)/KlpWmnfX">translations</a>);<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li><a href="/facts/Isometry/K5wX8ah6">Isometries</a> preserve angles and distances (e.g., <a href="/facts/Euclidean_transformation/0KzWYQXl">Euclidean transformations</a>);<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li><a href="/facts/Similarity_(geometry)/Sbd6hEAN">Similarities</a> preserve angles and ratios between distances (e.g., resizing);<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a></li> <li><a href="/facts/Affine_transformation/BD7yDr10">Affine transformations</a> preserve <a href="/facts/Parallel_(geometry)/16r5yq7D">parallelism</a> (e.g., <a href="/facts/Scaling_(geometry)/pFm6fvpu">scaling</a>, <a href="/facts/Shear_mapping/HVCYblfU">shear</a>);<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li> <li><a href="/facts/Projective_transformation/wCkNzUjd">Projective transformations</a> preserve <a href="/facts/Collinearity/SxbVBJYR">collinearity</a>;<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li></ul> <p>Each of these classes contains the previous one.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <ul><li><a href="/facts/M%25C3%25B6bius_transformation/YEkTTLfK">Möbius transformations</a> using complex coordinates on the plane (as well as <a href="/facts/Circle_inversion/VqFmbfDI">circle inversion</a>) preserve the set of all lines and circles, but may interchange lines and circles.</li></ul> <ul><li><a href="/facts/Conformal_transformation/cEpQnMad">Conformal transformations</a> preserve angles, and are, in the first order, similarities.</li> <li><a href="/facts/Equiareal_map/5MFDDkUr">Equiareal transformations</a>, preserve areas in the planar case or volumes in the three dimensional case.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> and are, in the first order, affine transformations of <a href="/facts/Determinant/eGYqJ2TF">determinant</a> 1.</li> <li><a href="/facts/Homeomorphism/jMfWg3Mn">Homeomorphisms</a> (bicontinuous transformations) preserve the neighborhoods of points.</li> <li><a href="/facts/Diffeomorphism/855N5YYj">Diffeomorphisms</a> (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.</li></ul> <p>Transformations of the same type form <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a> that may be sub-groups of other transformation groups. </p> <h2 id="opposite-group-actions">Opposite group actions</h2> <p class="note">Main articles: <a href="/facts/Group_action/Vh9VtUUw">Group action</a> and <a href="/facts/Opposite_group/MvwyDGt9">Opposite group</a></p> <p>Many geometric transformations are expressed with linear algebra. The bijective linear transformations are elements of a <a href="/facts/General_linear_group/B54gNILS">general linear group</a>. The <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> <i>A</i> is non-singular. For a <a href="/facts/Row_vector/pPuRr9Xk">row vector</a> <i>v</i>, the <a href="/facts/Matrix_product/lYDB6Ro2">matrix product</a> <i>vA</i> gives another row vector <i>w</i> = <i>vA</i>. </p><p>The <a href="/facts/Transpose/8wmsagGS">transpose</a> of a row vector <i>v</i> is a column vector <i>v</i>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 <i>A</i>T provides a left action on column vectors. </p><p>In transformation geometry there are <a href="/facts/Composition_of_relations/tD8joBlo">compositions</a> <i>AB</i>. Starting with a row vector <i>v</i>, the right action of the composed transformation is <i>w</i> = <i>vAB</i>. After transposition, </p> 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}.} <p>Thus for <i>AB</i> the associated left <a href="/facts/Group_action/Vh9VtUUw">group action</a> is B T A T . {\displaystyle B^{T}A^{T}.} In the study of <a href="/facts/Opposite_group/MvwyDGt9">opposite groups</a>, the distinction is made between opposite group actions because <a href="/facts/Commutative_group/FfWwmx4R">commutative groups</a> are the only groups for which these opposites are equal. </p> <h2 id="active-and-passive-transformations">Active and passive transformations</h2> <p class="note">This section is an excerpt from <a href="/facts/Active_and_passive_transformation/pxF9AaJ3">Active and passive transformation</a>.[<a href="https://en.wikipedia.org/w/index.php?title=Active_and_passive_transformation&action=edit">edit</a>]</p> <p>Geometric transformations can be distinguished into two types: <a href="/facts/Active_and_passive_transformation/pxF9AaJ3">active</a> or alibi transformations which change the physical position of a set of <a href="/facts/Point_(geometry)/6YPAvX2a">points</a> relative to a fixed <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> or <a href="/facts/Coordinate_system/Mtd0q4SP">coordinate system</a> (<i><a href="/facts/Alibi/4Yf2qtrP">alibi</a></i> meaning "being somewhere else at the same time"); and passive or alias transformations which leave points fixed but change the frame of reference or coordinate system relative to which they are described (<i><a href="/facts/Pseudonym/A5EjmWxQ">alias</a></i> meaning "going under a different name").<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> By <i>transformation</i>, <a href="/facts/Mathematician/eaFSR8Fx">mathematicians</a> usually refer to active transformations, while <a href="/facts/Physicist/cQcgvSIm">physicists</a> and <a href="/facts/Engineer/ph4TnHnG">engineers</a> could mean either. </p><p>For instance, active transformations are useful to describe successive positions of a <a href="/facts/Rigid_body/THezBlq2">rigid body</a>. On the other hand, passive transformations may be useful in human motion analysis to observe the motion of the <a href="/facts/Tibia/YXQ0F48L">tibia</a> relative to the <a href="/facts/Femur/uVyTwj4l">femur</a>, that is, its motion relative to a (<i>local</i>) coordinate system which moves together with the femur, rather than a (<i>global</i>) coordinate system which is fixed to the floor.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>In <a href="/facts/Three-dimensional_Euclidean_space/KAqKWUbS">three-dimensional Euclidean space</a>, any <a href="/facts/Rigid_transformation/0KzWYQXl">proper rigid transformation</a>, whether active or passive, can be represented as a <a href="/facts/Screw_displacement/Cos1gnJb">screw displacement</a>, the composition of a <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a> along an axis and a <a href="/facts/Rotation_(mathematics)/6gtkkcy0">rotation</a> about that axis. </p> The terms <i>active transformation</i> and <i>passive transformation</i> were first introduced in 1957 by <a href="/facts/Valentine_Bargmann/U8RzbDIr">Valentine Bargmann</a> for describing <a href="/facts/Lorentz_transformations/jotJL06G">Lorentz transformations</a> in <a href="/facts/Special_relativity/vsFFc63E">special relativity</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Coordinate_transformation/Mtd0q4SP">Coordinate transformation</a></li> <li><a href="/facts/Erlangen_program/HAVoCLHG">Erlangen program</a></li> <li><a href="/facts/Symmetry_(geometry)/TljpwtNQ">Symmetry (geometry)</a></li> <li><a href="/facts/Motion_(geometry)/x5zUzg7l">Motion</a></li> <li><a href="/facts/Reflection_(mathematics)/5JHWnrql">Reflection</a></li> <li><a href="/facts/Rigid_transformation/0KzWYQXl">Rigid transformation</a></li> <li><a href="/facts/Rotation_(mathematics)/6gtkkcy0">Rotation</a></li> <li><a href="/facts/Topology/2EqW47cU">Topology</a></li> <li><a href="/facts/Transformation_matrix/qAwPXmpR">Transformation matrix</a></li></ul> <h2 id="further-reading">Further reading</h2> Wikimedia Commons has media related to Transformations (geometry). <ul><li><a href="/facts/Irving_Adler/Av2ehyBj">Adler, Irving</a> (2012) [1966], <i>A New Look at Geometry</i>, Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-49851-5</li> <li><a href="/facts/Zolt%25C3%25A1n_P%25C3%25A1l_Dienes/8eg6xVUd">Dienes, Z. P.</a>; Golding, E. W. (1967) . <i>Geometry Through Transformations</i> (3 vols.): <i>Geometry of Distortion</i>, <i>Geometry of Congruence</i>, and <i>Groups and Coordinates</i>. New York: Herder and Herder.</li> <li><a href="/facts/David_Gans/pya2Nuwr">David Gans</a> – <i>Transformations and geometries</i>.</li> <li><a href="/facts/David_Hilbert/1vhlHn4b">Hilbert, David</a>; <a href="/facts/Stephan_Cohn-Vossen/mUfH8SzR">Cohn-Vossen, Stephan</a> (1952). <i>Geometry and the Imagination</i> (2nd ed.). Chelsea. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8284-1087-9. {{cite book}}: ISBN / Date incompatibility (help)</li> <li>John McCleary (2013) <i>Geometry from a Differentiable Viewpoint</i>, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a> <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-11607-7</li> <li>Modenov, P. S.; Parkhomenko, A. S. (1965) . <i>Geometric Transformations</i> (2 vols.): <i>Euclidean and Affine Transformations</i>, and <i>Projective Transformations</i>. New York: Academic Press.</li> <li>A. N. Pressley – <i>Elementary Differential Geometry</i>.</li> <li><a href="/facts/Isaak_Yaglom/5f6jeMN4">Yaglom, I. M.</a> (1962, 1968, 1973, 2009) . <i>Geometric Transformations</i> (4 vols.). <a href="/facts/Random_House/NcR9r1F1">Random House</a> (I, II & III), <a href="/facts/Mathematical_Association_of_America/G3BkG5YX">MAA</a> (I, II, III & IV).</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Usiskin, Zalman; Peressini, Anthony L.; Marchisotto, Elena; Stanley, Dick (2003). Mathematics for High School Teachers: An Advanced Perspective. Pearson Education. p. 84. ISBN 0-13-044941-5. OCLC 50004269. <a href="0-13-044941-5" target="_blank">0-13-044941-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Venema, Gerard A. (2006), Foundations of Geometry, Pearson Prentice Hall, p. 285, ISBN 9780131437005 <a href="9780131437005" target="_blank">9780131437005</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"Geometry Translation". www.mathsisfun.com. Retrieved 2020-05-02. <a href="https://www.mathsisfun.com/geometry/translation.html" target="_blank">https://www.mathsisfun.com/geometry/translation.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>"Geometric Transformations — Euclidean Transformations". pages.mtu.edu. Retrieved 2020-05-02. <a href="https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#euclidean" target="_blank">https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#euclidean</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Geometric transformation, p. 131, at Google Books <a href="https://books.google.com/books?id=pN0iAVavPR8C&pg=PA131" target="_blank">https://books.google.com/books?id=pN0iAVavPR8C&pg=PA131</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Transformations". www.mathsisfun.com. Retrieved 2020-05-02. <a href="https://www.mathsisfun.com/geometry/transformations.html" target="_blank">https://www.mathsisfun.com/geometry/transformations.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Geometric transformation, p. 131, at Google Books <a href="https://books.google.com/books?id=pN0iAVavPR8C&pg=PA131" target="_blank">https://books.google.com/books?id=pN0iAVavPR8C&pg=PA131</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Geometric Transformations — Affine Transformations". pages.mtu.edu. Retrieved 2020-05-02. <a href="https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#affine" target="_blank">https://pages.mtu.edu/~shene/COURSES/cs3621/NOTES/geometry/geo-tran.html#affine</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – 'Geometric transformation, p. 182, at Google Books <a href="https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA182" target="_blank">https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA182</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Leland Wilkinson, D. Wills, D. Rope, A. Norton, R. Dubbs – 'Geometric transformation, p. 182, at Google Books <a href="https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA182" target="_blank">https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA182</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Geometric transformation, p. 191, at Google Books Bruce E. Meserve – Fundamental Concepts of Geometry, page 191.] <a href="https://books.google.com/books?id=Y6jDAgAAQBAJ&pg=PA191" target="_blank">https://books.google.com/books?id=Y6jDAgAAQBAJ&pg=PA191</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Crampin, M.; Pirani, F.A.E. (1986). Applicable Differential Geometry. Cambridge University Press. p. 22. ISBN 978-0-521-23190-9. <a href="978-0-521-23190-9" target="_blank">978-0-521-23190-9</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Joseph K. Davidson, Kenneth Henderson Hunt (2004). "§4.4.1 The active interpretation and the active transformation". Robots and screw theory: applications of kinematics and statics to robotics. Oxford University Press. p. 74 ff. ISBN 0-19-856245-4. <a href="0-19-856245-4" target="_blank">0-19-856245-4</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Joseph K. Davidson, Kenneth Henderson Hunt (2004). "§4.4.1 The active interpretation and the active transformation". Robots and screw theory: applications of kinematics and statics to robotics. Oxford University Press. p. 74 ff. ISBN 0-19-856245-4. <a href="0-19-856245-4" target="_blank">0-19-856245-4</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Bargmann, Valentine (1957). "Relativity". Reviews of Modern Physics. 29 (2): 161–174. Bibcode:1957RvMP...29..161B. doi:10.1103/RevModPhys.29.161. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>