Test:Translation (geometry): Difference between revisions

Latest revision as of 11:21, 16 September 2022

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.^[1]

As a function

Note:This is a translation.

translation graphic — A translation moves every point of a figure or a space by the same amount in a given direction.

See also: Displacement (geometry)

If $\mathbf {v}$ is a fixed vector, known as the translation vector, and $\mathbf {p}$ is the initial position of some object, then the translation function $T_{\mathbf {v} }$ will work as $T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v}$ .

If $T$ is a translation, then the image of a subset $A$ under the function $T$ is the translate of $A$ by $T$ . The translate of Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '85ec0115befc9770913b93e5e28ad6f8' => (object) array( 'speakText' => 'upper T', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper T"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">T</mi></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle T}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.636ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 704.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">upper T</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true"> <use xlink:href="#E1-MJMATHI-54" x="0" y="0"></use></g></svg>', 'width' => '1.636ex', 'height' => '2.176ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'T', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="0">T</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.636ex; height:2.176ex;', 'sanetex' => '{\\displaystyle T}', 'speech' => 'upper T', ), )"): {\displaystyle A } by $T_{\mathbf {v} }$ is often written Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '85ec0115befc9770913b93e5e28ad6f8' => (object) array( 'speakText' => 'upper T', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper T"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">T</mi></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle T}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.636ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 704.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">upper T</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMATHI-54" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)" aria-hidden="true"> <use xlink:href="#E1-MJMATHI-54" x="0" y="0"></use></g></svg>', 'width' => '1.636ex', 'height' => '2.176ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'T', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="0">T</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.636ex; height:2.176ex;', 'sanetex' => '{\\displaystyle T}', 'speech' => 'upper T', ), )"): {\displaystyle A+\mathbf{v} } .

Horizontal and vertical translations

In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system.

Drawio

Attachments

External links

Translation Transform
Geometric Translation (Interactive Animation) at Math Is Fun
Understanding 2D Translation and Understanding 3D Translation by Roger Germundsson, The Wolfram Dmonstrations Project.

References

↑ Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmath

[1] Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmath

[1]

@@ Line 1: / Line 1: @@
-[[File:Test:Traslazione OK.svg|alt=translation graphic|thumb|<span style="color: rgb(32, 33, 34)">A translation moves every point of a figure or a space by the same amount in a given direction.</span>]]
+In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.<ref>Zazkis, R., Liljedahl, P., & Gadowsky, K. Conceptions of function translation: obstacles, intuitions, and rerouting. Journal of Mathematical Behavior, 22, 437-450. Retrieved April 29, 2014, from www.elsevier.com/locate/jmath</ref>
-{{Short description|Planar movement within a Euclidean space without rotation}}
-In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry.<ref>{{citation|title=Single Variable Calculus: Early Transcendentals|first1=Dennis|last1=Zill|first2=Warren S.|last2=Wright|publisher=Jones & Bartlett Learning|year=2009|isbn=9780763749651|page=269|url=https://books.google.com/books?id=0n0iPYKLo74C&pg=PA269}}.</ref>
-==As a function==
-{{see also|Displacement (geometry)}}
 == As a function ==
+{{Messagebox|boxtype=note|icon=|Note text=This is a translation.|bgcolor=}}[[File:Test:Traslazione OK.svg|alt=translation graphic|thumb|<span style="color: rgb(32, 33, 34)">A translation moves every point of a figure or a space by the same amount in a given direction.</span>]]
 See also: Displacement (geometry)
-<nowiki>If {\displaystyle \mathbf {v} }{\displaystyle \mathbf {v} } is a fixed vector, known as the translation vector, and {\displaystyle \mathbf {p} }\mathbf {p}  is the initial position of some object, then the translation function {\displaystyle T_{\mathbf {v} }}{\displaystyle T_{\mathbf {v} }} will work as {\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }{\displaystyle T_{\mathbf {v} }(\mathbf {p} )=\mathbf {p} +\mathbf {v} }.</nowiki>
+If <math>\mathbf{v} </math> is a fixed vector, known as the ''translation vector'', and <math>\mathbf{p}</math> is the initial position of some object, then the translation function <math>T_{\mathbf{v}} </math> will work as <math> T_{\mathbf{v}}(\mathbf{p})=\mathbf{p}+\mathbf{v}</math>.
-<nowiki>If {\displaystyle T} T is a translation, then the image of a subset {\displaystyle A}A under the function {\displaystyle T} T is the translate of {\displaystyle A}A by {\displaystyle T}T. The translate of {\displaystyle A}A by {\displaystyle T_{\mathbf {v} }}{\displaystyle T_{\mathbf {v} }} is often written {\displaystyle A+\mathbf {v} }{\displaystyle A+\mathbf {v} }.</nowiki>
+If <math> T</math> is a translation, then the image of a subset <math> A </math> under the function <math> T</math> is the '''translate''' of <math> A </math> by <math> T </math>. The translate of <math>A </math> by <math>T_{\mathbf{v}} </math> is often written <math>A+\mathbf{v} </math>.
 === Horizontal and vertical translations ===
 In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system.
+== Drawio ==
+<bs:drawio filename="Translation (geometry)-50289677" />
+== Attachments ==
+<attachments>
+* [[Media:Rasperry_pi.pdf]]
+</attachments>
 ==See also==
 * [[Advection]]
 * [[Parallel transport]]
-* [[Rotation matrix]]
-* [[Scaling (geometry)]]
-* [[Transformation matrix]]
-* [[Translational symmetry]]
 ==External links==
-{{Commons category|Translation (geometry)}}
-* [http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform] at [[cut-the-knot]]
+* [http://www.cut-the-knot.org/Curriculum/Geometry/Translation.shtml Translation Transform]
 * [http://www.mathsisfun.com/geometry/translation.html Geometric Translation (Interactive Animation)] at Math Is Fun
-* [http://demonstrations.wolfram.com/Understanding2DTranslation/ Understanding 2D Translation] and [http://demonstrations.wolfram.com/Understanding3DTranslation/ Understanding 3D Translation] by Roger Germundsson, [[The Wolfram Demonstrations Project]].
+* [http://demonstrations.wolfram.com/Understanding2DTranslation/ Understanding 2D Translation] and [http://demonstrations.wolfram.com/Understanding3DTranslation/ Understanding 3D Translation] by Roger Germundsson, The Wolfram Dmonstrations Project.
 ==References==
-{{reflist}}
 {{DEFAULTSORT:Translation (Geometry)}}
 [[Category:Euclidean symmetries]]