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[[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.
 
==As a function==
{{see also|Displacement (geometry)}}


== As a function ==
== 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)
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>.


 
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>.
<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>


=== Horizontal and vertical translations ===
=== 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.
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==
==See also==
* [[Advection]]
* [[Advection]]
* [[Parallel transport]]
* [[Parallel transport]]
* [[Rotation matrix]]
* [[Scaling (geometry)]]
* [[Transformation matrix]]
* [[Translational symmetry]]


==External links==
==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://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==
{{DEFAULTSORT:Translation (Geometry)}}
 
[[Category:Euclidean symmetries]]

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 is a fixed vector, known as the translation vector, and Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle \mathbf{p}} is the initial position of some object, then the translation function Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle T_{\mathbf{v}} } will work as .

If Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle T} is a translation, then the image of a subset Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle A } under the function Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle T} is the translate of Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle A } by Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle T } . The translate of Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle A } by Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\displaystyle T_{\mathbf{v}} } is often written Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '98b434d091ae15b7180667a0296aae31' => (object) array( 'speakText' => 'bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold" data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="bold" data-semantic-annotation="clearspeak:simple" data-semantic-id="0">v</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle \\mathbf {v} }</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.411ex" height="1.676ex" style="vertical-align: -0.338ex;" viewBox="0 -576.1 607.5 721.6" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">bold v</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMAINB-76" d="M401 444Q413 441 495 441Q568 441 574 444H580V382H510L409 156Q348 18 339 6Q331 -4 320 -4Q318 -4 313 -4T303 -3H288Q273 -3 264 12T221 102Q206 135 197 156L96 382H26V444H34Q49 441 145 441Q252 441 270 444H279V382H231L284 264Q335 149 338 149Q338 150 389 264T442 381Q442 382 418 382H394V444H401Z"></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-MJMAINB-76" x="0" y="0"></use></g></svg>', 'width' => '1.411ex', 'height' => '1.676ex', 'style' => 'vertical-align: -0.338ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'v', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="0">v</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.411ex; height:1.676ex;', 'sanetex' => '{\\displaystyle \\mathbf {v} }', 'speech' => 'bold v', ), )"): {\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

Translation (geometry)-50289677

Attachments

See also

External links


References

  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
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