Translation (geometry)

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 Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '2069d235e97876cd085dcb676e9ec1fb' => (object) array( 'speakText' => 'upper A', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper A"><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">A</mi></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle A}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 750.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">upper A</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></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-41" x="0" y="0"></use></g></svg>', 'width' => '1.743ex', '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' => 'A', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="0">A</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.743ex; height:2.176ex;', 'sanetex' => '{\\displaystyle A}', 'speech' => 'upper A', ), )"): {\displaystyle T } . The translate of Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '2069d235e97876cd085dcb676e9ec1fb' => (object) array( 'speakText' => 'upper A', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper A"><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">A</mi></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle A}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 750.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">upper A</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></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-41" x="0" y="0"></use></g></svg>', 'width' => '1.743ex', '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' => 'A', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="0">A</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.743ex; height:2.176ex;', 'sanetex' => '{\\displaystyle A}', 'speech' => 'upper A', ), )"): {\displaystyle A } by $T_{\mathbf {v} }$ is often written Failed to parse (Conversion error. Server ("cli") reported: "array ( 'nohash' => array ( ), 'success' => true, '2069d235e97876cd085dcb676e9ec1fb' => (object) array( 'speakText' => 'upper A', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper A"><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">A</mi></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle A}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.743ex" height="2.176ex" style="vertical-align: -0.338ex;" viewBox="0 -791.3 750.5 936.9" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" aria-labelledby="MathJax-SVG-1-Title"><title id="MathJax-SVG-1-Title">upper A</title><defs aria-hidden="true"><path stroke-width="1" id="E1-MJMATHI-41" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></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-41" x="0" y="0"></use></g></svg>', 'width' => '1.743ex', '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' => 'A', ), ), 'streeXml' => '<stree><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="0">A</identifier></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.338ex; width:1.743ex; height:2.176ex;', 'sanetex' => '{\\displaystyle A}', 'speech' => 'upper A', ), )"): {\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

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