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.

As a function

Note:This is a translation.

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, '8493c98e4ca7db8674c48bf0df019cb2' => (object) array( 'speakText' => 'upper T Subscript bold v', 'mml' => '<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="upper T Subscript bold v"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub data-semantic-type="subscript" data-semantic-role="latinletter" data-semantic-id="2" data-semantic-children="0,1"><mi data-semantic-type="identifier" data-semantic-role="latinletter" data-semantic-font="italic" data-semantic-annotation="clearspeak:simple" data-semantic-id="0" data-semantic-parent="2">T</mi><mrow class="MJX-TeXAtom-ORD"><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="1" data-semantic-parent="2">v</mi></mrow></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\\displaystyle T_{\\mathbf {v} }}</annotation></semantics></math>', 'svg' => '<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="2.588ex" height="2.509ex" style="vertical-align: -0.671ex;" viewBox="0 -791.3 1114.1 1080.4" 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 Subscript bold v</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><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-MJMATHI-54" x="0" y="0"></use> <use transform="scale(0.707)" xlink:href="#E1-MJMAINB-76" x="826" y="-213"></use></g></svg>', 'width' => '2.588ex', 'height' => '2.509ex', 'style' => 'vertical-align: -0.671ex;', 'streeJson' => (object) array( 'stree' => (object) array( 'type' => 'subscript', 'role' => 'latinletter', 'id' => '2', 'children' => array ( 0 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'italic', 'annotation' => 'clearspeak:simple', 'id' => '0', '$t' => 'T', ), 1 => (object) array( 'type' => 'identifier', 'role' => 'latinletter', 'font' => 'bold', 'annotation' => 'clearspeak:simple', 'id' => '1', '$t' => 'v', ), ), ), ), 'streeXml' => '<stree><subscript role="latinletter" id="2"><children><identifier role="latinletter" font="italic" annotation="clearspeak:simple" id="0">T</identifier><identifier role="latinletter" font="bold" annotation="clearspeak:simple" id="1">v</identifier></children></subscript></stree>', 'success' => true, 'log' => 'success', 'mathoidStyle' => 'vertical-align: -0.671ex; width:2.588ex; height:2.509ex;', 'sanetex' => '{\\displaystyle T_{\\mathbf {v} }}', 'speech' => 'upper T Subscript bold v', ), )"): {\displaystyle T } . The translate of $A$ by $T_{\mathbf {v} }$ is often written $A+\mathbf {v}$.

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

See also

• Advection
• Parallel transport

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.