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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;"><div class="noviewer noresize bksicon" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" aria-hidden="true" role="presentation"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></span></span></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> <i>Koordinate</i> ist eine Weiterleitung auf diesen Artikel. Ein ehemaliges litauisches Unternehmen hieß <a href="/wiki/Koordinat%C4%97" title="Koordinatė">Koordinatė</a>.</div> </div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Koordinaten-1d2d.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Koordinaten-1d2d.svg/260px-Koordinaten-1d2d.svg.png" decoding="async" width="260" height="327" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Koordinaten-1d2d.svg/390px-Koordinaten-1d2d.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Koordinaten-1d2d.svg/520px-Koordinaten-1d2d.svg.png 2x" data-file-width="295" data-file-height="371" /></a><figcaption>Zahlenstrahl (oben), ebene kartesische Koordinaten (unten)</figcaption></figure> <div class="tright"> <div class="thumbinner" style="width:254px;"> <table class="hintergrundfarbe-basis thumbimage" style="width:250px; border:none; border-spacing:0; text-align:center; line-height:100%;"> <tbody><tr style="line-height:125%;"> <td style="width:6%"> </td> <td style="width:11%">a</td> <td style="width:11%">b</td> <td style="width:11%">c</td> <td style="width:11%">d</td> <td style="width:11%">e</td> <td style="width:11%">f</td> <td style="width:11%">g</td> <td style="width:11%">h</td> <td style="width:6%"> </td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">8</td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/26px-Chess_--t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/39px-Chess_--t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Chess_--t45.svg/52px-Chess_--t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">8</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">7</td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">7</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">6</td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">6</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">5</td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">5</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">4</td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">4</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">3</td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">3</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">2</td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">2</td> </tr><tr style="background:#FFCE9E; color:#202122;"> <td class="hintergrundfarbe-basis" style="padding:0 4px;">1</td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td style="background:#D18B47; color:#202122;"><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/26px-Chess_t45.svg.png" decoding="async" width="26" height="26" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/39px-Chess_t45.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Chess_t45.svg/52px-Chess_t45.svg.png 2x" data-file-width="45" data-file-height="45" /></span></span></td> <td class="hintergrundfarbe-basis" style="padding:0 4px;">1</td> </tr><tr style="line-height:125%;"> <td style="width:6%"> </td> <td style="width:11%">a</td> <td style="width:11%">b</td> <td style="width:11%">c</td> <td style="width:11%">d</td> <td style="width:11%">e</td> <td style="width:11%">f</td> <td style="width:11%">g</td> <td style="width:11%">h</td> <td style="width:6%"> </td> </tr> </tbody></table> <div class="thumbcaption"> <p>Die Felder des Schachbretts werden mit einem Zahlen-Buchstaben-Paar bezeichnet. </p> </div> </div> </div> <p>Ein <b>Koordinatensystem</b> dient dazu, <a href="/wiki/Punkt_(Geometrie)" title="Punkt (Geometrie)">Punkte</a> mit Hilfe von Zahlen, den <b>Koordinaten</b>, in eindeutiger Weise zu beschreiben. Die einfachsten Beispiele sind ein <a href="/wiki/Zahlenstrahl" class="mw-redirect" title="Zahlenstrahl">Zahlenstrahl</a> und <a href="/wiki/Kartesische_Koordinaten" class="mw-redirect" title="Kartesische Koordinaten">kartesische Koordinaten</a> in der <a href="/wiki/Euklidische_Ebene" class="mw-redirect" title="Euklidische Ebene">Ebene</a>. Im ersten Fall wird einem Punkt einer Gerade eine <a href="/wiki/Reelle_Zahl" title="Reelle Zahl">reelle Zahl</a> zugeordnet. Im zweiten Fall wird ein Punkt in der Ebene durch zwei reelle Zahlen beschrieben. </p><p>Bei <a href="/wiki/Volumen" title="Volumen">räumlichen</a> Gebilden sind drei Koordinaten erforderlich, bei <a href="/wiki/Raum_(Physik)" title="Raum (Physik)">raum</a>-<a href="/wiki/Zeit" title="Zeit">zeitlichen</a> Gebilden vier. </p><p>Die Position eines Punktes im Raum kann in verschiedenen Koordinatensystemen dargestellt werden. Dabei wird die Position durch Koordinaten ausgedrückt. Je nach verwendetem Koordinatensystem hat derselbe Punkt unterschiedliche Koordinatenwerte. </p><p>Der Begriff <i>Koordinate</i> – in der Bedeutung „Lageangabe“ – wurde im 18. Jahrhundert aus dem Wort <i><a href="/wiki/Ordinate" class="mw-redirect" title="Ordinate">Ordinate</a></i> (Senkrechte) gebildet.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Koordinaten werden in verschiedenen Bereichen der <a href="/wiki/Mathematik" title="Mathematik">Mathematik</a> und <a href="/wiki/Physik" title="Physik">Physik</a> unterschiedlich bezeichnet. So heißen die Koordinaten eines Elements (<a href="/wiki/Vektor" title="Vektor">Vektors</a>) eines endlichdimensionalen <a href="/wiki/Vektorraum" title="Vektorraum">Vektorraums</a> seine Komponenten, die Koordinaten in einem <a href="/wiki/Mengenprodukt" class="mw-redirect" title="Mengenprodukt">Produkt von Mengen</a> sind die <a href="/wiki/Projektion_(Mengenlehre)" title="Projektion (Mengenlehre)">Projektionen</a> auf einen der Faktoren. Oft gibt es zahlreiche Möglichkeiten, ein Koordinatensystem einzuführen. Beim Beispiel des Zahlenstrahls hat man beliebig viele Möglichkeiten einen Punkt auszuwählen, dem die Koordinate 0 zugeordnet werden soll. In der Ebene ist die Situation sogar noch komplizierter. Selbst nach Wahl eines Punktes, der die Koordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> erhält, lässt sich jedes (verschiedene) Paar von Zahlenstrahlen durch diesen Punkt als <a href="/wiki/Koordinatenachse" title="Koordinatenachse">Koordinatenachsen</a> wählen. </p><p>Je nach Beschaffenheit der Menge, auf der man ein Koordinatensystem wählen möchte, benötigt man auch mehr als ein oder zwei Koordinaten. Die <a href="/wiki/Geordnete_Menge" class="mw-redirect" title="Geordnete Menge">geordnete Menge</a> der Koordinaten wird meist als ein <a href="/wiki/N-Tupel" class="mw-redirect" title="N-Tupel">n-Tupel</a> bezeichnet. Der Punkt des Zahlenstrahls mit der Koordinate 0 und der Punkt der Ebene mit den Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d630d3e781a53b0a3559ae7e5b45f9479a3141a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (0,0)}"></span> beziehungsweise der ausgezeichnete Punkt einer Menge, dessen Koordinaten alle 0 sind, wird als <b>Koordinatenursprung</b> (kurz: <i>Ursprung</i>) bezeichnet. </p><p>Neben den weit verbreiteten kartesischen (rechtwinkligen) Koordinaten gibt es auch andere Arten, Koordinatensysteme zu definieren. Möchte man beispielsweise auf der <a href="/wiki/Kreis" title="Kreis">Kreisfläche</a> Koordinaten einführen, so würden sich <a href="/wiki/Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a> anbieten. Der Kreismittelpunkt wäre dann der Ursprung und jeder Punkt der Kreisfläche würde durch Angabe der Entfernung vom Mittelpunkt und eines Winkels eindeutig beschrieben. In diesem Fall lässt sich im Vergleich zu den kartesischen Koordinaten nur eine der beiden Koordinaten als Länge interpretieren. Ein anderes Beispiel ist das eines <a href="/wiki/Schachbrett" title="Schachbrett">Schachbrettes</a>. Hier wird eine Kombination aus Buchstaben und <a href="/wiki/Nat%C3%BCrliche_Zahl" title="Natürliche Zahl">natürlichen Zahlen</a> genutzt, um die Felder des Brettes zu benennen. </p><p>In vielen Situationen ist es unmöglich, hinreichend sinnvolle und bequeme globale Koordinaten auf der gesamten Menge einzuführen. Zum Beispiel können die Punkte einer <a href="/wiki/Kugel" title="Kugel">Kugeloberfläche</a>, anders als die einer Ebene, nicht in eine kontinuierliche Eins-zu-Eins-Korrespondenz mit Zahlenpaaren gebracht werden. Daher wurde das Konzept der <a href="/w/index.php?title=Lokale_Koordinaten&action=edit&redlink=1" class="new" title="Lokale Koordinaten (Seite nicht vorhanden)">lokalen Koordinaten</a> eingeführt. Dies ist zum Beispiel die Situation in der Theorie der <a href="/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeiten</a>. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Gebräuchliche_Koordinatensysteme"><span class="tocnumber">1</span> <span class="toctext">Gebräuchliche Koordinatensysteme</span></a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Zahlengerade"><span class="tocnumber">1.1</span> <span class="toctext">Zahlengerade</span></a></li> <li class="toclevel-2 tocsection-3"><a href="#Kartesisches_Koordinatensystem"><span class="tocnumber">1.2</span> <span class="toctext">Kartesisches Koordinatensystem</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Affines_Koordinatensystem"><span class="tocnumber">1.3</span> <span class="toctext">Affines Koordinatensystem</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Polarkoordinaten"><span class="tocnumber">1.4</span> <span class="toctext">Polarkoordinaten</span></a></li> <li class="toclevel-2 tocsection-6"><a href="#Kugel-_und_Zylinderkoordinaten"><span class="tocnumber">1.5</span> <span class="toctext">Kugel- und Zylinderkoordinaten</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Elliptische_Koordinaten"><span class="tocnumber">1.6</span> <span class="toctext">Elliptische Koordinaten</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-8"><a href="#Parameterdarstellung"><span class="tocnumber">2</span> <span class="toctext">Parameterdarstellung</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Lokales_Koordinatensystem"><span class="tocnumber">3</span> <span class="toctext">Lokales Koordinatensystem</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Koordinatentransformationen"><span class="tocnumber">4</span> <span class="toctext">Koordinatentransformationen</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#Homogene_Koordinaten_in_der_Ebene"><span class="tocnumber">5</span> <span class="toctext">Homogene Koordinaten in der Ebene</span></a></li> <li class="toclevel-1 tocsection-12"><a href="#Weitere_Koordinatensysteme"><span class="tocnumber">6</span> <span class="toctext">Weitere Koordinatensysteme</span></a></li> <li class="toclevel-1 tocsection-13"><a href="#Literatur"><span class="tocnumber">7</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-14"><a href="#Weblinks"><span class="tocnumber">8</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-15"><a href="#Einzelnachweise"><span class="tocnumber">9</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Gebräuchliche_Koordinatensysteme"><span id="Gebr.C3.A4uchliche_Koordinatensysteme"></span>Gebräuchliche Koordinatensysteme</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=1" title="Abschnitt bearbeiten: Gebräuchliche Koordinatensysteme" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Gebräuchliche Koordinatensysteme"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Zahlengerade">Zahlengerade</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=2" title="Abschnitt bearbeiten: Zahlengerade" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Zahlengerade"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Zahlengerade" title="Zahlengerade">Zahlengerade</a></i></div> <p>Das einfachste Beispiel eines Koordinatensystems ist die Identifikation von Punkten auf einer Gerade mit der reellen Zahlengerade. In diesem System wird ein beliebiger Punkt <i>O</i> (der <i>Ursprung</i>) auf einer gegebenen Geraden gewählt. Die Koordinate eines Punktes <i>P</i> ist definiert als der vorzeichenbehaftete Abstand von <i>O</i> zu <i>P</i>, wobei der vorzeichenbehaftete Abstand als positiv oder negativ angenommen wird, je nachdem, auf welcher Seite der Linie <i>P</i> liegt. Jeder Punkt erhält eine eindeutige Koordinate und jede reelle Zahl ist die Koordinate eines eindeutigen Punktes.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/Datei:Number-line.svg" class="mw-file-description" title="Die Zahlenlinie"><img alt="Die Zahlenlinie" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/750px-Number-line.svg.png" decoding="async" width="750" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/1125px-Number-line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/1500px-Number-line.svg.png 2x" data-file-width="750" data-file-height="50" /></a><figcaption>Die Zahlenlinie</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Kartesisches_Koordinatensystem">Kartesisches Koordinatensystem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=3" title="Abschnitt bearbeiten: Kartesisches Koordinatensystem" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Kartesisches Koordinatensystem"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Rectangular_coordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Rectangular_coordinates.svg/220px-Rectangular_coordinates.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Rectangular_coordinates.svg/330px-Rectangular_coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Rectangular_coordinates.svg/440px-Rectangular_coordinates.svg.png 2x" data-file-width="429" data-file-height="425" /></a><figcaption>Das kartesische Koordinatensystem im dreidimensionalen Raum</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Cartesian_coordinate_system_handedness.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Cartesian_coordinate_system_handedness.svg/220px-Cartesian_coordinate_system_handedness.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Cartesian_coordinate_system_handedness.svg/330px-Cartesian_coordinate_system_handedness.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Cartesian_coordinate_system_handedness.svg/440px-Cartesian_coordinate_system_handedness.svg.png 2x" data-file-width="400" data-file-height="245" /></a><figcaption>Links- und rechtshändiges (rechts) dreidimensionales Koordinatensystem</figcaption></figure> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Kartesisches_Koordinatensystem" title="Kartesisches Koordinatensystem">Kartesisches Koordinatensystem</a></i></div> <p>Eines der bekanntesten Beispiele für ein Koordinatensystem ist das kartesische Koordinatensystem. In der Ebene werden zwei <a href="/wiki/Senkrechte" class="mw-redirect mw-disambig" title="Senkrechte">Senkrechte</a> gewählt und die Koordinaten eines Punktes als die vorzeichenbehafteten Abstände zu den Geraden aufgefasst. In drei Dimensionen wählt man drei zueinander <a href="/wiki/Orthogonalit%C3%A4t" title="Orthogonalität">orthogonale</a> Ebenen und die drei Koordinaten eines Punktes sind die vorzeichenbehafteten Abstände zu jeder der Ebenen. Dies kann verallgemeinert werden, um <i>n</i>-Koordinaten für jeden Punkt im <i>n</i>-dimensionalen euklidischen Raum zu erzeugen. </p><p>Es ist nach dem <a href="/wiki/Latinisierung" title="Latinisierung">latinisierten</a> Namen <i>Cartesius</i> des französischen Mathematikers <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> benannt, der das Konzept der „kartesischen Koordinaten“ bekannt gemacht hat.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Je nach Anordnung der Koordinatenachsen kann das dreidimensionale Koordinatensystem ein <a href="/wiki/Rechtssystem_(Mathematik)" title="Rechtssystem (Mathematik)">Rechts-</a> oder ein Linkssystem sein. </p> <div class="mw-heading mw-heading3"><h3 id="Affines_Koordinatensystem">Affines Koordinatensystem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=4" title="Abschnitt bearbeiten: Affines Koordinatensystem" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Affines Koordinatensystem"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Datei:Koordinaten-aff-2d.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Koordinaten-aff-2d.svg/220px-Koordinaten-aff-2d.svg.png" decoding="async" width="220" height="207" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Koordinaten-aff-2d.svg/330px-Koordinaten-aff-2d.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Koordinaten-aff-2d.svg/440px-Koordinaten-aff-2d.svg.png 2x" data-file-width="295" data-file-height="277" /></a><figcaption>Affine Koordinaten</figcaption></figure> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Affine_Koordinaten" title="Affine Koordinaten">Affine Koordinaten</a></i></div> <p>Wählt man in der euklidischen Ebene drei nicht auf einer Gerade liegende Punkte <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0},P_{1},P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0},P_{1},P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4447961e9bf558b71aa478c4080eddb92e9b7446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.707ex; height:2.509ex;" alt="{\displaystyle P_{0},P_{1},P_{2}}"></span> aus, so sind die beiden Vektoren <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576805c4661cbfa3b11e38380634aa8db236b957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:11.48ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}}"></span> linear unabhängig. Mit dem Punkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> als Ursprung lässt sich der <a href="/wiki/Ortsvektor" title="Ortsvektor">Ortsvektor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {P_{0}P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {P_{0}P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcabed85015bb95db8e055219a1b40e8dbc3dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:4.587ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {P_{0}P}}}"></span> eines beliebigen Punktes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> so schreiben: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {P_{0}P}}=u{\overrightarrow {P_{0}P_{1}}}+v{\overrightarrow {P_{0}P_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>P</mi> </mrow> <mo>→</mo> </mover> </mrow> <mo>=</mo> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>+</mo> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {P_{0}P}}=u{\overrightarrow {P_{0}P_{1}}}+v{\overrightarrow {P_{0}P_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14f8d2e0f2a33ed4dc267723387937504c4f5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:23.429ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {P_{0}P}}=u{\overrightarrow {P_{0}P_{1}}}+v{\overrightarrow {P_{0}P_{2}}}}"></span></dd></dl> <p>und dem Punkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> das Zahlenpaar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u,v)_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u,v)_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f0d74394fcb802e0a4a48b7b3be2d26b10dc14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.402ex; height:2.843ex;" alt="{\displaystyle (u,v)_{a}}"></span> als <i>affine Koordinaten</i><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> bezüglich den <i>Basispunkten</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0},P_{1},P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0},P_{1},P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4447961e9bf558b71aa478c4080eddb92e9b7446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.707ex; height:2.509ex;" alt="{\displaystyle P_{0},P_{1},P_{2}}"></span> zuordnen. </p><p>Bilden die Vektoren <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>→</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/576805c4661cbfa3b11e38380634aa8db236b957" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-top: -0.449ex; width:11.48ex; height:4.176ex;" alt="{\displaystyle {\overrightarrow {P_{0}P_{1}}},{\overrightarrow {P_{0}P_{2}}}}"></span> eine <a href="/wiki/Orthonormalbasis" title="Orthonormalbasis">Orthonormalbasis</a>, so ergeben sich die zuvor genannten kartesischen Koordinaten. In diesem Fall sind für einen Punkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{0},v_{0})_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (u_{0},v_{0})_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2172940670ab44480e5073e9c5b83c83f18ec29d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle (u_{0},v_{0})_{a}}"></span> die Punktmengen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=u_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u=u_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ffea3ef298f4e8ccd41b4c77b931c7e9ad356f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.812ex; height:2.009ex;" alt="{\displaystyle u=u_{0}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=v_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=v_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600f1fced4041372dcb68ef2f69706559e45c024" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.408ex; height:2.009ex;" alt="{\displaystyle v=v_{0}}"></span> Geraden, die sich orthogonal schneiden. Sind die Basisvektoren nicht orthogonal (siehe Bild), spricht man von <a href="/wiki/Schiefwinklige_Koordinaten" class="mw-redirect" title="Schiefwinklige Koordinaten">schiefwinkligen Koordinaten</a>. </p><p>Entsprechend sind affine Koordinaten für <i>höhere Dimensionen</i> erklärt. Koordinaten auf diese Weise zu definieren ist für jeden n-dimensionalen <a href="/wiki/Affine_R%C3%A4ume" class="mw-redirect" title="Affine Räume">affinen Raum</a> über einem Körper möglich, ist also nicht auf einen euklidischen Raum beschränkt. </p> <div class="mw-heading mw-heading3"><h3 id="Polarkoordinaten">Polarkoordinaten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=5" title="Abschnitt bearbeiten: Polarkoordinaten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Polarkoordinaten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Kreis-orthog-traj.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Kreis-orthog-traj.svg/220px-Kreis-orthog-traj.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Kreis-orthog-traj.svg/330px-Kreis-orthog-traj.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b1/Kreis-orthog-traj.svg/440px-Kreis-orthog-traj.svg.png 2x" data-file-width="324" data-file-height="324" /></a><figcaption>Polarkoordinaten</figcaption></figure> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Polarkoordinaten" title="Polarkoordinaten">Polarkoordinaten</a></i></div> <p>Ein weiteres häufig genutztes Koordinatensystem ist das der Polarkoordinaten. Dieses kann nur in der Ebene eingeführt werden. Für den dreidimensionalen Raum gibt es mit den Kugel- und den Zylinderkoordinaten zwei unterschiedliche Verallgemeinerungen. Im Gegensatz zu den zuvor genannten Systemen sind dieses Koordinatensystem und seine zwei Verallgemeinerungen keine Spezialfälle affiner Koordinatensysteme. </p><p>Für die Definition dieses Koordinatensystems wird ein Punkt als Pol und ein Strahl von diesem Punkt als Polachse gewählt. Für einen gegebenen Winkel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> gibt es eine einzige Linie durch den Pol, deren Winkel mit der Polachse <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> ist (gemessen gegen den Uhrzeigersinn von der Achse zur Linie). Dann gibt es einen einzigen Punkt auf dieser Linie, dessen Abstand vom Ursprung den Wert <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> ist. Für ein gegebenes Koordinatenpaar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389994f381dc6fcbbf41cef5bb3f694b9bfef296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.277ex; height:2.843ex;" alt="{\displaystyle (r,\phi )}"></span> gibt es einen einzigen Punkt, aber jeder Punkt wird durch viele Koordinatenpaare dargestellt. Zum Beispiel sind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389994f381dc6fcbbf41cef5bb3f694b9bfef296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.277ex; height:2.843ex;" alt="{\displaystyle (r,\phi )}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\phi +2\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϕ</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\phi +2\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d230044e67dd33f95c9d28fd2257d441eab18854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.612ex; height:2.843ex;" alt="{\displaystyle (r,\phi +2\pi )}"></span> Polarkoordinaten für denselben Punkt. Der Pol wird durch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb54b9d1707dad5039ba76ec0e94d1e4391c084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.391ex; height:2.843ex;" alt="{\displaystyle (0,\phi )}"></span> für einen beliebigen Wert von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span> dargestellt. </p> <div class="mw-heading mw-heading3"><h3 id="Kugel-_und_Zylinderkoordinaten">Kugel- und Zylinderkoordinaten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=6" title="Abschnitt bearbeiten: Kugel- und Zylinderkoordinaten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Kugel- und Zylinderkoordinaten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Cylindrical_Coordinates.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/150px-Cylindrical_Coordinates.svg.png" decoding="async" width="150" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/225px-Cylindrical_Coordinates.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Cylindrical_Coordinates.svg/300px-Cylindrical_Coordinates.svg.png 2x" data-file-width="748" data-file-height="745" /></a><figcaption>Zylinderkoordinaten</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Kugelkoord-def.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/150px-Kugelkoord-def.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/225px-Kugelkoord-def.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/69/Kugelkoord-def.svg/300px-Kugelkoord-def.svg.png 2x" data-file-width="261" data-file-height="261" /></a><figcaption>Kugelkoordinaten</figcaption></figure> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> und <a href="/wiki/Zylinderkoordinaten" class="mw-redirect" title="Zylinderkoordinaten">Zylinderkoordinaten</a></i></div> <p>Es gibt zwei übliche Methoden zur Erweiterung der Polarkoordinaten für den dreidimensionalen Raum. </p><p>Bei zylindrischen Koordinatensystem wird eine z-Koordinate mit der gleichen Bedeutung wie bei kartesischen Koordinaten zu den Polarkoordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\phi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\phi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389994f381dc6fcbbf41cef5bb3f694b9bfef296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.277ex; height:2.843ex;" alt="{\displaystyle (r,\phi )}"></span> hinzugefügt, was ein Tripel <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\phi ,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>ϕ</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\phi ,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666edd2256f224bd278bf7ff7c04b98ad12b47a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.399ex; height:2.843ex;" alt="{\displaystyle (r,\phi ,z)}"></span> ergibt. </p><p>Bei Kugelkoordinaten oder räumlichen Polarkoordinaten wird ein Punkt im dreidimensionalen Raum durch seinen Abstand vom Ursprung und durch zwei Winkel angegeben. Ein bekanntes Beispiel eines Kugelkoordinatensystems ist das <a href="/wiki/Geographische_Koordinaten" title="Geographische Koordinaten">System der Geographischen Koordinaten</a> mit deren Hilfe die <a href="/wiki/Erde" title="Erde">Erde</a> in <a href="/wiki/L%C3%A4ngengrad" class="mw-redirect" title="Längengrad">Längen-</a> und <a href="/wiki/Breitengrad" class="mw-redirect" title="Breitengrad">Breitengrade</a> unterteilt wird. Die dritte Koordinate also der Abstand vom Erdmittelpunkt ist bei diesem System nicht relevant. </p> <div class="mw-heading mw-heading3"><h3 id="Elliptische_Koordinaten">Elliptische Koordinaten</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=7" title="Abschnitt bearbeiten: Elliptische Koordinaten" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Elliptische Koordinaten"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Elliptische_Koordinaten" title="Elliptische Koordinaten">Elliptische Koordinaten</a></i></div> <p>Elliptische Koordinaten verwenden sich senkrecht schneidende Systeme von <a href="/wiki/Konfokale_Kegelschnitte" title="Konfokale Kegelschnitte">konfokalen Ellipsen</a> und Hyperbeln. Diese Koordinaten sind für die Brennpunkte und die Punkte dazwischen nicht definiert. </p><p>Den (ebenen) elliptischen Koordinaten entsprechen die <a href="/wiki/Konfokale_Kegelschnitte" title="Konfokale Kegelschnitte">Ellipsoid-Koordinaten</a>. Das hier verwendete orthogonale Flächensystem besteht aus konfokalen Ellipsoiden, einschaligen und zweischaligen Hyperboloiden. </p><p>Weiterhin gibt es noch die <a href="/wiki/Ellipsoidische_Koordinaten" title="Ellipsoidische Koordinaten">ellipsoidische Koordinaten</a>, die zur Beschreibung von Punkten eines Rotations-Ellipsoids (Erde) verwendet werden. </p> <div class="mw-heading mw-heading2"><h2 id="Parameterdarstellung">Parameterdarstellung</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=8" title="Abschnitt bearbeiten: Parameterdarstellung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Parameterdarstellung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Parameterdarstellung" title="Parameterdarstellung">Parameterdarstellung</a></i></div> <p>Parameterdarstellungen von Flächen kann man als Koordinatensysteme dieser Flächen ansehen. Z. B. die Parameterdarstellung einer Ebene, die übliche Parameterdarstellung einer Kugeloberfläche mit geographischer Länge und Breite oder die Paramerdarstellung eines <a href="/wiki/Ellipsoid" title="Ellipsoid">Ellipsoids</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Lokales_Koordinatensystem">Lokales Koordinatensystem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=9" title="Abschnitt bearbeiten: Lokales Koordinatensystem" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Lokales Koordinatensystem"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:Kugelkoord-lokale-Basis-s.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Kugelkoord-lokale-Basis-s.svg/200px-Kugelkoord-lokale-Basis-s.svg.png" decoding="async" width="200" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Kugelkoord-lokale-Basis-s.svg/300px-Kugelkoord-lokale-Basis-s.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Kugelkoord-lokale-Basis-s.svg/400px-Kugelkoord-lokale-Basis-s.svg.png 2x" data-file-width="260" data-file-height="256" /></a><figcaption>Kugelkoordinaten mit zugehöriger lokaler Basis <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{r},\mathbf {e} _{\theta },\mathbf {e} _{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{r},\mathbf {e} _{\theta },\mathbf {e} _{\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/763be2f9a2a05e4af8be6990f8a0acefc439ec25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.028ex; height:2.343ex;" alt="{\displaystyle \mathbf {e} _{r},\mathbf {e} _{\theta },\mathbf {e} _{\varphi }}"></span></figcaption></figure> <p>Ein lokales Koordinatensystem oder auch (Koordinaten-)Karte<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> ist ein Koordinatensystem für eine Teilmenge eines geometrischen Objekts. Das Konzept der Koordinatenkarten ist zentral für die Theorie der <a href="/wiki/Mannigfaltigkeit" title="Mannigfaltigkeit">Mannigfaltigkeiten</a>. Eine Mannigfaltigkeit ist ein geometrisches Objekt, so dass für jeden Punkt ein lokales Koordinatensystem existiert, das mit den benachbarten Koordinatensystemen verträglich ist. Genauer gesagt ist eine Koordinatenkarte ein <a href="/wiki/Hom%C3%B6omorphismus" title="Homöomorphismus">Homöomorphismus</a> von einer offenen Teilmenge eines Raumes zu einer <a href="/wiki/Offene_Teilmenge" class="mw-redirect" title="Offene Teilmenge">offenen Teilmenge</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Oft ist es nicht möglich, ein einziges konsistentes Koordinatensystem für einen ganzen Raum bereitzustellen. In diesem Fall wird eine Sammlung von Koordinatenkarten zu einem <a href="/wiki/Atlas_(Mathematik)" title="Atlas (Mathematik)">Atlas</a> zusammengesetzt, der den ganzen Raum abdeckt.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Um die lokalen Basisvektoren zu bestimmen, geht man von den <a href="/wiki/Koordinatenlinie" title="Koordinatenlinie">Koordinatenlinien</a> aus. Dabei handelt es sich um <a href="/wiki/Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurven</a>, die entstehen, wenn an einem Punkt alle Koordinaten bis auf jeweils eine konstant sind. Die lokalen Basisvektoren an einem Punkt sind dann die Tangentenvektoren an diese Koordinatenlinien und können durch Ableitung nach dem Kurvenparameter berechnet werden (<a href="/wiki/Kugelkoordinaten#Transformation_der_Vektorraumbasis" title="Kugelkoordinaten">siehe Berechnung für Kugelkoordinaten</a>). Für <a href="/wiki/Kugelkoordinaten" title="Kugelkoordinaten">Kugelkoordinaten</a> mit den Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (r,\theta ,\varphi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mo>,</mo> <mi>θ</mi> <mo>,</mo> <mi>φ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (r,\theta ,\varphi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0fbf2e667b8c94c169165fc44e0d73e66c901a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.536ex; height:2.843ex;" alt="{\displaystyle (r,\theta ,\varphi )}"></span> sind die Koordinatenlinien </p> <ul><li>Halbgeraden, die im Koordinatenursprung beginnen (Kurvenparameter r)</li> <li>Halbkreise („Meridiane“) mit dem Koordinatenursprung als Mittelpunkt (Kurvenparameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span>)</li> <li>Kreise („Breitenkreise“) senkrecht zur z-Achse (Kurvenparameter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>).</li></ul> <p>Durch Normierung der Tangentenvektoren erhält man die Einheitsvektoren <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {e}}_{r},{\vec {e}}_{\theta },{\vec {e}}_{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {e}}_{r},{\vec {e}}_{\theta },{\vec {e}}_{\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff2f52eb3659ad50faebc6ee8a3417d8373d90e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.021ex; height:3.009ex;" alt="{\displaystyle {\vec {e}}_{r},{\vec {e}}_{\theta },{\vec {e}}_{\varphi }}"></span>. Diese stehen paarweise senkrecht aufeinander, die Kugelkoordinaten sind somit ein orthogonales Koordinatensystem. </p> <div class="mw-heading mw-heading2"><h2 id="Koordinatentransformationen">Koordinatentransformationen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=10" title="Abschnitt bearbeiten: Koordinatentransformationen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=10" title="Quellcode des Abschnitts bearbeiten: Koordinatentransformationen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hauptartikel" role="navigation"><span class="hauptartikel-pfeil" title="siehe" aria-hidden="true" role="presentation">→ </span><i><span class="hauptartikel-text">Hauptartikel</span>: <a href="/wiki/Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformation</a></i></div> <p>Für das Umrechnen von Koordinaten eines Systems in die Koordinaten eines anderen Systems gibt es die Koordinatentransformationen. </p> <div class="mw-heading mw-heading2"><h2 id="Homogene_Koordinaten_in_der_Ebene">Homogene Koordinaten in der Ebene</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=11" title="Abschnitt bearbeiten: Homogene Koordinaten in der Ebene" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=11" title="Quellcode des Abschnitts bearbeiten: Homogene Koordinaten in der Ebene"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Die euklidische Ebene lässt sich auch mit <a href="/wiki/Homogene_Koordinaten" title="Homogene Koordinaten">homogenen Koordinaten</a> beschreiben. Dabei werden einem Punkt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/099c643e67f1973c546d7e001eae4ab0ec7b2415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.172ex; height:2.843ex;" alt="{\displaystyle P=(x,y)}"></span> drei homogene Koordinaten <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}:x_{2}:x_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1}:x_{2}:x_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0162ad6314cc077b39aadeda69f8d8561a8db291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.835ex; height:2.843ex;" alt="{\displaystyle (x_{1}:x_{2}:x_{3})}"></span> so zugeordnet, dass auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(cx_{1}:cx_{2}:cx_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>c</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>c</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(cx_{1}:cx_{2}:cx_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e72c2e023f66093e9eab725185c7cd10d736da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.7ex; height:2.843ex;" alt="{\displaystyle P=(cx_{1}:cx_{2}:cx_{3})}"></span> für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be27396bd0e62003728d08329a8767eee94409e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.676ex;" alt="{\displaystyle c\neq 0}"></span> gilt. Eine Standardzuordnung ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x:y:1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mi>y</mi> <mo>:</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x:y:1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f760870e37a534ba8cccd1b990c5fc904b53ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.175ex; height:2.843ex;" alt="{\displaystyle P=(x:y:1)}"></span>. Setzt man <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=(x:y:1-x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>:</mo> <mi>y</mi> <mo>:</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=(x:y:1-x-y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b37e001f96dce2cb4ae3599f8a04bdecdd42e0ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.341ex; height:2.843ex;" alt="{\displaystyle P=(x:y:1-x-y)}"></span> erhält man <a href="/wiki/Baryzentrische_Koordinaten" title="Baryzentrische Koordinaten">baryzentrische Koordinaten</a>. Der große Vorteil homogener Koordinaten ist, dass Punkte der Ferngerade einfach zu beschreiben sind: Im Standardfall durch die Gleichung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{3}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55547972561662fda8d82465866be57f729271c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{3}=0}"></span>, im baryzentrischen Fall durch die Gleichung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+x_{2}+x_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+x_{2}+x_{3}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a37e9c6b819f4da6042404adb9a11094fcf97a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.093ex; height:2.509ex;" alt="{\displaystyle x_{1}+x_{2}+x_{3}=0}"></span>. Die bei affinen Koordinaten nötigen Grenzwert-Überlegungen werden im Standardfall zum einfachen Setzen von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{3}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{3}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55547972561662fda8d82465866be57f729271c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{3}=0}"></span>. </p><p>In der <a href="/wiki/Dreiecksgeometrie" title="Dreiecksgeometrie">Dreiecksgeometrie</a> werden auch <a href="/wiki/Trilineare_Koordinaten" title="Trilineare Koordinaten">trilineare Koordinaten</a> verwendet. </p> <div class="mw-heading mw-heading2"><h2 id="Weitere_Koordinatensysteme">Weitere Koordinatensysteme</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=12" title="Abschnitt bearbeiten: Weitere Koordinatensysteme" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=12" title="Quellcode des Abschnitts bearbeiten: Weitere Koordinatensysteme"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Einige nur in Fachgebieten (z. B. <a href="/wiki/Geod%C3%A4sie" title="Geodäsie">Geodäsie</a>, <a href="/wiki/Kartografie" title="Kartografie">Kartografie</a>, <a href="/wiki/Geographie" title="Geographie">Geographie</a>, <a href="/wiki/Fernerkundung" title="Fernerkundung">Fernerkundung</a>, <a href="/wiki/Astronomie" title="Astronomie">Astronomie</a>, <a href="/wiki/Amateurfunkdienst" title="Amateurfunkdienst">Amateurfunk</a>) gebräuchliche Koordinatensysteme sind: </p> <ul><li><a href="/wiki/Inertialsystem" title="Inertialsystem">Inertialsystem</a></li> <li><a href="/wiki/Geographische_Koordinaten" title="Geographische Koordinaten">Geographisches Koordinatensystem</a></li> <li><a href="/wiki/Geod%C3%A4tisches_Gitter" title="Geodätisches Gitter">Geodätische Gitter</a>: <ul><li><a href="/wiki/Soldner-Koordinatensystem" title="Soldner-Koordinatensystem">Soldner-Koordinatensystem</a></li> <li><a href="/wiki/Gau%C3%9F-Kr%C3%BCger-Koordinatensystem" title="Gauß-Krüger-Koordinatensystem">Gauß-Krüger-Koordinatensystem</a></li> <li><a href="/wiki/UTM-Koordinatensystem" title="UTM-Koordinatensystem">UTM-Koordinatensystem</a> <ul><li><a href="/wiki/UTM-Referenzsystem" title="UTM-Referenzsystem">UTM-Referenzsystem</a> auch MGRS</li></ul></li></ul></li> <li><a href="/wiki/Astronomische_Koordinatensysteme" title="Astronomische Koordinatensysteme">Astronomische Koordinatensysteme</a> wie das <a href="/wiki/Ekliptikales_Koordinatensystem" title="Ekliptikales Koordinatensystem">ekliptikale</a> oder <a href="/wiki/Galaktisches_Koordinatensystem" title="Galaktisches Koordinatensystem">galaktische</a></li> <li><a href="/wiki/Parallele_Koordinaten" title="Parallele Koordinaten">Parallele Koordinaten</a></li> <li>Bewegte Koordinatensysteme</li> <li>Rotierende Koordinatensysteme</li> <li><a href="/wiki/Fahrzeugkoordinatensystem" title="Fahrzeugkoordinatensystem">Fahrzeugkoordinatensystem</a></li> <li><a href="/wiki/Weltkoordinatensystem" title="Weltkoordinatensystem">Weltkoordinatensystem</a></li> <li><a href="/wiki/QTH-Locator" title="QTH-Locator">QTH-Locator</a> (Amateurfunk)</li> <li><a href="/wiki/Marinequadrat" title="Marinequadrat">Marinequadrate</a> und <a href="/wiki/Jagdgradnetzmeldeverfahren" title="Jagdgradnetzmeldeverfahren">Gradnetze</a> (aus dem Zweiten Weltkrieg)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=13" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=13" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>W. Werner: <cite style="font-style:italic">Vektoren und Tensoren als universelle Sprache in Physik und Technik</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>1</span>. Springer Vieweg, <a href="/wiki/Spezial:ISBN-Suche/9783658252717" class="internal mw-magiclink-isbn">ISBN 978-3-658-25271-7</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Koordinatensystem&rft.au=W.+Werner&rft.btitle=Vektoren+und+Tensoren+als+universelle+Sprache+in+Physik+und+Technik&rft.genre=book&rft.isbn=9783658252717&rft.pub=Springer+Vieweg&rft.volume=1" style="display:none"> </span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=14" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=14" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /></span></span></span><b><a href="https://de.wiktionary.org/wiki/Koordinate" class="extiw" title="wikt:Koordinate">Wiktionary: Koordinate</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/16px-Wiktfavicon_en.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/24px-Wiktfavicon_en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c3/Wiktfavicon_en.svg/32px-Wiktfavicon_en.svg.png 2x" data-file-width="16" data-file-height="16" /></span></span></span><b><a href="https://de.wiktionary.org/wiki/Koordinatensystem" class="extiw" title="wikt:Koordinatensystem">Wiktionary: Koordinatensystem</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div> <ul><li><a rel="nofollow" class="external text" href="https://www.mathe-online.at/mathint/zeich/i.html">Einfache und verständliche Erklärung</a> (hpts. durch Abbildungen)</li> <li><a href="/wiki/Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/CoordinateSystem.html"><i>Coordinate System</i>.</a> In: <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Koordinatensystem&veaction=edit&section=15" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Koordinatensystem&action=edit&section=15" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Etymologie nach <i>Kluge Etymologisches Wörterbuch der deutschen Sprache</i>, 24. Auflage, 2002.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">James B. Stewart, Lothar Redlin, Saleem Watson: <cite style="font-style:italic">College Algebra</cite>. Hrsg.: <a href="/w/index.php?title=Brooks_Cole&action=edit&redlink=1" class="new" title="Brooks Cole (Seite nicht vorhanden)">Brooks Cole</a>. 5. Auflage. 2008, <a href="/wiki/Spezial:ISBN-Suche/9780495565215" class="internal mw-magiclink-isbn">ISBN 978-0-495-56521-5</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>13–19</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Koordinatensystem&rft.au=James+B.+Stewart%2C+Lothar+Redlin%2C+Saleem+Watson&rft.btitle=College+Algebra&rft.date=2008&rft.edition=5.&rft.genre=book&rft.isbn=9780495565215&rft.pages=13-19" style="display:none"> </span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><cite style="font-style:italic">Kartesische Koordinaten</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, <a href="/wiki/Spezial:ISBN-Suche/3827404398" class="internal mw-magiclink-isbn">ISBN 3-8274-0439-8</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Koordinatensystem&rft.atitle=Kartesische+Koordinaten&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1&rft.genre=book&rft.isbn=3827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">Bronstein, Semendjajew: Taschenbuch der Mathematik, Teubner-Verlag, Leipzig, 1979, <a href="/wiki/Spezial:ISBN-Suche/3871444928" class="internal mw-magiclink-isbn">ISBN 3 87144 492 8</a>, S. 606</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><cite style="font-style:italic">Komplexe Mannigfaltigkeit</cite>. In: Guido Walz (Hrsg.): <cite style="font-style:italic">Lexikon der Mathematik</cite>. 1. Auflage. Spektrum Akademischer Verlag, Mannheim/Heidelberg 2000, <a href="/wiki/Spezial:ISBN-Suche/3827404398" class="internal mw-magiclink-isbn">ISBN 3-8274-0439-8</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Koordinatensystem&rft.atitle=Komplexe+Mannigfaltigkeit&rft.btitle=Lexikon+der+Mathematik&rft.date=2000&rft.edition=1&rft.genre=book&rft.isbn=3827404398&rft.place=Mannheim%2FHeidelberg&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text">John M. Lee: <i>Introduction to Smooth Manifolds</i> (= <i>Graduate Texts in Mathematics</i> 218). 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href="https://ba.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0%D2%BB%D1%8B" title="Координаталар системаһы – Baschkirisch" lang="ba" hreflang="ba" data-title="Координаталар системаһы" data-language-autonym="Башҡортса" data-language-local-name="Baschkirisch" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%96%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Сістэма каардынат – Belarussisch" lang="be" hreflang="be" data-title="Сістэма каардынат" data-language-autonym="Беларуская" data-language-local-name="Belarussisch" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A1%D1%8B%D1%81%D1%82%D1%8D%D0%BC%D0%B0_%D0%BA%D0%B0%D0%B0%D1%80%D0%B4%D1%8B%D0%BD%D0%B0%D1%82" title="Сыстэма каардынат – Weißrussisch (Taraschkewiza)" lang="be-tarask" hreflang="be-tarask" data-title="Сыстэма каардынат" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Weißrussisch (Taraschkewiza)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0" title="Координатна система – Bulgarisch" lang="bg" hreflang="bg" data-title="Координатна система" data-language-autonym="Български" data-language-local-name="Bulgarisch" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%95%E0%A5%8B%E0%A4%86%E0%A4%B0%E0%A5%8D%E0%A4%A1%E0%A4%BF%E0%A4%A8%E0%A5%87%E0%A4%9F_%E0%A4%B8%E0%A4%BF%E0%A4%B8%E0%A5%8D%E0%A4%9F%E0%A4%AE" title="कोआर्डिनेट सिस्टम – Bhojpuri" lang="bh" hreflang="bh" data-title="कोआर्डिनेट सिस्टम" data-language-autonym="भोजपुरी" data-language-local-name="Bhojpuri" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE%E0%A6%A8%E0%A6%BE%E0%A6%99%E0%A7%8D%E0%A6%95_%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%AC%E0%A6%B8%E0%A7%8D%E0%A6%A5%E0%A6%BE" title="স্থানাঙ্ক ব্যবস্থা – Bengalisch" lang="bn" hreflang="bn" data-title="স্থানাঙ্ক ব্যবস্থা" data-language-autonym="বাংলা" data-language-local-name="Bengalisch" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Koordinatni_sistem" title="Koordinatni sistem – Bosnisch" lang="bs" hreflang="bs" data-title="Koordinatni sistem" data-language-autonym="Bosanski" data-language-local-name="Bosnisch" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Sistema_de_coordenades" title="Sistema de coordenades – Katalanisch" lang="ca" hreflang="ca" data-title="Sistema de coordenades" data-language-autonym="Català" data-language-local-name="Katalanisch" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%B3%DB%8C%D8%B3%D8%AA%D9%85%DB%8C_%D9%BE%DB%86%D8%AA%D8%A7%D9%86%DB%95%DA%A9%D8%A7%D9%86" title="سیستمی پۆتانەکان – Zentralkurdisch" lang="ckb" hreflang="ckb" data-title="سیستمی پۆتانەکان" data-language-autonym="کوردی" data-language-local-name="Zentralkurdisch" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Soustava_sou%C5%99adnic" title="Soustava souřadnic – Tschechisch" lang="cs" hreflang="cs" data-title="Soustava souřadnic" data-language-autonym="Čeština" data-language-local-name="Tschechisch" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%81%D0%B5%D0%BD_%D1%82%D1%8B%D1%82%C4%83%D0%BC%C4%95" title="Координатсен тытăмĕ – Tschuwaschisch" lang="cv" hreflang="cv" data-title="Координатсен тытăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Tschuwaschisch" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/System_gyfesurynnau" title="System gyfesurynnau – Walisisch" lang="cy" hreflang="cy" data-title="System gyfesurynnau" data-language-autonym="Cymraeg" data-language-local-name="Walisisch" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Koordinatsystem" title="Koordinatsystem – Dänisch" lang="da" hreflang="da" data-title="Koordinatsystem" data-language-autonym="Dansk" data-language-local-name="Dänisch" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8D%CF%83%CF%84%CE%B7%CE%BC%CE%B1_%CE%B1%CE%BD%CE%B1%CF%86%CE%BF%CF%81%CE%AC%CF%82" title="Σύστημα αναφοράς – Griechisch" lang="el" hreflang="el" data-title="Σύστημα αναφοράς" data-language-autonym="Ελληνικά" data-language-local-name="Griechisch" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Coordinate_system" title="Coordinate system – Englisch" lang="en" hreflang="en" data-title="Coordinate system" data-language-autonym="English" data-language-local-name="Englisch" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Koordinatsistemo" title="Koordinatsistemo – Esperanto" lang="eo" hreflang="eo" data-title="Koordinatsistemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas – Spanisch" lang="es" hreflang="es" data-title="Sistema de coordenadas" data-language-autonym="Español" data-language-local-name="Spanisch" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Koordinaadis%C3%BCsteem" title="Koordinaadisüsteem – Estnisch" lang="et" hreflang="et" data-title="Koordinaadisüsteem" data-language-autonym="Eesti" data-language-local-name="Estnisch" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Koordenatu_sistema" title="Koordenatu sistema – Baskisch" lang="eu" hreflang="eu" data-title="Koordenatu sistema" data-language-autonym="Euskara" data-language-local-name="Baskisch" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%B3%D8%AA%DA%AF%D8%A7%D9%87_%D9%85%D8%AE%D8%AA%D8%B5%D8%A7%D8%AA" title="دستگاه مختصات – Persisch" lang="fa" hreflang="fa" data-title="دستگاه مختصات" data-language-autonym="فارسی" data-language-local-name="Persisch" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Koordinaatisto" title="Koordinaatisto – Finnisch" lang="fi" hreflang="fi" data-title="Koordinaatisto" data-language-autonym="Suomi" data-language-local-name="Finnisch" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_de_coordonn%C3%A9es" title="Système de coordonnées – Französisch" lang="fr" hreflang="fr" data-title="Système de coordonnées" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Koordinaatens%C3%BCsteem" title="Koordinaatensüsteem – Nordfriesisch" lang="frr" hreflang="frr" data-title="Koordinaatensüsteem" data-language-autonym="Nordfriisk" data-language-local-name="Nordfriesisch" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E5%BA%A7%E6%A8%99" title="座標 – Gan" lang="gan" hreflang="gan" data-title="座標" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Sist%C3%A8m_di_ko%C3%B2rdon%C3%A9" title="Sistèm di koòrdoné – Französisch-Guayana Kreolisch" lang="gcr" hreflang="gcr" data-title="Sistèm di koòrdoné" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Französisch-Guayana Kreolisch" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Siostaman_cho-chomharran" title="Siostaman cho-chomharran – Gälisch (Schottland)" lang="gd" hreflang="gd" data-title="Siostaman cho-chomharran" data-language-autonym="Gàidhlig" data-language-local-name="Gälisch (Schottland)" class="interlanguage-link-target"><span>Gàidhlig</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas – Galicisch" lang="gl" hreflang="gl" data-title="Sistema de coordenadas" data-language-autonym="Galego" data-language-local-name="Galicisch" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" title="קואורדינטות – Hebräisch" lang="he" hreflang="he" data-title="קואורדינטות" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A4%BF%E0%A4%B0%E0%A5%8D%E0%A4%A6%E0%A5%87%E0%A4%B6%E0%A4%BE%E0%A4%82%E0%A4%95_%E0%A4%AA%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%A4%E0%A4%BF" title="निर्देशांक पद्धति – Hindi" lang="hi" hreflang="hi" data-title="निर्देशांक पद्धति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Coordinate_system" title="Coordinate system – Fidschi-Hindi" lang="hif" hreflang="hif" data-title="Coordinate system" data-language-autonym="Fiji Hindi" data-language-local-name="Fidschi-Hindi" class="interlanguage-link-target"><span>Fiji Hindi</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Koordinatni_sustav" title="Koordinatni sustav – Kroatisch" lang="hr" hreflang="hr" data-title="Koordinatni sustav" data-language-autonym="Hrvatski" data-language-local-name="Kroatisch" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Sist%C3%A8m_kouw%C3%B2done" title="Sistèm kouwòdone – Haiti-Kreolisch" lang="ht" hreflang="ht" data-title="Sistèm kouwòdone" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haiti-Kreolisch" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Koordin%C3%A1ta-rendszer" title="Koordináta-rendszer – Ungarisch" lang="hu" hreflang="hu" data-title="Koordináta-rendszer" data-language-autonym="Magyar" data-language-local-name="Ungarisch" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BF%D5%B8%D5%B8%D6%80%D5%A4%D5%AB%D5%B6%D5%A1%D5%BF%D5%A1%D5%B5%D5%AB%D5%B6_%D5%B0%D5%A1%D5%B4%D5%A1%D5%AF%D5%A1%D6%80%D5%A3" title="Կոորդինատային համակարգ – Armenisch" lang="hy" hreflang="hy" data-title="Կոորդինատային համակարգ" data-language-autonym="Հայերեն" data-language-local-name="Armenisch" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sistem_koordinat" title="Sistem koordinat – Indonesisch" lang="id" hreflang="id" data-title="Sistem koordinat" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesisch" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Hnit_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Hnit (stærðfræði) – Isländisch" lang="is" hreflang="is" data-title="Hnit (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="Isländisch" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sistema_di_coordinate" title="Sistema di coordinate – Italienisch" lang="it" hreflang="it" data-title="Sistema di coordinate" data-language-autonym="Italiano" data-language-local-name="Italienisch" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BA%A7%E6%A8%99" title="座標 – Japanisch" lang="ja" hreflang="ja" data-title="座標" data-language-autonym="日本語" data-language-local-name="Japanisch" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Kwaadinet_(matimatix)" title="Kwaadinet (matimatix) – Jamaikanisch-Kreolisch" lang="jam" hreflang="jam" data-title="Kwaadinet (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaikanisch-Kreolisch" class="interlanguage-link-target"><span>Patois</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D1%82%D0%B0%D1%80_%D0%B6%D2%AF%D0%B9%D0%B5%D1%81%D1%96" title="Координаттар жүйесі – Kasachisch" lang="kk" hreflang="kk" data-title="Координаттар жүйесі" data-language-autonym="Қазақша" data-language-local-name="Kasachisch" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A2%8C%ED%91%9C%EA%B3%84" title="좌표계 – Koreanisch" lang="ko" hreflang="ko" data-title="좌표계" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B0%D0%BB%D0%B0%D1%80_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%B4%D0%B0)" title="Координаталар (математикада) – Kirgisisch" lang="ky" hreflang="ky" data-title="Координаталар (математикада)" data-language-autonym="Кыргызча" data-language-local-name="Kirgisisch" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Systema_coordinatarum" title="Systema coordinatarum – Latein" lang="la" hreflang="la" data-title="Systema coordinatarum" data-language-autonym="Latina" data-language-local-name="Latein" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Koordinatesystem" title="Koordinatesystem – Luxemburgisch" lang="lb" hreflang="lb" data-title="Koordinatesystem" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxemburgisch" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Koordina%C4%8Di%C5%B3_sistema" title="Koordinačių sistema – Litauisch" lang="lt" hreflang="lt" data-title="Koordinačių sistema" data-language-autonym="Lietuvių" data-language-local-name="Litauisch" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Koordin%C4%81tu_sist%C4%93ma" title="Koordinātu sistēma – Lettisch" lang="lv" hreflang="lv" data-title="Koordinātu sistēma" data-language-autonym="Latviešu" data-language-local-name="Lettisch" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5%D0%BD_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC" title="Координатен систем – Mazedonisch" lang="mk" hreflang="mk" data-title="Координатен систем" data-language-autonym="Македонски" data-language-local-name="Mazedonisch" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%A8%E0%B4%BF%E0%B5%BC%E0%B4%A6%E0%B5%87%E0%B4%B6%E0%B4%BE%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%B5%E0%B4%B8%E0%B5%8D%E0%B4%A5" title="നിർദേശാങ്കവ്യവസ്ഥ – Malayalam" lang="ml" hreflang="ml" data-title="നിർദേശാങ്കവ്യവസ്ഥ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Sistem_koordinat" title="Sistem koordinat – Malaiisch" lang="ms" hreflang="ms" data-title="Sistem koordinat" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%AF%E1%80%A9%E1%80%92%E1%80%AD%E1%80%94%E1%80%AD%E1%80%90%E1%80%BA%E1%80%85%E1%80%94%E1%80%85%E1%80%BA" title="ကိုဩဒိနိတ်စနစ် – Birmanisch" lang="my" hreflang="my" data-title="ကိုဩဒိနိတ်စနစ်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Birmanisch" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Koordinatensysteem" title="Koordinatensysteem – Niederdeutsch" lang="nds" hreflang="nds" data-title="Koordinatensysteem" data-language-autonym="Plattdüütsch" data-language-local-name="Niederdeutsch" class="interlanguage-link-target"><span>Plattdüütsch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Co%C3%B6rdinatenstelsel" title="Coördinatenstelsel – Niederländisch" lang="nl" hreflang="nl" data-title="Coördinatenstelsel" data-language-autonym="Nederlands" data-language-local-name="Niederländisch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Koordinatsystem" title="Koordinatsystem – Norwegisch (Nynorsk)" lang="nn" hreflang="nn" data-title="Koordinatsystem" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegisch (Nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Koordinatsystem" title="Koordinatsystem – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Koordinatsystem" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Sist%C3%A8ma_de_coordenadas" title="Sistèma de coordenadas – Okzitanisch" lang="oc" hreflang="oc" data-title="Sistèma de coordenadas" data-language-autonym="Occitan" data-language-local-name="Okzitanisch" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Sirna_Diriiroo_Ko%E2%80%99oordineetotaa" title="Sirna Diriiroo Ko’oordineetotaa – Oromo" lang="om" hreflang="om" data-title="Sirna Diriiroo Ko’oordineetotaa" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%95_%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%AC%E0%A9%B0%E0%A8%A7" title="ਗੁਣਕ ਪ੍ਰਬੰਧ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੁਣਕ ਪ੍ਰਬੰਧ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych" title="Układ współrzędnych – Polnisch" lang="pl" hreflang="pl" data-title="Układ współrzędnych" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Coordin%C3%A0" title="Coordinà – Piemontesisch" lang="pms" hreflang="pms" data-title="Coordinà" data-language-autonym="Piemontèis" data-language-local-name="Piemontesisch" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%A9%D9%88%D8%A2%D8%B1%DA%88%DB%8C%D9%86%DB%8C%D9%B9_%D9%BE%D8%B1%D8%A8%D9%86%D8%AF%DA%BE" title="کوآرڈینیٹ پربندھ – Westliches Panjabi" lang="pnb" hreflang="pnb" data-title="کوآرڈینیٹ پربندھ" data-language-autonym="پنجابی" data-language-local-name="Westliches Panjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sistema_de_coordenadas" title="Sistema de coordenadas – Portugiesisch" lang="pt" hreflang="pt" data-title="Sistema de coordenadas" data-language-autonym="Português" data-language-local-name="Portugiesisch" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Sistem_de_coordonate" title="Sistem de coordonate – Rumänisch" lang="ro" hreflang="ro" data-title="Sistem de coordonate" data-language-autonym="Română" data-language-local-name="Rumänisch" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Система координат – Russisch" lang="ru" hreflang="ru" data-title="Система координат" data-language-autonym="Русский" data-language-local-name="Russisch" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%DA%AA%D9%88%D8%A2%D8%B1%DA%8A%D9%8A%D9%86%D9%8A%D9%BD_%D8%B3%D8%B3%D9%BD%D9%85" title="ڪوآرڊينيٽ سسٽم – Sindhi" lang="sd" hreflang="sd" data-title="ڪوآرڊينيٽ سسٽم" data-language-autonym="سنڌي" data-language-local-name="Sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Koordinate" title="Koordinate – Serbokroatisch" lang="sh" hreflang="sh" data-title="Koordinate" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbokroatisch" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Coordinate_system" title="Coordinate system – einfaches Englisch" lang="en-simple" hreflang="en-simple" data-title="Coordinate system" data-language-autonym="Simple English" data-language-local-name="einfaches Englisch" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/S%C3%BAstava_s%C3%BAradn%C3%ADc" title="Sústava súradníc – Slowakisch" lang="sk" hreflang="sk" data-title="Sústava súradníc" data-language-autonym="Slovenčina" data-language-local-name="Slowakisch" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Koordinatni_sistem" title="Koordinatni sistem – Slowenisch" lang="sl" hreflang="sl" data-title="Koordinatni sistem" data-language-autonym="Slovenščina" data-language-local-name="Slowenisch" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Kupanga_Mazano" title="Kupanga Mazano – Shona" lang="sn" hreflang="sn" data-title="Kupanga Mazano" data-language-autonym="ChiShona" data-language-local-name="Shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Sistemi_koordinativ" title="Sistemi koordinativ – Albanisch" lang="sq" hreflang="sq" data-title="Sistemi koordinativ" data-language-autonym="Shqip" data-language-local-name="Albanisch" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82%D0%B5" title="Координате – Serbisch" lang="sr" hreflang="sr" data-title="Координате" data-language-autonym="Српски / srpski" data-language-local-name="Serbisch" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Koordinatsystem" title="Koordinatsystem – Schwedisch" lang="sv" hreflang="sv" data-title="Koordinatsystem" data-language-autonym="Svenska" data-language-local-name="Schwedisch" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Majiranukta" title="Majiranukta – Suaheli" lang="sw" hreflang="sw" data-title="Majiranukta" data-language-autonym="Kiswahili" data-language-local-name="Suaheli" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Uk%C5%82od_wsp%C5%AF%C5%82rzyndnych" title="Ukłod wspůłrzyndnych – Schlesisch (Wasserpolnisch)" lang="szl" hreflang="szl" data-title="Ukłod wspůłrzyndnych" data-language-autonym="Ślůnski" data-language-local-name="Schlesisch (Wasserpolnisch)" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%86%E0%AE%B3%E0%AF%8D%E0%AE%95%E0%AF%82%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81_%E0%AE%AE%E0%AF%81%E0%AE%B1%E0%AF%88%E0%AE%AE%E0%AF%88" title="ஆள்கூற்று முறைமை – Tamil" lang="ta" hreflang="ta" data-title="ஆள்கூற்று முறைமை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B0%E0%B8%9A%E0%B8%9A%E0%B8%9E%E0%B8%B4%E0%B8%81%E0%B8%B1%E0%B8%94" title="ระบบพิกัด – Thailändisch" lang="th" hreflang="th" data-title="ระบบพิกัด" data-language-autonym="ไทย" data-language-local-name="Thailändisch" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Sistema_ng_koordinado" title="Sistema ng koordinado – Tagalog" lang="tl" hreflang="tl" data-title="Sistema ng koordinado" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Koordinat_sistemi" title="Koordinat sistemi – Türkisch" lang="tr" hreflang="tr" data-title="Koordinat sistemi" data-language-autonym="Türkçe" data-language-local-name="Türkisch" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B0_%D0%BA%D0%BE%D0%BE%D1%80%D0%B4%D0%B8%D0%BD%D0%B0%D1%82" title="Система координат – Ukrainisch" lang="uk" hreflang="uk" data-title="Система координат" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AA%D9%86%D8%A7%D8%B3%D9%82_%D9%86%D8%B8%D8%A7%D9%85" title="متناسق نظام – Urdu" lang="ur" hreflang="ur" data-title="متناسق نظام" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Koordinatalar" title="Koordinatalar – Usbekisch" lang="uz" hreflang="uz" data-title="Koordinatalar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%E1%BB%87_t%E1%BB%8Da_%C4%91%E1%BB%99" title="Hệ tọa độ – Vietnamesisch" lang="vi" hreflang="vi" data-title="Hệ tọa độ" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamesisch" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Sistema_coordinata" title="Sistema coordinata – Waray" lang="war" hreflang="war" data-title="Sistema coordinata" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9D%90%E6%A0%87%E7%B3%BB" title="坐标系 – Wu" lang="wuu" hreflang="wuu" data-title="坐标系" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A7%D7%90%D7%90%D7%A8%D7%93%D7%99%D7%A0%D7%90%D7%98%D7%9F_%D7%A1%D7%99%D7%A1%D7%98%D7%A2%D7%9D" title="קאארדינאטן סיסטעם – Jiddisch" lang="yi" hreflang="yi" data-title="קאארדינאטן סיסטעם" data-language-autonym="ייִדיש" data-language-local-name="Jiddisch" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9D%90%E6%A8%99%E7%B3%BB" title="坐標系 – Chinesisch" lang="zh" hreflang="zh" data-title="坐標系" data-language-autonym="中文" data-language-local-name="Chinesisch" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ch%C5%8D-piau-h%C4%93" title="Chō-piau-hē – Min Nan" lang="nan" hreflang="nan" data-title="Chō-piau-hē" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Min Nan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%9D%90%E6%A8%99" title="坐標 – Kantonesisch" lang="yue" hreflang="yue" data-title="坐標" data-language-autonym="粵語" data-language-local-name="Kantonesisch" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q11210#sitelinks-wikipedia" title="Links auf Artikel in 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