DE-9IM - Wikipedia

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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Topological model</div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/File:DE-9IM-logoSmall.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/DE-9IM-logoSmall.png/250px-DE-9IM-logoSmall.png" decoding="async" width="230" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/DE-9IM-logoSmall.png/500px-DE-9IM-logoSmall.png 1.5x" data-file-width="976" data-file-height="862" /></a><figcaption></figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=DE-9IM&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">May 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The <b>Dimensionally Extended 9-Intersection Model</b> (<b>DE-9IM</b>) is a <a href="/wiki/Topological" class="mw-redirect" title="Topological">topological</a> <a href="/wiki/Interpretation_(logic)" title="Interpretation (logic)">model</a> and a <a href="/wiki/Specification_(technical_standard)" title="Specification (technical standard)">standard</a> used to describe the <a href="/wiki/Spatial_relation" title="Spatial relation">spatial relations</a> of two regions (two <a href="/wiki/2D_geometric_model" title="2D geometric model">geometries in two-dimensions</a>, <b>R</b><sup>2</sup>), in <a href="/wiki/Geometry" title="Geometry">geometry</a>, <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a>, <a href="/wiki/Geospatial_topology" title="Geospatial topology">geospatial topology</a>, and fields related to <a href="/wiki/Spatial_analysis" title="Spatial analysis">computer spatial analysis</a>. The spatial relations expressed by the model are invariant to <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>, <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translation</a> and <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">scaling</a> transformations. </p><p>The matrix provides an approach for classifying geometry relations. Roughly speaking, with a true/false matrix domain, there are 512 possible 2D topologic relations, that can be grouped into <i>binary classification schemes</i>. The English language contains about 10 schemes (relations), such as "intersects", "touches" and "equals". When testing two geometries against a scheme, the result is a <i><b>spatial predicate</b></i> named by the scheme. </p><p>The model was developed by Clementini and others<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><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> based on the seminal works of Egenhofer and others.<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><sup id="cite_ref-sdh1990_4-0" class="reference"><a href="#cite_note-sdh1990-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It has been used as a basis for standards of <i><a href="/wiki/Information_retrieval" title="Information retrieval">queries</a></i> and <i><a href="/wiki/First-order_logic" title="First-order logic">assertions</a></i> in <a href="/wiki/Geographic_information_systems" class="mw-redirect" title="Geographic information systems">geographic information systems</a> (GIS) and <a href="/wiki/Spatial_database" title="Spatial database">spatial databases</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Matrix_model">Matrix model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=1" title="Edit section: Matrix model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The DE-9IM model is based on a 3×3 <a href="/wiki/Intersection" title="Intersection">intersection</a> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> with the form: </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \operatorname {DE9IM} (a,b)={\begin{bmatrix}\dim(I(a)\cap I(b))&\dim(I(a)\cap B(b))&\dim(I(a)\cap E(b))\\\dim(B(a)\cap I(b))&\dim(B(a)\cap B(b))&\dim(B(a)\cap E(b))\\\dim(E(a)\cap I(b))&\dim(E(a)\cap B(b))&\dim(E(a)\cap E(b))\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>DE9IM</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DE9IM} (a,b)={\begin{bmatrix}\dim(I(a)\cap I(b))&\dim(I(a)\cap B(b))&\dim(I(a)\cap E(b))\\\dim(B(a)\cap I(b))&\dim(B(a)\cap B(b))&\dim(B(a)\cap E(b))\\\dim(E(a)\cap I(b))&\dim(E(a)\cap B(b))&\dim(E(a)\cap E(b))\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecbbb04eaa44d40afee0b6d3799ea9d9080c6abb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:76.535ex; height:9.843ex;" alt="{\displaystyle \operatorname {DE9IM} (a,b)={\begin{bmatrix}\dim(I(a)\cap I(b))&\dim(I(a)\cap B(b))&\dim(I(a)\cap E(b))\\\dim(B(a)\cap I(b))&\dim(B(a)\cap B(b))&\dim(B(a)\cap E(b))\\\dim(E(a)\cap I(b))&\dim(E(a)\cap B(b))&\dim(E(a)\cap E(b))\end{bmatrix}}}" /></span> </td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table> <p>where <span class="nowrap">⁠<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 \dim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66115c83c4bb19068adb45849f0f596647c18f2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.875ex; height:2.176ex;" alt="{\displaystyle \dim }" /></span>⁠</span> is the <a href="/wiki/Dimension" title="Dimension">dimension</a> of the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> (∩) of the <a href="/wiki/Interior_of_a_manifold" class="mw-redirect" title="Interior of a manifold">interior</a> (I), <a href="/wiki/Boundary_of_a_manifold" class="mw-redirect" title="Boundary of a manifold">boundary</a> (B), and <a href="/wiki/Exterior_(topology)" class="mw-redirect" title="Exterior (topology)">exterior</a> (E) of geometries <i>a</i> and <i>b</i>. </p><p>The terms <i>interior</i> and <i>boundary</i> in this article are used in the sense used in algebraic topology and manifold theory, not in the sense used in general topology: for example, the interior of a line segment is the line segment without its endpoints, and its boundary is just the two endpoints (in general topology, the interior of a line segment in the plane is empty and the line segment is its own boundary). </p><p>In the notation of topological space operators, the matrix elements can be expressed also as </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml"><i>I</i>(<i>a</i>)=<i>a</i><sup>o</sup>    <i>B</i>(<i>a</i>)=∂<i>a</i>    <i>E</i>(<i>a</i>)=<i>a</i><sup><i>e</i></sup></span></td> <td></td> <td class="nowrap"><span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table> <p>The dimension of <a href="/wiki/Empty_set" title="Empty set">empty sets</a> (∅) are denoted as −1 or <style data-mw-deduplicate="TemplateStyles:r886049734">.mw-parser-output .monospaced{font-family:monospace,monospace}</style><span class="monospaced">F</span> (false). The dimension of non-empty sets (¬∅) are denoted with the maximum number of dimensions of the intersection, specifically <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span> for <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span> for <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">2</span> for <a href="/wiki/Area" title="Area">areas</a>. Then, the <a href="/wiki/Data_domain" title="Data domain">domain</a> of the model is {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">2</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>}. </p><p>A simplified version of <span class="nowrap">⁠<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 \dim(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86019a45bd75e31fd79c705de1b1c2eb1cc237b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.014ex; height:2.843ex;" alt="{\displaystyle \dim(x)}" /></span>⁠</span> values are obtained mapping the values {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0,1,2</span>} to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">T</span> (true), so using the <a href="/wiki/Boolean_domain" title="Boolean domain">boolean domain</a> {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">T</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>}. The matrix, denoted with operators, can be expressed as </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 \operatorname {bin} (\operatorname {DE9IM} (a,b))=\operatorname {9IM} (a,b)={\begin{bmatrix}a^{o}\cap b^{o}\neq \emptyset &a^{o}\cap \partial {b}\neq \emptyset &a^{o}\cap b^{e}\neq \emptyset \\\partial {a}\cap b^{o}\neq \emptyset &\partial {a}\cap \partial {b}\neq \emptyset &\partial {a}\cap b^{e}\neq \emptyset \\a^{e}\cap b^{o}\neq \emptyset &a^{e}\cap \partial {b}\neq \emptyset &a^{e}\cap b^{e}\neq \emptyset \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>bin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>DE9IM</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>9IM</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>∩</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo>∩</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo>∩</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mo>∩</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msup> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>∩</mo> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>∩</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>≠</mo> <mi mathvariant="normal">∅</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {bin} (\operatorname {DE9IM} (a,b))=\operatorname {9IM} (a,b)={\begin{bmatrix}a^{o}\cap b^{o}\neq \emptyset &a^{o}\cap \partial {b}\neq \emptyset &a^{o}\cap b^{e}\neq \emptyset \\\partial {a}\cap b^{o}\neq \emptyset &\partial {a}\cap \partial {b}\neq \emptyset &\partial {a}\cap b^{e}\neq \emptyset \\a^{e}\cap b^{o}\neq \emptyset &a^{e}\cap \partial {b}\neq \emptyset &a^{e}\cap b^{e}\neq \emptyset \end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048f66f466b16dcb79c73e2912132bc443f41971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:76.014ex; height:9.509ex;" alt="{\displaystyle \operatorname {bin} (\operatorname {DE9IM} (a,b))=\operatorname {9IM} (a,b)={\begin{bmatrix}a^{o}\cap b^{o}\neq \emptyset &a^{o}\cap \partial {b}\neq \emptyset &a^{o}\cap b^{e}\neq \emptyset \\\partial {a}\cap b^{o}\neq \emptyset &\partial {a}\cap \partial {b}\neq \emptyset &\partial {a}\cap b^{e}\neq \emptyset \\a^{e}\cap b^{o}\neq \emptyset &a^{e}\cap \partial {b}\neq \emptyset &a^{e}\cap b^{e}\neq \emptyset \end{bmatrix}}}" /></span> </td> <td></td> <td class="nowrap"><span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span></td></tr></tbody></table> <p>The elements of the matrix can be named as shown below: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {\begin{bmatrix}II&IB&IE\\BI&BB&BE\\EI&EB&EE\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>I</mi> <mi>I</mi> </mtd> <mtd> <mi>I</mi> <mi>B</mi> </mtd> <mtd> <mi>I</mi> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> <mi>I</mi> </mtd> <mtd> <mi>B</mi> <mi>B</mi> </mtd> <mtd> <mi>B</mi> <mi>E</mi> </mtd> </mtr> <mtr> <mtd> <mi>E</mi> <mi>I</mi> </mtd> <mtd> <mi>E</mi> <mi>B</mi> </mtd> <mtd> <mi>E</mi> <mi>E</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}II&IB&IE\\BI&BB&BE\\EI&EB&EE\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244681cfb37bfc62a2dfcc929371d639d67d7d09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.536ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}II&IB&IE\\BI&BB&BE\\EI&EB&EE\end{bmatrix}}}" /></span> </td> <td></td> <td class="nowrap"><span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span></td></tr></tbody></table> <p>Both matrix forms, with dimensional and boolean domains, can be <a href="/wiki/Serialization" title="Serialization">serialized</a> as "<i>DE-9IM string codes</i>", which represent them in a single-line string pattern. Since 1999 the <i>string codes</i> have a <a href="#Standards">standard</a><sup id="cite_ref-firstStd_5-0" class="reference"><a href="#cite_note-firstStd-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> format. </p><p>For output checking or pattern analysis, a matrix value (or a string code) can be checked by a "<a href="/wiki/Mask_(computing)" title="Mask (computing)">mask</a>": a desired output value with optional <a href="/wiki/Asterisk" title="Asterisk">asterisk</a> symbols as <a href="/wiki/Wildcard_character" title="Wildcard character">wildcards</a> — that is, "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">*</span>" indicating output positions that the designer does not care about (free values or "don't-care positions"). The domain of the mask elements is {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">2</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">*</span>}, or {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">T</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">*</span>} for the boolean form. </p><p>The simpler models <i>4-Intersection</i> and <i>9-Intersection</i> were proposed before DE-9IM for expressing <i>spatial relations</i><sup id="cite_ref-4vs9_6-0" class="reference"><a href="#cite_note-4vs9-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (and originated the terms <i>4IM</i> and <i>9IM</i>). They can be used instead of the DE-9IM to optimize computation when input conditions satisfy specific constraints. </p> <div class="mw-heading mw-heading3"><h3 id="Illustration">Illustration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=2" title="Edit section: Illustration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Visually, for two overlapping polygonal geometries, the result of the function <b>DE_9IM(<i>a</i>,<i>b</i>)</b> looks like:<sup id="cite_ref-PostGISch4_7-0" class="reference"><a href="#cite_note-PostGISch4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table border="0" align="center"> <tbody><tr> <td> </td> <td align="center"> <table border="0" summary="manufactured viewport for HTML img" cellspacing="0" cellpadding="0"> <tbody><tr> <td align="center" valign="middle"><i>b</i>   <span typeof="mw:File"><a href="/wiki/File:DE9IM_b.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/DE9IM_b.svg/120px-DE9IM_b.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e5/DE9IM_b.svg/150px-DE9IM_b.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e5/DE9IM_b.svg/200px-DE9IM_b.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td></tr></tbody></table> </td></tr> <tr> <td align="center" valign="middle"> <table border="0" summary="manufactured viewport for HTML img" cellspacing="0" cellpadding="0"> <tbody><tr> <td align="center" valign="middle"><i>a</i><br /><span typeof="mw:File"><a href="/wiki/File:DE9IM_a.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/DE9IM_a.svg/100px-DE9IM_a.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/DE9IM_a.svg/150px-DE9IM_a.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/DE9IM_a.svg/200px-DE9IM_a.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> </td></tr></tbody></table> </td> <td> <table border="0" class="wikitable"> <tbody><tr> <th align="center"> </th> <th align="center"><b>Interior</b> </th> <th align="center"><b>Boundary</b> </th> <th align="center"><b>Exterior</b> </th></tr> <tr> <td align="center"><b>Interior</b> </td> <td align="center" style="border-right: 2px solid #BCB; border-bottom: 2px solid #BCB;"><span typeof="mw:File"><a href="/wiki/File:DE9IM_II.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/DE9IM_II.svg/100px-DE9IM_II.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/DE9IM_II.svg/150px-DE9IM_II.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/DE9IM_II.svg/200px-DE9IM_II.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <p>  <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 \dim[I(a){\color {red}\cap }I(b)]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[I(a){\color {red}\cap }I(b)]=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e17e1d2938ec6f4c1598820d5afca1e57ce2c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.17ex; height:2.843ex;" alt="{\displaystyle \dim[I(a){\color {red}\cap }I(b)]=2}" /></span>   </p> </td> <td align="center" style="border-right: 2px solid #BCB; border-bottom: 2px solid #BCB;"><span typeof="mw:File"><a href="/wiki/File:DE9IM_IB.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/DE9IM_IB.svg/100px-DE9IM_IB.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/DE9IM_IB.svg/150px-DE9IM_IB.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/DE9IM_IB.svg/200px-DE9IM_IB.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <p>  <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 \dim[I(a){\color {red}\cap }B(b)]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[I(a){\color {red}\cap }B(b)]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc870a326b37aaf3114d3286875417333b16fb5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.762ex; height:2.843ex;" alt="{\displaystyle \dim[I(a){\color {red}\cap }B(b)]=1}" /></span>   </p> </td> <td align="center" style="border-bottom: 2px solid #BCB;"><span typeof="mw:File"><a href="/wiki/File:DE9IM_IE.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/DE9IM_IE.svg/100px-DE9IM_IE.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/DE9IM_IE.svg/150px-DE9IM_IE.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/DE9IM_IE.svg/200px-DE9IM_IE.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span> <p>  <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 \dim[I(a){\color {red}\cap }E(b)]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[I(a){\color {red}\cap }E(b)]=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44a3c38cbce5b22fc9f3c0b3da98fca7074fcbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.773ex; height:2.843ex;" alt="{\displaystyle \dim[I(a){\color {red}\cap }E(b)]=2}" /></span>   </p> </td></tr> <tr> <td align="center"><span class="bold"><b>Boundary</b></span> </td> <td align="center" style="border-right: 2px solid #BCB; border-bottom: 2px solid #BCB;"><div class="informalfigure"><div><span typeof="mw:File"><a href="/wiki/File:DE9IM_BI.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/DE9IM_BI.svg/120px-DE9IM_BI.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/DE9IM_BI.svg/150px-DE9IM_BI.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/DE9IM_BI.svg/200px-DE9IM_BI.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></div> <p>  <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 \dim[B(a){\color {red}\cap }I(b)]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[B(a){\color {red}\cap }I(b)]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0903c4c326c3885a8b3d95740af8e35eda09e257" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.762ex; height:2.843ex;" alt="{\displaystyle \dim[B(a){\color {red}\cap }I(b)]=1}" /></span>   </p> </td> <td align="center" style="border-right: 2px solid #BCB; border-bottom: 2px solid #BCB;"><div class="informalfigure"><div><span typeof="mw:File"><a href="/wiki/File:DE9IM_BB.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/DE9IM_BB.svg/120px-DE9IM_BB.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/DE9IM_BB.svg/150px-DE9IM_BB.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/DE9IM_BB.svg/200px-DE9IM_BB.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></div> <p>  <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 \dim[B(a){\color {red}\cap }B(b)]=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[B(a){\color {red}\cap }B(b)]=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3bb8821030e73c27dd93e4a8efd044ce36c7c30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.354ex; height:2.843ex;" alt="{\displaystyle \dim[B(a){\color {red}\cap }B(b)]=0}" /></span>   </p> </td> <td align="center" style="border-bottom: 2px solid #BCB;"><div class="informalfigure"><div><span typeof="mw:File"><a href="/wiki/File:DE9IM_BE.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/DE9IM_BE.svg/100px-DE9IM_BE.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/DE9IM_BE.svg/150px-DE9IM_BE.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/DE9IM_BE.svg/200px-DE9IM_BE.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></div> <p>  <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 \dim[B(a){\color {red}\cap }E(b)]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[B(a){\color {red}\cap }E(b)]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b208644a96773d199ef5f5576bcd93294daeb134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.366ex; height:2.843ex;" alt="{\displaystyle \dim[B(a){\color {red}\cap }E(b)]=1}" /></span>   </p> </td></tr> <tr> <td align="center"><span class="bold"><b>Exterior</b></span> </td> <td align="center" style="border-right: 2px solid #BCB;"><div class="informalfigure"><div><span typeof="mw:File"><a href="/wiki/File:DE9IM_EI.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/DE9IM_EI.svg/100px-DE9IM_EI.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/DE9IM_EI.svg/150px-DE9IM_EI.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/DE9IM_EI.svg/200px-DE9IM_EI.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></div> <p>  <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 \dim[E(a){\color {red}\cap }I(b)]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[E(a){\color {red}\cap }I(b)]=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c976aad45223c32a4a2c1bf2da9bd509e4e0ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.773ex; height:2.843ex;" alt="{\displaystyle \dim[E(a){\color {red}\cap }I(b)]=2}" /></span>   </p> </td> <td align="center" style="border-right: 2px solid #BCB;"><div class="informalfigure"><div><span typeof="mw:File"><a href="/wiki/File:DE9IM_EB.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/DE9IM_EB.svg/100px-DE9IM_EB.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/DE9IM_EB.svg/150px-DE9IM_EB.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/DE9IM_EB.svg/200px-DE9IM_EB.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></div> <p>  <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 \dim[E(a){\color {red}\cap }B(b)]=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[E(a){\color {red}\cap }B(b)]=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caaf90fed3f1761b438b92d3562c484b9043ac5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.366ex; height:2.843ex;" alt="{\displaystyle \dim[E(a){\color {red}\cap }B(b)]=1}" /></span>   </p> </td> <td align="center"><div class="informalfigure"><div><span typeof="mw:File"><a href="/wiki/File:DE9IM_EE.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/DE9IM_EE.svg/120px-DE9IM_EE.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/DE9IM_EE.svg/150px-DE9IM_EE.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/DE9IM_EE.svg/200px-DE9IM_EE.svg.png 2x" data-file-width="200" data-file-height="200" /></a></span></div></div> <p>  <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 \dim[E(a){\color {red}\cap }E(b)]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">[</mo> <mi>E</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mo>∩</mo> </mstyle> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim[E(a){\color {red}\cap }E(b)]=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2e8465051e0b0a9dccf42e620fa0a92de86caa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.377ex; height:2.843ex;" alt="{\displaystyle \dim[E(a){\color {red}\cap }E(b)]=2}" /></span>   </p> </td></tr></tbody></table> </td></tr></tbody></table> <p>This matrix can be <a href="/wiki/Serialization" title="Serialization">serialized</a>. Reading from left-to-right and top-to-bottom, the result is <span class="nowrap">⁠<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 II=2,\,IB=1,\,IE=2,\,BI=1,\,BB=0,\,BE=1,\,EI=2,\,EB=1,\,EE=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>I</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>I</mi> <mi>B</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>I</mi> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mi>I</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mi>B</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mi>I</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mi>B</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mi>E</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle II=2,\,IB=1,\,IE=2,\,BI=1,\,BB=0,\,BE=1,\,EI=2,\,EB=1,\,EE=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bec65cf10c755bdd96b9eaf49d45495b645a55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:77.985ex; height:2.509ex;" alt="{\displaystyle II=2,\,IB=1,\,IE=2,\,BI=1,\,BB=0,\,BE=1,\,EI=2,\,EB=1,\,EE=2}" /></span>⁠</span>.  So, in a compact representation as string code is '<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced"><b>212101212</b></span>'. </p> <div class="mw-heading mw-heading2"><h2 id="Spatial_predicates">Spatial predicates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=3" title="Edit section: Spatial predicates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/Topological_property" title="Topological property">topological property</a> based on a DE-9IM <a href="/wiki/Binary_relation" title="Binary relation">binary</a> <a href="/wiki/Spatial_relation" title="Spatial relation">spatial relation</a> is a <b>spatial predicate</b>. For ease of use "named spatial predicates" have been defined for some common relations, which later became standard predicates. The <i>spatial predicate <a href="/wiki/Subroutine" class="mw-redirect" title="Subroutine">functions</a></i> that can be derived from DE-9IM include:<sup id="cite_ref-sdh1990_4-1" class="reference"><a href="#cite_note-sdh1990-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd>Predicates defined with masks of domain {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">T</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">*</span>}:</dd></dl> <table class="wikitable"> <tbody><tr> <th style="width:8em">Name (synonym) </th> <th colspan="4">Intersection matrix and mask code string<br />(<a href="/wiki/Boolean_algebra#Basic_operations" title="Boolean algebra">boolean OR</a> between matrices) </th> <th>Meaning and definition<sup id="cite_ref-sdh1990_4-2" class="reference"><a href="#cite_note-sdh1990-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </th> <th>Equivalent </th></tr> <tr> <th rowspan="2">Equals </th> <td colspan="4"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7586db813711ea86b0ecde20b6fde35a6bf9cdf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.826ex; height:5.009ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml"><i>II</i> ∧ ~<i>IE</i> ∧ ~<i>BE</i> ∧ ~<i>EI</i> ∧ ~<i>EB</i></span></td> <td></td> <td class="nowrap"><span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span></td></tr></tbody></table><i>a</i> and <i>b</i> are topologically <a href="/wiki/Equality_(relational_operator)" class="mw-redirect" title="Equality (relational operator)">equal</a>. "Two geometries are topologically equal if their interiors intersect and no part of the interior or boundary of one geometry intersects the exterior of the other".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </td> <td rowspan="2"><i>Within</i> & <i>Contains</i> </td></tr> <tr> <td><code>T*F**FFF*</code> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th rowspan="2">Disjoint </th> <td colspan="4"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {F} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {F} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cadfa6d63e4c3ac3e51e04d320b51b9bf62b67b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.557ex; margin-bottom: -0.281ex; width:7.462ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {F} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml"><i>~II</i> ∧ ~<i>IB</i> ∧ ~<i>BI</i> ∧ ~<i>BB</i></span></td> <td></td> <td class="nowrap"><span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span></td></tr></tbody></table><i>a</i> and <i>b</i> are <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a>: they have no points in common. They form a set of <a href="/wiki/Disconnected_(topology)#Disconnected_spaces" class="mw-redirect" title="Disconnected (topology)">disconnected</a> geometries. </td> <td rowspan="2"><i>not Intersects</i> </td></tr> <tr> <td><code>FF*FF****</code> </td> <td> </td> <td> </td> <td> </td></tr> <tr> <th rowspan="2">Touches<br />(meets) </th> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c9a224bf36c57a655dab756a0b6ba6f60cfd1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6efc4b0b9c381240407b6981813097b6056b343a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.324ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td colspan="2"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e124785d99f66e8b4a884feb66ba75ff65e235c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {F} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml"><i>~II</i> ∧ (<i>IB</i> ∨ <i>BI</i> ∨ <i>BB</i>)</span></td> <td></td> <td class="nowrap"><span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span></td></tr></tbody></table><i>a</i> touches <i>b</i>: they have at least one point in common, but their interiors do not intersect. </td> <td rowspan="2"> </td></tr> <tr> <td><code>FT*******</code> </td> <td><code>F**T*****</code> </td> <td colspan="2"><code>F***T****</code> </td></tr> <tr> <th rowspan="2">Contains </th> <td colspan="4"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d8bb25b663aacd92e78e5d79f6e21f1c1e0fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml"><i>II</i> ∧ ~<i>EI</i> ∧ ~<i>EB</i></span></td> <td></td> <td class="nowrap"><span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span></td></tr></tbody></table><i>a</i> contains <i>b</i>: geometry <i>b</i> lies in <i>a</i>, and the interiors intersect. Another definition: "<i>a</i> contains <i>b</i> <a href="/wiki/If_and_only_if" title="If and only if">iff</a> no points of <i>b</i> lie in the exterior of <i>a</i>, and at least one point of the interior of <i>b</i> lies in the interior of <i>a</i>".<sup id="cite_ref-davis2007_10-0" class="reference"><a href="#cite_note-davis2007-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </td> <td rowspan="2"><i>Within</i>(<i>b</i>,<i>a</i>) </td></tr> <tr> <td colspan="4"><code>T*****FF*</code> </td></tr> <tr> <th rowspan="2">Covers </th> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d8bb25b663aacd92e78e5d79f6e21f1c1e0fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11dd55de574f24671d3ec0f2f21eb3ef77c0205b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc045b3009973f60c3b1d33526b6e2cc8a0b7ca4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f02c4d6acd00929b68a482b7d4882391934b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {F} &\mathrm {F} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml">(<i>II</i> ∨ <i>IB</i> ∨ <i>BI</i> ∨ <i>BB</i>) ∧ ~<i>EI</i> ∧ ~<i>EB</i></span></td> <td></td> <td class="nowrap"><span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span></td></tr></tbody></table><i>a</i> covers <i>b</i>: geometry <i>b</i> lies in <i>a</i>. Other definitions: "At least one point of <i>b</i> lies in <i>a</i>, and no points of <i>b</i> lie in the exterior of <i>a</i>", or "Every point of <i>b</i> is a point of (the interior or boundary of) <i>a</i>". </td> <td rowspan="2"><i>CoveredBy</i>(<i>b</i>,<i>a</i>) </td></tr> <tr> <td><code>T*****FF*</code> </td> <td><code>*T****FF*</code> </td> <td><code>***T**FF*</code> </td> <td><code>****T*FF*</code> </td></tr></tbody></table> <dl><dd>Predicates that can be obtained from the above by <a href="/wiki/Negation#Definition" title="Negation">logical negation</a> or parameter inversion (<a href="/wiki/Transposed_matrix" class="mw-redirect" title="Transposed matrix">matrix transposition</a>), as indicated by the last column:</dd></dl> <table class="wikitable"> <tbody><tr> <th rowspan="2" style="width:8em">Intersects </th> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/366e2a07dd38dc5332fc0218bc27fceee0ffc7eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.324ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89a3ef2f1e4d4b44eca1bb22126a42e3ac095662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.324ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ee49e959010debadad86188963207fd39e2bd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.324ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60154536d270939c76a035e11c4318bf5984ae38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.324ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {T} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><i>a</i> <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersects</a> <i>b</i>: geometries <i>a</i> and <i>b</i> have at least one point in common. </td> <td rowspan="2"><i>not Disjoint</i> </td></tr> <tr> <td><code>T********</code> </td> <td><code>*T*******</code> </td> <td><code>***T*****</code> </td> <td><code>****T****</code> </td></tr> <tr> <th rowspan="2">Within <br />(inside) </th> <td colspan="4"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc321d8035064d37880f4a87cdb50321ecd180a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.557ex; margin-bottom: -0.281ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><i>a</i> is within <i>b</i>: <i>a</i> lies in the interior of <i>b</i>. </td> <td rowspan="2"><i>Contains</i>(<i>b</i>,<i>a</i>) </td></tr> <tr> <td colspan="4"><code>T*F**F***</code> </td></tr> <tr> <th rowspan="2">CoveredBy </th> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc321d8035064d37880f4a87cdb50321ecd180a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.557ex; margin-bottom: -0.281ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac41547656355cd1896942c2226b307ea4360fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.557ex; margin-bottom: -0.281ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {T} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac9abff3dcc4a2765c100d65989c45b4e0dd19a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.557ex; margin-bottom: -0.281ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {T} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {T} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {T} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c7890ffb06d92cac8cf52ce3e43aaead54d97ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.557ex; margin-bottom: -0.281ex; width:7.575ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {*} &\mathrm {*} &\mathrm {F} \\\mathrm {*} &\mathrm {T} &\mathrm {F} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2"><i>a</i> is covered by <i>b</i> (extends <i>Within</i>): geometry <i>a</i> lies in <i>b</i>. Other definitions: "At least one point of <i>a</i> lies in <i>b</i>, and no points of <i>a</i> lie in the exterior of <i>b</i>", or "Every point of <i>a</i> is a point of (the interior or boundary of) <i>b</i>". </td> <td rowspan="2"><i>Covers</i>(<i>b</i>,<i>a</i>) </td></tr> <tr> <td><code>T*F**F***</code> </td> <td><code>*TF**F***</code> </td> <td><code>**FT*F***</code> </td> <td><code>**F*TF***</code> </td></tr></tbody></table> <dl><dd>Predicates that utilize the input dimensions, and are defined with masks of domain {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">T</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">*</span>}:</dd></dl> <table class="wikitable"> <tbody><tr> <th rowspan="2" style="width:8em"><b>Crosses</b><span style="font-weight:normal"><br /><span class="nowrap">⁠<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 \dim(a)\neq \dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)\neq \dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa31b4ceb767600a4ea814e7e9165460f8b988f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)\neq \dim(b)}" /></span>⁠</span> or <br /><span class="texhtml">dim(any) = 1</span></span> </th> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe532477bb720127bd54a5e8f9f5e36f5237ab45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.688ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa67592ee02c4f63cdf22d4b69d76a5772617757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.552ex; margin-bottom: -0.286ex; width:7.324ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td colspan="2"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {0} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {0} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a51b91577eab17c9001c5c886b901032510d17b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.959ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {0} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2" colspan="2"><i>a</i> crosses <i>b</i>: they have some but not all interior points in common, and the dimension of the intersection is less than that of at least one of them. The following mask selection rules must <b>only</b> be checked when <span class="nowrap">⁠<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 \dim(a)\neq \dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)\neq \dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa31b4ceb767600a4ea814e7e9165460f8b988f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)\neq \dim(b)}" /></span>⁠</span> (except for line / line inputs, which <i>are</i> allowed), otherwise the predicate is <b>false</b>:<sup id="cite_ref-ST_Crosses_11-0" class="reference"><a href="#cite_note-ST_Crosses-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml">(<i>II</i>=0)</span> for lines,   <span class="texhtml">(<i>II</i> ∧ <i>IE</i>)</span> when <span class="nowrap">⁠<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 \dim(a)<\dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)<\dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b9f2e4c9477008bce8ba6729b0cbc341950aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)<\dim(b)}" /></span>⁠</span>,   <span class="texhtml">(<i>II</i> ∧ <i>EI</i>)</span> when <span class="nowrap">⁠<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 \dim(a)>\dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)>\dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088752ca7fec0101890da544cf05644435a4e3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)>\dim(b)}" /></span>⁠</span></td> <td></td> <td class="nowrap"><span id="math_10" class="reference nourlexpansion" style="font-weight:bold;">10</span></td></tr></tbody></table> </td></tr> <tr> <td><code>T*T******</code> <small><br /><span class="nowrap">⁠<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 \dim(a)<\dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)<\dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b9f2e4c9477008bce8ba6729b0cbc341950aad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)<\dim(b)}" /></span>⁠</span></small> </td> <td><code>T*****T**</code> <small><br /><span class="nowrap">⁠<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 \dim(a)>\dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)>\dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/088752ca7fec0101890da544cf05644435a4e3fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)>\dim(b)}" /></span>⁠</span></small> </td> <td colspan="2"><code>0********</code> <small><br /><span class="texhtml">dim(any) = 1</span></small> </td></tr> <tr> <th rowspan="2">Overlaps<span style="font-weight:normal"><br /><span class="nowrap">⁠<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 \dim(a)=\dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)=\dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c3c81d28342acf5f9c60afd8c8a475647fd9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)=\dim(b)}" /></span>⁠</span></span> </th> <td><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 {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a0c75230be0eda779214e6f2a433182fc89a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.552ex; margin-bottom: -0.286ex; width:7.688ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {T} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td colspan="3"><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 {\Bigl [}{\begin{smallmatrix}\mathrm {1} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </mtd> </mtr> </mtable> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {1} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75d59c8c66be427d0c7e0deed17c48dca60c73e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.552ex; margin-bottom: -0.286ex; width:7.688ex; height:4.843ex;" alt="{\displaystyle {\Bigl [}{\begin{smallmatrix}\mathrm {1} &\mathrm {*} &\mathrm {T} \\\mathrm {*} &\mathrm {*} &\mathrm {*} \\\mathrm {T} &\mathrm {*} &\mathrm {*} \end{smallmatrix}}{\Bigr ]}}" /></span> </td> <td rowspan="2" colspan="2"><i>a</i> overlaps <i>b</i>: they have some but not all points in common, they have the same dimension, and the intersection of the interiors of the two geometries has the same dimension as the geometries themselves. The following mask selection rules must <b>only</b> be checked when <span class="nowrap">⁠<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 \dim(a)=\dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)=\dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c3c81d28342acf5f9c60afd8c8a475647fd9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.695ex; height:2.843ex;" alt="{\displaystyle \dim(a)=\dim(b)}" /></span>⁠</span>, otherwise the predicate is <b>false</b>: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="texhtml">(<i>II</i> ∧ <i>IE</i> ∧ <i>EI</i>)</span> for points or surfaces,   <span class="texhtml">(<i>II</i>=1 ∧ <i>IE</i> ∧ <i>EI</i>)</span> for lines</td> <td></td> <td class="nowrap"><span id="math_11" class="reference nourlexpansion" style="font-weight:bold;">11</span></td></tr></tbody></table> </td></tr> <tr> <td><code>T*T***T**</code> <small><br /><span class="texhtml">dim = 0 or 2</span></small> </td> <td colspan="2"><code>1*T***T**</code> <small><br /><span class="texhtml">dim = 1</span></small> </td></tr></tbody></table> <p>Notice that: </p> <ul><li>The <i>topologically equal</i> definition does not imply that they have the same points or even that they are of the same class.</li></ul> <ul><li>The output of <span class="nowrap">⁠<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 \operatorname {DE-9IM} (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">E</mi> <mtext>-</mtext> <mn>9</mn> <mi mathvariant="normal">I</mi> <mi mathvariant="normal">M</mi> </mrow> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {DE-9IM} (a,b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f3d5be476f8b470ab0c837657fb54fadc14b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.337ex; height:2.843ex;" alt="{\displaystyle \operatorname {DE-9IM} (a,b)}" /></span>⁠</span> have the information contained in a list of all interpretable predicates about geometries <i>a</i> and <i>b</i>.</li></ul> <ul><li>All predicates are computed by masks. Only <i>Crosses</i> and <i>Overlaps</i> have additional conditions about <span class="nowrap">⁠<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 \dim(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc072c03308ecaae0cc3225ea58a83dbc619ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.914ex; height:2.843ex;" alt="{\displaystyle \dim(a)}" /></span>⁠</span> and <span class="nowrap">⁠<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 \dim(b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a204d7a71c8a2a41606ddd10e0a898e591679fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.682ex; height:2.843ex;" alt="{\displaystyle \dim(b)}" /></span>⁠</span>.</li></ul> <ul><li>All mask string codes end with <code>*</code>. This is because <i>EE</i> is trivially true, and thus provides no useful information.</li></ul> <ul><li>The <i>Equals</i> mask, <code>T*F**FFF*</code>, is the "merge" of <i>Contains</i> (<code>T*****FF*</code>) and <i>Within</i> (<code>T*F**F***</code>): <span class="texhtml">(<i>II</i> ∧ ~<i>EI</i> ∧ ~<i>EB</i>) ∧ (<i>II</i> ∧ ~<i>IE</i> ∧ ~<i>BE</i>)</span>.</li></ul> <ul><li>The mask <code>T*****FF*</code> occurs in the definition of both <i>Contains</i> and <i>Covers</i>. <i>Covers</i> is a more inclusive relation. In particular, unlike <i>Contains</i> it does not distinguish between points in the boundary and in the interior of geometries. For most situations, <i>Covers</i> should be used in preference to <i>Contains</i>.</li></ul> <ul><li>Similarly, the mask <code>T*F**F***</code> occurs in the definition of both <i>Within</i> and <i>CoveredBy</i>. For most situations, <i>CoveredBy</i> should be used in preference to <i>Within</i>.</li></ul> <ul><li>Historically, other terms and other formal approaches have been used to express <i>spatial predicates</i>; for example <a href="/wiki/Region_connection_calculus" title="Region connection calculus">region connection calculus</a> was introduced in 1992 by<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Randell, Cohn and Cohn.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The spatial predicates have the following properties of <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>: </p> <ul><li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a>: Equals, Contains, Covers, CoveredBy, Intersects, Within</li> <li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Anti-reflexive</a>: Disjoint</li> <li><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a>: Equals, Intersects, Crosses, Touches, Overlaps</li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a>: Equals, Contains, Covers, CoveredBy, Within</li></ul> <div class="mw-heading mw-heading3"><h3 id="Interpretation">Interpretation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=5" title="Edit section: Interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:TopologicSpatialRelarions2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/TopologicSpatialRelarions2.png/400px-TopologicSpatialRelarions2.png" decoding="async" width="400" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/TopologicSpatialRelarions2.png/600px-TopologicSpatialRelarions2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/55/TopologicSpatialRelarions2.png 2x" data-file-width="789" data-file-height="561" /></a><figcaption>Examples of spatial relations.</figcaption></figure> <p>The choice of terminology and semantics for the spatial predicates is based on reasonable conventions and the tradition of topological studies.<sup id="cite_ref-sdh1990_4-3" class="reference"><a href="#cite_note-sdh1990-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Relationships such as <i>Intersects</i>, <i>Disjoint</i>, <i>Touches</i>, <i>Within</i>, <i>Equals</i> (between two geometries <i>a</i> and <i>b</i>) have an obvious semantic:<sup id="cite_ref-davis2007_10-1" class="reference"><a href="#cite_note-davis2007-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-cafrca_13-0" class="reference"><a href="#cite_note-cafrca-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <dl><dt><i>Equals</i></dt> <dd><i>a</i> = <i>b</i> that is (<i>a</i> ∩ <i>b</i> = <i>a</i>) ∧ (<i>a</i> ∩ <i>b</i> = <i>b</i>)</dd> <dt><i>Within</i></dt> <dd><i>a</i> ∩ <i>b</i> = <i>a</i></dd> <dt><i>Intersects</i></dt> <dd><i>a</i> ∩ <i>b</i> ≠ ∅</dd> <dt><i>Touches</i></dt> <dd>(<i>a</i> ∩ <i>b</i> ≠ ∅) ∧ (<i>a</i><sup>ο</sup> ∩ <i>b</i><sup>ο</sup> = ∅)</dd></dl> <p>The predicates <i>Contains</i> and <i>Within</i> have subtle aspects to their definition which are contrary to intuition. For example,<sup id="cite_ref-davis2007_10-2" class="reference"><a href="#cite_note-davis2007-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> a line <i>L</i> which is completely contained in the boundary of a polygon <i>P</i> is <i>not</i> considered to be contained in <i>P</i>. This quirk can be expressed as "Polygons do not contain their boundary". This issue is caused by the final clause of the <i>Contains</i> definition above: "at least one point of the interior of B lies in the interior of A". For this case, the predicate <i>Covers</i> has more intuitive semantics (see definition), avoiding boundary considerations. </p><p>For better understanding, the dimensionality of inputs can be used as justification for a gradual introduction of semantic complexity: </p> <dl><dd><table class="wikitable"> <tbody><tr> <th>Relations between </th> <th>Appropriate predicates </th> <th>Semantic added </th></tr> <tr> <td>point/point </td> <td><i>Equals</i>, <i>Disjoint</i> </td> <td>Other valid predicates collapses into <i>Equals</i>. </td></tr> <tr> <td>point/line </td> <td>adds <i>Intersects</i> </td> <td><i>Intersects</i> is a refinement of <i>Equals</i>: "some equal point at the line". </td></tr> <tr> <td>line/line </td> <td width="180">adds <i>Touches</i>, <i>Crosses</i>, ... </td> <td><i>Touches</i> is a refinement of <i>Intersects</i>, about "boundaries" only. <i>Crosses</i> is about "only one point". </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Coverage_on_possible_matrix_results">Coverage on possible matrix results</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=6" title="Edit section: Coverage on possible matrix results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The number of possible results in a boolean <i>9IM</i> matrix is 2<sup>9</sup>=512, and in a DE-9IM matrix is 3<sup>9</sup>=6561. The percentage of these results that satisfy a specific predicate is determined as following, </p> <table class="wikitable"><tbody><tr style="vertical-align:top"><th scope="col" style="text-align:right;">Probability</th><th scope="col">Name</th></tr><tr style="vertical-align:top"><td style="text-align:right;"> 93.7%</td><td><i>Intersects</i></td></tr><tr style="vertical-align:top"><td style="text-align:right;"> 43.8%</td><td><i>Touches</i></td></tr><tr style="vertical-align:top"><td style="text-align:right;"> 25%</td><td><i>Crosses</i> (for valid inputs, 0% otherwise)</td></tr><tr style="vertical-align:top"><td style="text-align:right;"> 23.4%</td><td><i>Covers</i> and <i>CoveredBy</i></td></tr><tr style="vertical-align:top"><td style="text-align:right;"> 12.5%</td><td><i>Contains</i>, <i>Overlaps</i> (for valid inputs, 0% otherwise) and <i>Within</i></td></tr><tr style="vertical-align:top"><td style="text-align:right;"> 6.3%</td><td><i>Disjoint</i></td></tr><tr style="vertical-align:top"><td style="text-align:right;"> 3.1%</td><td><i>Equals</i></td></tr></tbody></table> <p>On usual applications the geometries intersects <i>a priori</i>, and the other relations are checked. </p><p>The composite predicates "<i>Intersects</i> OR <i>Disjoint</i>" and "<i>Equals</i> OR <i>Different</i>" have the sum 100% (always true predicates), but "<i>Covers</i> OR <i>CoveredBy</i>" have 41%, that is not the sum, because they are neither logical complements or independent relations; similarly "<i>Contains</i> OR <i>Within</i>", have 21%. The sum 25 % + 12.5 % = 37.5 % is obtained when ignoring overlapping lines in "<i>Crosses</i> OR <i>Overlaps</i>", because the valid input sets are disjoint. </p> <div class="mw-heading mw-heading2"><h2 id="Queries_and_assertions">Queries and assertions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=7" title="Edit section: Queries and assertions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <i>DE-9IM</i> offers a full descriptive assertion about the two input geometries. It is a mathematical function that represents a <a href="/wiki/Functional_completeness" title="Functional completeness">complete set</a> of all possible relations about two entities, like a <a href="/wiki/Truth_table" title="Truth table">Truth table</a>, the <a href="/wiki/Three-way_comparison" title="Three-way comparison">Three-way comparison</a>, a <a href="/wiki/Karnaugh_map#2-variable_map_examples" title="Karnaugh map">Karnaugh map</a> or a <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a>. Each output value is like a truth table line, that represent relations of specific inputs. </p><p>As illustrated above, the output '212101212' resulted from <i>DE-9IM</i>(<i>a</i>,<i>b</i>) is a complete description of all topologic relations between specific geometries <i>a</i> and <i>b</i>. It says to us that <span class="nowrap">⁠<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 II=2,\,IB=1,\,IE=2,\,BI=1,\,BB=0,\,BE=1,\,EI=2,\,EB=1,\,EE=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>I</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>I</mi> <mi>B</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>I</mi> <mi>E</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mi>I</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mi>B</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mi>E</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mi>I</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mi>B</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>E</mi> <mi>E</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle II=2,\,IB=1,\,IE=2,\,BI=1,\,BB=0,\,BE=1,\,EI=2,\,EB=1,\,EE=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bec65cf10c755bdd96b9eaf49d45495b645a55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:77.985ex; height:2.509ex;" alt="{\displaystyle II=2,\,IB=1,\,IE=2,\,BI=1,\,BB=0,\,BE=1,\,EI=2,\,EB=1,\,EE=2}" /></span>⁠</span>. </p><p>By other hand, if we check predicates like <i>Intersects</i>(<i>a</i>,<i>b</i>) or <i>Touches</i>(<i>a</i>,<i>b</i>) — for the same example we have "<i>Intersects</i>=<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">true</span> and <i>Touches</i>=<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">true</span>" — it is an incomplete description of "all topologic relations". Predicates also do not say any thing about the dimensionality of the geometries (it doesn't matter if <i>a</i> and <i>b</i> are lines, areas or points). </p><p>This independence of geometry-type and the lack of <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">completeness</a>, on <i>predicates</i>, are useful for <a href="/wiki/Query_language" title="Query language">general queries</a> about two geometries: </p> <dl><dd><table border="0" class="wikitable"> <tbody><tr> <td> </td> <th>interior/boundary/exterior semantic </th> <th>usual semantic </th></tr> <tr> <th>Assertions </th> <td style="background-color:#DDC" align="center"><b>more descriptive</b><br /> " <i>a</i> and <i>b</i> have <span class="texhtml">DE-9IM(<i>a</i>,<i>b</i>)='212101212'</span> " </td> <td style="background-color:#DDC" align="center"><b>less descriptive</b><br /> " <i>a Touches b</i> " </td></tr> <tr> <th>Queries </th> <td style="background-color:#DDC" align="center"><b>more restrictive</b><br />" Show all pair of geometries where <span class="texhtml">DE-9IM(<i>a</i>,<i>b</i>)='212101212'</span> " </td> <td style="background-color:#DDC" align="center"><b>more general</b><br />" Show all pair of geometries where <i>Touches</i>(<i>a</i>,<i>b</i>) " </td></tr></tbody></table></dd></dl> <p>For usual applications, the use of <i>spatial predicates</i> also is justified by being more <a href="/wiki/Human-readable_medium" class="mw-redirect" title="Human-readable medium">human-readable</a> than <i>DE-9IM</i> descriptions: a typical user have better intuition about predicates (than a set of interiors/border/exterior intersections). </p><p>Predicates have useful <a href="/wiki/Semantics_(computer_science)" title="Semantics (computer science)">semantic</a> into usual applications, so it is useful the translation of a <i>DE-9IM</i> description into a list of all associated predicates,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-SDO_RELATE_15-0" class="reference"><a href="#cite_note-SDO_RELATE-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> that is like a <a href="/wiki/Type_conversion" title="Type conversion">casting process</a> between the two different semantic types. Examples: </p> <ul><li>The string codes "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0F1F00102</span>" and "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0F1FF0102</span>" have the semantic of "<i>Intersects & Crosses & Overlaps</i>".</li> <li>The string code "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1FFF0FFF2</span>" have the semantic of "<i>Equals</i>".</li> <li>The string codes "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F01FF0102</span>", "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">FF10F0102</span>", "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">FF1F00102</span>", "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F01FFF102</span>", and "<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">FF1F0F1F2</span>" have the semantic of "<i>Intersects & Touches</i>".</li></ul> <div class="mw-heading mw-heading3"><h3 id="Standards">Standards</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=8" title="Edit section: Standards"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Open_Geospatial_Consortium" title="Open Geospatial Consortium">Open Geospatial Consortium</a> (OGC) has standardized the typical spatial predicates (Contains, Crosses, Intersects, Touches, etc.) as boolean functions, and the DE-9IM model,<sup id="cite_ref-ogs1_16-0" class="reference"><a href="#cite_note-ogs1-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> as a function that returns a string (the DE-9IM code), with domain of {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">2</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>}, meaning <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span>=point, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span>=line, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">2</span>=area, and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>="empty set". This DE-9IM string code is a standardized format for data interchange. </p><p>The <a href="/wiki/Simple_Feature_Access" class="mw-redirect" title="Simple Feature Access">Simple Feature Access</a> (ISO 19125) standard,<sup id="cite_ref-SFS2007_17-0" class="reference"><a href="#cite_note-SFS2007-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> in the chapter 7.2.8, "SQL routines on type Geometry", recommends as supported routines the <i>SQL/MM Spatial</i><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> (ISO 13249-3 Part 3: Spatial) <i>ST_Dimension</i>, <i>ST_GeometryType</i>, <i>ST_IsEmpty</i>, <i>ST_IsSimple</i>, <i>ST_Boundary</i> for all Geometry Types. The same standard, consistent with the definitions of relations in "Part 1, Clause 6.1.2.3" of the SQL/MM, recommends (shall be supported) the function labels: <i>ST_Equals</i>, <i>ST_Disjoint</i>, <i>ST_Intersects</i>, <i>ST_Touches</i>, <i>ST_Crosses</i>, <i>ST_Within</i>, <i>ST_Contains</i>, <i>ST_Overlaps</i> and <i>ST_Relate</i>. </p><p>The DE-9IM in the OGC standards use the following definitions of Interior and Boundary, for the main OGC standard geometry types:<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable" align="center"> <tbody><tr> <th>Subtypes </th> <th>Dim </th> <th>Interior (<span class="texhtml mvar" style="font-style:italic;">I</span>) </th> <th>boundary (<span class="texhtml mvar" style="font-style:italic;">B</span>) </th></tr> <tr> <td>Point, MultiPoint </td> <td>0 </td> <td>Point, Points </td> <td>Empty </td></tr> <tr> <td>LineString, Line </td> <td>1 </td> <td>Points that are left when the boundary points are removed. </td> <td>Two end points. </td></tr> <tr> <td>LinearRing </td> <td>1 </td> <td>All points along the geometry. </td> <td>Empty. </td></tr> <tr> <td>MultilineString </td> <td>1 </td> <td>Points that are left when the boundary points are removed. </td> <td>Those points that are in the boundaries of an odd number of its elements (curves). </td></tr> <tr> <td>Polygon </td> <td>2 </td> <td>Points within the rings. </td> <td>Set of rings. </td></tr> <tr> <td>MultiPolygon </td> <td>2 </td> <td>Points within the rings. </td> <td>Set of rings of its elements (polygons). </td></tr> <tr> <td colspan="4">NOTICE: <b>exterior points (E)</b> are points <i>p</i> not in the <i>interior</i> or <i>boundary</i>, so not need extra interpretation, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">E(p)=not(I(p) or B(p))</span>. </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Implementation_and_practical_use">Implementation and practical use</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=9" title="Edit section: Implementation and practical use"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most spatial databases, such as <a href="/wiki/PostGIS" title="PostGIS">PostGIS</a>, implements the <i>DE-9IM()</i> model by the standard functions:<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <code>ST_Relate</code>, <code>ST_Equals</code>, <code>ST_Intersects</code>, etc. The function <code>ST_Relate(a,b)</code> outputs the standard OGC's <i>DE-9IM string code</i>. </p><p>Examples: two geometries, <i>a</i> and <i>b</i>, that intersects and touches with a point (for instance with <span class="nowrap">⁠<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 \dim(B(a)\cap I(b))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(B(a)\cap I(b))=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c747825f3fa5fadd43bc786342e6096ffbe53f03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.31ex; height:2.843ex;" alt="{\displaystyle \dim(B(a)\cap I(b))=0}" /></span>⁠</span> and <span class="nowrap">⁠<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 \dim(I(a)\cap I(b))=F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dim(I(a)\cap I(b))=F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/276e8457d30e6155b8a0d71d2ab2e06cb9f9019f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.296ex; height:2.843ex;" alt="{\displaystyle \dim(I(a)\cap I(b))=F}" /></span>⁠</span>), can be <code>ST_Relate(a,b)='FF1F0F1F2'</code> or <code>ST_Relate(a,b)='FF10F0102'</code> or <code>ST_Relate(a,b)='FF1F0F1F2'</code>. It also satisfies <code>ST_Intersects(a,b)=true</code> and <code>ST_Touches(a,b)=true</code>. When <code>ST_Relate(a,b)='0FFFFF212'</code>, the returned DE-9IM code have the semantic of "Intersects(a,b) & Crosses(a,b) & Within(a,b) & CoveredBy(a,b)", that is, returns <code>true</code> on the boolean expression <code>ST_Intersects(a,b) AND ST_Crosses(a,b) AND ST_Within(a,b) AND ST_Coveredby(a,b)</code>. </p><p>The use of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">ST_Relate()</span> is faster than direct computing of a set of correspondent predicates.<sup id="cite_ref-PostGISch4_7-1" class="reference"><a href="#cite_note-PostGISch4-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> There are cases where using <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">ST_Relate()</span> is the only way to compute a complex predicate — see the example of the code <code>0FFFFF0F2</code>,<sup id="cite_ref-test1case4_21-0" class="reference"><a href="#cite_note-test1case4-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> of a point that not "crosses" a multipoint (an object that is a set of points), but predicate <i>Crosses</i> (when defined by a mask) returns <i>true</i>. </p><p>It is usual to <a href="/wiki/Function_overloading" title="Function overloading">overload</a> the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">ST_Relate()</span> by adding a mask parameter, or use a returned <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">ST_Relate(a,b)</span> string into the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">ST_RelateMatch()</span> function.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> When using <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">ST_Relate(a,b,mask)</span>, it returns a boolean. Examples: </p> <ul><li><code>ST_Relate(a,b,'*FF*FF212')</code> returns <i>true</i> when <code>ST_Relate(a,b)</code> is <code>0FFFFF212</code> or <code>01FFFF212</code>, and returns <i>false</i> when <code>01FFFF122</code> or <code>0FF1FFFFF</code>.</li> <li><code>ST_RelateMatch('0FFFFF212','*FF*FF212')</code> and <code>ST_RelateMatch('01FFFF212','TTF*FF212')</code> are <i>true</i>, <code>ST_RelateMatch('01FFFF122','*FF*FF212')</code> is <i>false</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Synonyms">Synonyms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=10" title="Edit section: Synonyms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>"Egenhofer-Matrix" is a synonym for the <i>9IM</i> 3x3 matrix of boolean domain.<sup id="cite_ref-encygis_23-0" class="reference"><a href="#cite_note-encygis-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>"Clementini-Matrix" is a synonym for the <a href="#Matrix_model">DE-9IM</a> 3x3 matrix of {<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">0</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">1</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">2</span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">F</span>} domain.<sup id="cite_ref-encygis_23-1" class="reference"><a href="#cite_note-encygis-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup></li> <li>"Egenhofer operators" and "Clementini operators" are sometimes a reference to matrix elements as <i>II</i>, <i>IE</i>, etc. that can be used in boolean operations. Example: the predicate "<i>G<sub>1</sub></i> contains <i>G<sub>2</sub></i>" can be expressed by "<span class="texhtml"><span class="nowrap">⟨<i>G<sub>1</sub></i>| II ∧ ~EI ∧ ~EB |<i>G<sub>1</sub></i>⟩</span></span>", that can be translated to mask syntax, <code>T*****FF*</code>.</li> <li><a href="#Spatial_predicates">Predicates</a> "meets" is a synonym for <i>touches</i>; "inside" is a synonym for <i>within</i></li> <li>Oracle's<sup id="cite_ref-SDO_RELATE_15-1" class="reference"><a href="#cite_note-SDO_RELATE-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> "ANYINTERACT" is a synonym for <i>intersects</i> and "OVERLAPBDYINTERSECT" is a synonym for <i>overlaps</i>. Its "OVERLAPBDYDISJOINT" does not have a corresponding named predicate.</li> <li>In <a href="/wiki/Region_connection_calculus" title="Region connection calculus">Region connection calculus</a> operators offer some synonyms for <a href="#Spatial_predicates">predicates</a>: <i>disjoint</i> is DC (disconnected), <i>touches</i> is EC (externally connected), <i>equals</i> is EQ. Other, like <i>Overlaps</i> as PO (partially overlapping), need context analysis or composition.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table> <tbody><tr> <td valign="top">Standards: <ul><li><a href="/wiki/Simple_feature_access" class="mw-redirect" title="Simple feature access">Simple feature access</a> (ISO 19125)</li> <li><a href="/wiki/Open_Geospatial_Consortium" title="Open Geospatial Consortium">Open Geospatial Consortium</a></li> <li><a href="/wiki/GeoSPARQL" title="GeoSPARQL">GeoSPARQL</a></li></ul> </td> <td>      </td> <td valign="top">Software: <ul><li><a href="/wiki/Spatial_database" title="Spatial database">Spatial database</a></li> <li><a href="/wiki/Object-based_spatial_database" title="Object-based spatial database">Object-based spatial database</a></li> <li><a href="/wiki/Free_and_open-source_software" title="Free and open-source software">FOSS</a>: <ul><li><a href="/wiki/JTS_Topology_Suite" title="JTS Topology Suite">JTS Topology Suite</a></li> <li><a href="/wiki/GRASS_GIS" title="GRASS GIS">GRASS GIS</a></li> <li><a href="/wiki/GDAL" title="GDAL">GDAL</a> (library)</li> <li><a href="/wiki/PostGIS" title="PostGIS">PostGIS</a></li></ul></li></ul> </td> <td>      </td> <td valign="top">Related topics: <ul><li><a href="/wiki/Geospatial_topology" title="Geospatial topology">Geospatial topology</a></li> <li><a href="/wiki/Karnaugh_map#2-variable_map_examples" title="Karnaugh map">Karnaugh (2-variable) map</a></li> <li><a href="/wiki/Relational_operator" title="Relational operator">Relational operator</a></li> <li><a href="/wiki/Spatial_analysis" title="Spatial analysis">Spatial analysis</a></li> <li><b><a href="/wiki/Spatial_relation" title="Spatial relation">Spatial relation</a></b></li> <li><a href="/wiki/Spatial%E2%80%93temporal_reasoning" title="Spatial–temporal reasoning">Spatial–temporal reasoning</a></li></ul> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFClementiniDi_Felicevan_Oosterom1993" class="citation book cs1 cs1-prop-long-vol">Clementini, Eliseo; Di Felice, Paolino; van Oosterom, Peter (1993). "A small set of formal topological relationships suitable for end-user interaction". In Abel, David; Ooi, Beng Chin (eds.). <a rel="nofollow" class="external text" href="http://resolver.tudelft.nl/uuid:a2db9ae8-f768-4bff-ada3-966a6c8e9db6"><i>Advances in Spatial Databases: Third International Symposium, SSD '93 Singapore, June 23–25, 1993 Proceedings</i></a>. Lecture Notes in Computer Science. Vol. 692/1993. Springer. pp. <span class="nowrap">277–</span>295. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-56869-7_16">10.1007/3-540-56869-7_16</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-56869-8" title="Special:BookSources/978-3-540-56869-8"><bdi>978-3-540-56869-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+small+set+of+formal+topological+relationships+suitable+for+end-user+interaction&rft.btitle=Advances+in+Spatial+Databases%3A+Third+International+Symposium%2C+SSD+%2793+Singapore%2C+June+23%E2%80%9325%2C+1993+Proceedings&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E277-%3C%2Fspan%3E295&rft.pub=Springer&rft.date=1993&rft_id=info%3Adoi%2F10.1007%2F3-540-56869-7_16&rft.isbn=978-3-540-56869-8&rft.aulast=Clementini&rft.aufirst=Eliseo&rft.au=Di+Felice%2C+Paolino&rft.au=van+Oosterom%2C+Peter&rft_id=http%3A%2F%2Fresolver.tudelft.nl%2Fuuid%3Aa2db9ae8-f768-4bff-ada3-966a6c8e9db6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFClementiniSharmaEgenhofer1994" class="citation journal cs1">Clementini, Eliseo; Sharma, Jayant; Egenhofer, Max J. (1994). "Modelling topological spatial relations: Strategies for query processing". <i>Computers & Graphics</i>. <b>18</b> (6): <span class="nowrap">815–</span>822. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0097-8493%2894%2990007-8">10.1016/0097-8493(94)90007-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+%26+Graphics&rft.atitle=Modelling+topological+spatial+relations%3A+Strategies+for+query+processing&rft.volume=18&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E815-%3C%2Fspan%3E822&rft.date=1994&rft_id=info%3Adoi%2F10.1016%2F0097-8493%2894%2990007-8&rft.aulast=Clementini&rft.aufirst=Eliseo&rft.au=Sharma%2C+Jayant&rft.au=Egenhofer%2C+Max+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEgenhoferFranzosa1991" class="citation journal cs1">Egenhofer, M.J.; Franzosa, R.D. (1991). <a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02693799108927841">"Point-set topological spatial relations"</a>. <i>Int. J. GIS</i>. <b>5</b> (2): <span class="nowrap">161–</span>174. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F02693799108927841">10.1080/02693799108927841</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Int.+J.+GIS&rft.atitle=Point-set+topological+spatial+relations&rft.volume=5&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E161-%3C%2Fspan%3E174&rft.date=1991&rft_id=info%3Adoi%2F10.1080%2F02693799108927841&rft.aulast=Egenhofer&rft.aufirst=M.J.&rft.au=Franzosa%2C+R.D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1080%252F02693799108927841&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-sdh1990-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-sdh1990_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-sdh1990_4-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-sdh1990_4-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-sdh1990_4-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEgenhoferHerring1990" class="citation conference cs1">Egenhofer, M.J.; Herring, J.R. (1990). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100614161335/http://www.spatial.maine.edu/~max/MJEJRH-SDH1990.pdf">"A Mathematical Framework for the Definition of Topological Relationships"</a> <span class="cs1-format">(PDF)</span>. <i>Proceedings of the 4th International Symposium on Spatial Data Handling</i>. Archived from <a rel="nofollow" class="external text" href="http://www.spatial.maine.edu/~max/MJEJRH-SDH1990.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2010-06-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=A+Mathematical+Framework+for+the+Definition+of+Topological+Relationships&rft.btitle=Proceedings+of+the+4th+International+Symposium+on+Spatial+Data+Handling&rft.date=1990&rft.aulast=Egenhofer&rft.aufirst=M.J.&rft.au=Herring%2C+J.R.&rft_id=http%3A%2F%2Fwww.spatial.maine.edu%2F~max%2FMJEJRH-SDH1990.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-firstStd-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-firstStd_5-0">^</a></b></span> <span class="reference-text">The "<a href="/wiki/OpenGIS" class="mw-redirect" title="OpenGIS">OpenGIS</a> Simple Features Specification For SQL", <a rel="nofollow" class="external text" href="http://portal.opengeospatial.org/files/?artifact_id=829">Revision 1.1</a>, was released at May 5, 1999. It was the first international standard to establish the format conventions for <i>DE-9IM string codes</i>, and the names of the "Named Spatial Relationship predicates based on the DE-9IM" (see section with this title).</span> </li> <li id="cite_note-4vs9-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-4vs9_6-0">^</a></b></span> <span class="reference-text">M. J. Egenhofer, J. Sharma, and D. Mark (1993) "<a rel="nofollow" class="external text" href="http://www.spatial.maine.edu/~max/4Vs9.pdf">A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100614023708/http://www.spatial.maine.edu/~max/4Vs9.pdf">Archived</a> 2010-06-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>", In: <a rel="nofollow" class="external text" href="http://mapcontext.com/autocarto/proceedings/auto-carto-11/index.html">Auto-Carto XI</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140925121313/http://mapcontext.com/autocarto/proceedings/auto-carto-11/index.html">Archived</a> 2014-09-25 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-PostGISch4-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-PostGISch4_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-PostGISch4_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.postgis.org/docs/using_postgis_dbmanagement.html#DE-9IM">Chapter 4. Using PostGIS: Data Management and Queries</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110321145301/http://www.vividsolutions.com/jts/javadoc/com/vividsolutions/jts/geom/IntersectionMatrix.html"><i>JTS: Class IntersectionMatrix</i></a>, Vivid Solutions, Inc., archived from <a rel="nofollow" class="external text" href="http://www.vividsolutions.com/jts/javadoc/com/vividsolutions/jts/geom/IntersectionMatrix.html">the original</a> on 2011-03-21</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=JTS%3A+Class+IntersectionMatrix&rft.pub=Vivid+Solutions%2C+Inc.&rft_id=http%3A%2F%2Fwww.vividsolutions.com%2Fjts%2Fjavadoc%2Fcom%2Fvividsolutions%2Fjts%2Fgeom%2FIntersectionMatrix.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">JTS Technical Specifications of 2003.</span> </li> <li id="cite_note-davis2007-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-davis2007_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-davis2007_10-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-davis2007_10-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">M. Davis (2007), "<a rel="nofollow" class="external text" href="http://lin-ear-th-inking.blogspot.com.br/2007/06/subtleties-of-ogc-covers-spatial.html">Quirks of the 'Contains' Spatial Predicate</a>".</span> </li> <li id="cite_note-ST_Crosses-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-ST_Crosses_11-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://postgis.org/docs/ST_Crosses.html">ST_Crosses</a></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="randell92" class="citation conference cs1">Randell, D.A.; Cui, Z; Cohn, A.G. (1992). "A spatial logic based on regions and connection". <i>3rd Int. Conf. on Knowledge Representation and Reasoning</i>. Morgan Kaufmann. pp. <span class="nowrap">165–</span>176.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=A+spatial+logic+based+on+regions+and+connection&rft.btitle=3rd+Int.+Conf.+on+Knowledge+Representation+and+Reasoning&rft.pages=%3Cspan+class%3D%22nowrap%22%3E165-%3C%2Fspan%3E176&rft.pub=Morgan+Kaufmann&rft.date=1992&rft.au=Randell%2C+D.A.&rft.au=Cui%2C+Z&rft.au=Cohn%2C+A.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-cafrca-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-cafrca_13-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCâmaraFreitasCasanova1995" class="citation conference cs1">Câmara, G.; Freitas, U. M.; Casanova, M. A. (1995). "Fields and Objects Algebras for GIS Operations". <i>Proceedings of III Brazilian Symposium on GIS</i>. pp. <span class="nowrap">407–</span>424. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.991">10.1.1.17.991</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Fields+and+Objects+Algebras+for+GIS+Operations&rft.btitle=Proceedings+of+III+Brazilian+Symposium+on+GIS&rft.pages=%3Cspan+class%3D%22nowrap%22%3E407-%3C%2Fspan%3E424&rft.date=1995&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.17.991%23id-name%3DCiteSeerX&rft.aulast=C%C3%A2mara&rft.aufirst=G.&rft.au=Freitas%2C+U.+M.&rft.au=Casanova%2C+M.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">A <a rel="nofollow" class="external text" href="https://github.com/ppKrauss/postgis-st-relate-summary"><i>DE-9IM</i> translator</a>, of all associated predicates of a spatial relation.</span> </li> <li id="cite_note-SDO_RELATE-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-SDO_RELATE_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-SDO_RELATE_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Note. The <a rel="nofollow" class="external text" href="http://docs.oracle.com/cd/B19306_01/appdev.102/b14255/sdo_operat.htm#i78531">Oracle's spatial function <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r886049734" /><span class="monospaced">SDO_RELATE()</span></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130721054305/http://docs.oracle.com/cd/B19306_01/appdev.102/b14255/sdo_operat.htm#i78531">Archived</a> 2013-07-21 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> do only a partial translation, internally, offering to user a mask for a or-list of predicates to be checked, instead the DE-9IM string.</span> </li> <li id="cite_note-ogs1-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-ogs1_16-0">^</a></b></span> <span class="reference-text">"OpenGIS Implementation Specification for Geographic information - Simple feature access - Part 2: SQL option", <a href="/wiki/Open_Geospatial_Consortium" title="Open Geospatial Consortium">OGC</a>, <a rel="nofollow" class="external free" href="http://www.opengeospatial.org/standards/sfs">http://www.opengeospatial.org/standards/sfs</a></span> </li> <li id="cite_note-SFS2007-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-SFS2007_17-0">^</a></b></span> <span class="reference-text"> <a href="/wiki/Open_Geospatial_Consortium" title="Open Geospatial Consortium">Open Geospatial Consortium Inc.</a> (2007), "OpenGIS® Implementation Standard for Geographic information - Simple feature access - Part 2: SQL option", <a rel="nofollow" class="external text" href="http://www.opengeospatial.org/standards/sfs">OGC document</a> <i>06-104r4</i> version 1.2.1 (review of 2010-08-04).</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">ISO 13249-3 Part 3: Spatial, summarized in <a rel="nofollow" class="external text" href="http://www.sigmod.org/record/issues/0112/standards.pdf">SQL Multimedia and Application Packages (SQL/MM)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100214215605/http://www.sigmod.org/record/issues/0112/standards.pdf">Archived</a> 2010-02-14 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">"Encyclopedia of GIS", edited by Shashi Shekhar and Hui Xiong. SpringerScience 2008. pg. 242</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"> ST_Relate() <a href="/wiki/PostGIS" title="PostGIS">PostGIS</a> function <a rel="nofollow" class="external text" href="http://www.postgis.org/docs/ST_Relate.html">online documentation</a>.</span> </li> <li id="cite_note-test1case4-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-test1case4_21-0">^</a></b></span> <span class="reference-text">JTS test case of "point A within one of B points", <a rel="nofollow" class="external free" href="http://www.vividsolutions.com/jts/tests/Run1Case4.html">http://www.vividsolutions.com/jts/tests/Run1Case4.html</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304032952/http://www.vividsolutions.com/jts/tests/Run1Case4.html">Archived</a> 2016-03-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"> ST_RelateMatch() <a href="/wiki/PostGIS" title="PostGIS">PostGIS</a> function <a rel="nofollow" class="external text" href="http://www.postgis.org/docs/ST_RelateMatch.html">online documentation</a>.</span> </li> <li id="cite_note-encygis-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-encygis_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-encygis_23-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">"Encyclopedia of GIS", S. Shekhar, H. Xiong. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-35975-5" title="Special:BookSources/978-0-387-35975-5">978-0-387-35975-5</a>.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">"Multidimensional Region Connection Calculus" (2017), <a rel="nofollow" class="external free" href="http://qrg.northwestern.edu/qr2017/papers/QR2017_paper_8.pdf">http://qrg.northwestern.edu/qr2017/papers/QR2017_paper_8.pdf</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSabharwalLeopold2013" class="citation book cs1">Sabharwal, Chaman L.; Leopold, Jennifer L. (2013). "Identification of Relations in Region Connection Calculus: 9-Intersection Reduced to 3 + -Intersection Predicates". <i>Advances in Soft Computing and Its Applications</i>. Lecture Notes in Computer Science. Vol. 8266. pp. <span class="nowrap">362–</span>375. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-45111-9_32">10.1007/978-3-642-45111-9_32</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-45110-2" title="Special:BookSources/978-3-642-45110-2"><bdi>978-3-642-45110-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Identification+of+Relations+in+Region+Connection+Calculus%3A+9-Intersection+Reduced+to+3+%2B+-Intersection+Predicates&rft.btitle=Advances+in+Soft+Computing+and+Its+Applications&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=%3Cspan+class%3D%22nowrap%22%3E362-%3C%2Fspan%3E375&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-3-642-45111-9_32&rft.isbn=978-3-642-45110-2&rft.aulast=Sabharwal&rft.aufirst=Chaman+L.&rft.au=Leopold%2C+Jennifer+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADE-9IM" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=DE-9IM&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://docs.geotools.org/latest/userguide/library/jts/dim9.html">Point Set Theory and the DE-9IM Matrix</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20131206005929/http://edndoc.esri.com/arcsde/9.1/general_topics/understand_spatial_relations.htm">Illustrated Tutorial for DE-9IM</a></li></ul>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=DE-9IM&oldid=1283736899">https://en.wikipedia.org/w/index.php?title=DE-9IM&oldid=1283736899</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Matrices" title="Category:Matrices">Matrices</a></li><li><a href="/wiki/Category:Geometric_topology" title="Category:Geometric topology">Geometric topology</a></li><li><a href="/wiki/Category:Geographic_data_and_information" title="Category:Geographic data and information">Geographic data and information</a></li><li><a href="/wiki/Category:Binary_operations" title="Category:Binary operations">Binary operations</a></li><li><a href="/wiki/Category:Geometric_intersection" title="Category:Geometric intersection">Geometric intersection</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_that_are_too_technical_from_May_2019" title="Category:Wikipedia articles that are too technical from May 2019">Wikipedia articles that are too technical from May 2019</a></li><li><a href="/wiki/Category:All_articles_that_are_too_technical" title="Category:All articles that are too technical">All articles that are too technical</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 3 April 2025, at 09:55<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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DE-9IM - Wikipedia