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<title>Spatial relation</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Spatial relation</span></span>

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<p>A <b>spatial relation</b><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-Mark_2-0" class="reference"><a href="#cite_note-Mark-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> specifies how some object is located in space in relation to some reference object. When the reference object is much bigger than the object to locate, the latter is often represented by a point. The reference object is often represented by a <a href="Bounding_box" class="mw-redirect" title="Bounding box">bounding box</a>.

</p><p>In <a href="Anatomical_terms_of_location" title="Anatomical terms of location">Anatomy</a> it might be the case that a spatial relation is not fully applicable. Thus, the degree of applicability is defined which specifies from 0 till 100% how strongly a spatial relation holds. Often researchers concentrate on defining the applicability function for various spatial relations.

</p><p>In <a href="Spatial_database" title="Spatial database">spatial databases</a> and <a href="Geospatial_topology" title="Geospatial topology">geospatial topology</a> the <i>spatial relations</i> are used for <a href="Spatial_analysis" title="Spatial analysis">spatial analysis</a> and constraint specifications.

</p><p>In <a href="Cognitive_development" title="Cognitive development">cognitive development</a> for walk and for catch objects, or <a href="Water-level_task" title="Water-level task">for understand objects-behaviour</a>; in <a href="Automated_Guided_Vehicle" class="mw-redirect" title="Automated Guided Vehicle">robotic Natural Features Navigation</a>; and many other areas, <i>spatial relations</i> plays a central role.

</p><p>Commonly used types of <i>spatial relations</i> are: <i>topological</i>, <i>directional</i> and <i>distance</i> relations.

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<div class="mw-heading mw-heading2"><h2 id="Topological_relations">Topological relations</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spatial_topology" class="mw-redirect" title="Spatial topology">Spatial topology</a></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Spatial_predicates" class="mw-redirect" title="Spatial predicates">Spatial predicates</a></div>

<p>The <a href="DE-9IM" title="DE-9IM">DE-9IM</a> model expresses important <i>space relations</i> which are invariant to <a href="Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>, <a href="Translation_(geometry)" title="Translation (geometry)">translation</a> and <a href="Scaling_(geometry)" title="Scaling (geometry)">scaling</a> transformations.

</p><p>For any two spatial objects <i>a</i> and <i>b</i>, that can be points, lines and/or polygonal areas, there are 9 relations derived from <i>DE-9IM</i>:

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<th valign="top"><i>Equals</i>

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<td><i>a</i> = <i>b</i><br>Topologically <a href="Equality_(relational_operator)" class="mw-redirect" title="Equality (relational operator)">equal</a>. Also (<i>a</i> ∩ <i>b</i> = <i>a</i>) ∧ (<i>a</i> ∩ <i>b</i> = <i>b</i>)

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<th valign="top"><i>Disjoint</i>

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<td><i>a</i> ∩ <i>b</i> = ∅ <br> <i>a</i> and <i>b</i> are disjoint, have no point in common. They form a set of <a href="Disconnected_(topology)" class="mw-redirect" title="Disconnected (topology)">disconnected</a> geometries.

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<th valign="middle"><i>Intersects</i>   

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<td><i>a</i> ∩ <i>b</i> ≠ ∅

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<th valign="top"><i>Touches</i>

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<td>(<i>a</i> ∩ <i>b</i> ≠ ∅) ∧ (<i>a</i><sup>ο</sup> ∩ <i>b</i><sup>ο</sup> = ∅) <br> <i>a</i> touches <i>b</i>, they have at least one boundary point in common, but no interior points.

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<th valign="top"><i>Contains</i>

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<th valign="top"><i>Covers</i>

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<td><i>a</i><sup>ο</sup> ∩ <i>b</i> = <i>b</i> <br> <i>b</i> lies in the interior of <i>a</i> (extends <i>Contains</i>). Other definitions: "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 of) <i>a</i>".

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<th valign="middle"><i>CoveredBy</i>  

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<td><i>Covers(b,a)</i>

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<th valign="top"><i>Within</i>

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<div class="mw-heading mw-heading2"><h2 id="Directional_relations">Directional relations</h2></div>

<p>Directional relations can again be differentiated into external directional relations and internal directional relations. An internal directional relation specifies where an object is located inside the reference object while an external relations specifies where the object is located outside of the reference objects.

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<ul><li>Examples for internal directional relations: left; on the back; athwart, abaft</li>

<li>Examples for external directional relations: on the right of; behind; in front of, abeam, astern</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Distance_relations">Distance relations</h2></div>

<p>Distance relations specify how far is the object away from the reference object.

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<ul><li>Examples are: at; nearby; in the vicinity; far away</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Relations_by_class">Relations by class</h2></div>

<p>Reference objects represented by a <a href="Bounding_box" class="mw-redirect" title="Bounding box">bounding box</a> or another kind of "spatial envelope" that encloses its borders, can be denoted with the maximum number of <a href="Dimension_(mathematics_and_physics)" class="mw-redirect" title="Dimension (mathematics and physics)">dimensions</a> of this envelope: '0' for <a href="Point_(geometry)" title="Point (geometry)">punctual objects</a>, '1' for <a href="Line_(geometry)" title="Line (geometry)">linear objects</a>, '2' for <a href="Area" title="Area">planar objects</a>, '3' for <a href="Volume" title="Volume">volumetric objects</a>. So, any object, in a <a href="2D_geometric_model" title="2D geometric model">2D modeling</a>, can by classified as <i>point</i>, <i>line</i> or <i>area</i> according to its delimitation. Then, a <i>type of spatial relation</i> can be expressed by the class of the objects that participate in the relation:

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<ul><li>point-point relations: ...</li>

<li>point-line relations:</li>

<li>point-area relations:</li>

<li>line-line relations:</li>

<li>line-area relations:</li>

<li>area-area relations:</li></ul>

<p>More <i>complex</i> modeling schemas can represent an object as a composition of <i>simple sub-objects</i>. Examples: represent in an <a href="Astronomical_map" class="mw-redirect" title="Astronomical map">astronomical map</a> a star by a <i>point</i> and a <a href="Binary_star" title="Binary star">binary star</a> by <i>two points</i>; represent in <a href="Geographical_map" class="mw-redirect" title="Geographical map">geographical map</a> a river with a <i>line</i>, for its <a href="River_source" title="River source">source</a> <a href="Stream" title="Stream">stream</a>, and with an strip-<i>area</i>, for the rest of the river. These schemas can use the above classes, uniform composition classes (<i>multi-point</i>, <i>multi-line</i> and <i>multi-area</i>) and heterogeneous composition (<i>points</i>+<i>lines</i> as "object of dimension 1", <i>points</i>+<i>lines</i>+<i>areas</i> as "object of dimension 2").

</p><p>Two internal components of a <i>complex object</i> can express (the above) <a href="Binary_operation" title="Binary operation">binary relations</a> between them, and <a href="Ternary_operation" title="Ternary operation">ternary relations</a>, using the whole object as a <a href="Frame_of_reference" title="Frame of reference">frame of reference</a>. Some relations can be expressed by an abstract component, such the <a href="Center_of_mass" title="Center of mass">center of mass</a> of the binary star, or a center line of the river.

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<div class="mw-heading mw-heading2"><h2 id="Temporal_references">Temporal references</h2></div>

<p>For human thinking, spatial relations include qualities like size, distance, volume, order, and, also, time:

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<p>Time is spatial: it requires understanding ordered sequences such as days of the week, months of the year, and seasons. A person with spatial difficulties may have problems understanding “yesterday,” “last week,” and “next month”.

Time expressed digitally is just as spatial as time expressed by moving clock hands, but digital clocks remove the need to translate the hand position into numbers.

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</blockquote><div class="templatequotecite"><p style="display: inline; padding-left: 2.3em;">— Stockdale and Possin</p></div>

<p>Stockdale and Possin<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> discusses the many ways in which people with difficulty establishing spatial and temporal relationships can face problems in ordinary situations.

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<ul><li><a href="Anatomical_terms_of_location" title="Anatomical terms of location">Anatomical terms of location</a></li>

<li><a href="DE-9IM" title="DE-9IM">Dimensionally Extended nine-Intersection Model</a> (DE-9IM)</li>

<li><a href="Water-level_task" title="Water-level task">Water-level task</a></li>

<li><a href="Allen's_interval_algebra" title="Allen's interval algebra">Allen's interval algebra</a> (temporal analog)</li>

<li><a href="Commonsense_reasoning" title="Commonsense reasoning">Commonsense reasoning</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">J Freeman (1975), "The modelling of spatial relations", Computer Graphics and Image Processing, Elsevier. <a href="https://doi.org/10.1016/S0146-664X(75)80007-4" class="extiw external" title="doi:10.1016/S0146-664X(75)80007-4">doi:10.1016/S0146-664X(75)80007-4</a></span>

</li>

<li id="cite_note-Mark-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Mark_2-0">^</a></b></span> <span class="reference-text">D. M. Mark and M. J. Egenhofer (1994), "Modeling Spatial Relations Between Lines and Regions: Combining Formal Mathematical Models and Human Subjects Testing". <a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.25.9493&rep=rep1&type=pdf">PDF</a></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">C. Stockdale and C. Possin (1998) <a rel="nofollow" class="external text" href="http://impactofspecialneeds.weebly.com/uploads/3/4/1/9/3419723/spatial.pdf">Spatial Relations and Learning</a>.</span>

</li>

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