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<title>Largest empty sphere</title>
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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Largest empty sphere</span></span>
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<p>In <a href="Computational_geometry" title="Computational geometry">computational geometry</a>, the <b>largest empty sphere</b> problem is the problem of finding a <a href="Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a> of largest radius in <i>d</i>-dimensional space whose interior does not overlap with any given obstacles.
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<div class="mw-heading mw-heading2"><h2 id="Two_dimensions">Two dimensions</h2></div>
<p>The <b>largest empty circle</b> problem is the problem of finding a <a href="Circle" title="Circle">circle</a> of largest radius in <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">the plane</a> whose interior does not overlap with any given obstacles.
</p><p>A common special case is as follows. Given <i>n</i> points in the plane, find a largest circle centered within their <a href="Convex_hull" title="Convex hull">convex hull</a> and enclosing none of them. The problem may be solved using <a href="Voronoi_diagram" title="Voronoi diagram">Voronoi diagrams</a> in <a href="Time_complexity" title="Time complexity">optimal time</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta (n\,\log \,n)}">
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<annotation encoding="application/x-tex">{\displaystyle \Theta (n\,\log \,n)}</annotation>
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</math></span><img src="./45788c266a88ed5eaf2ca2f8ec1e2699c8c40af7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.927ex; height:2.843ex;" alt="{\displaystyle \Theta (n\,\log \,n)}" loading="lazy"></span>.<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>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Bounding_sphere" title="Bounding sphere">Bounding sphere</a></li>
<li><a href="Farthest-first_traversal" title="Farthest-first traversal">Farthest-first traversal</a></li>
<li><a href="Largest_empty_rectangle" title="Largest empty rectangle">Largest empty rectangle</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">G. T. Toussaint, "Computing largest empty circles with location constraints," <i>International Journal of Computer and Information Sciences</i>, vol. 12, No. 5, October, 1983, pp. 347-358.</span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Megan Schuster, <a rel="nofollow" class="external text" href="https://www.cs.swarthmore.edu/~adanner/cs97/s08/papers/schuster.pdf">"The Largest Empty Circle Problem"</a></span>
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