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[[File:Espace octaedrique cubique faces centrees.svg|thumb|The dashed circle is the outline of the largest empty sphere in the [[Close-packing of equal spheres|close-packing of spheres]]. See also ''[[Interstitial defect]]''.]]
[[File:Plus grand cercle vide voronoi.svg|thumb|Finding the largest empty circle using the Voronoi diagram (two solutions).]]
In [[computational geometry]], the '''largest empty sphere''' problem is the problem of finding a [[hypersphere]] of largest radius in ''d''-dimensional space whose interior does not overlap with any given obstacles.
==Two dimensions==
The '''largest empty circle''' problem is the problem of finding a [[circle]] of largest radius in [[the plane]] whose interior does not overlap with any given obstacles.
A common special case is as follows. Given ''n'' points in the plane, find a largest circle centered within their [[convex hull]] and enclosing none of them. The problem may be solved using [[Voronoi diagram]]s in [[time complexity|optimal time]] <math>\Theta(n\, \log\, n)</math>.<ref>G. T. Toussaint, "Computing largest empty circles with location constraints," ''International Journal of Computer and Information Sciences'', vol. 12, No. 5, October, 1983, pp. 347-358.</ref>
==See also==
*[[Bounding sphere]]
*[[Largest empty rectangle]]
==References==
{{reflist}}
[[Category:Geometric algorithms]]
Revision as of 14:42, 16 September 2013
The dashed circle is the outline of the largest empty sphere in the close-packing of spheres. See also Interstitial defect.Finding the largest empty circle using the Voronoi diagram (two solutions).
In computational geometry, the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d-dimensional space whose interior does not overlap with any given obstacles.
Two dimensions
The largest empty circle problem is the problem of finding a circle of largest radius in the plane whose interior does not overlap with any given obstacles.
A common special case is as follows. Given n points in the plane, find a largest circle centered within their convex hull and enclosing none of them. The problem may be solved using Voronoi diagrams in optimal time.[1]
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↑G. T. Toussaint, "Computing largest empty circles with location constraints," International Journal of Computer and Information Sciences, vol. 12, No. 5, October, 1983, pp. 347-358.
