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| [[Image:Orthogonal-convex-hull.svg|thumb|The orthogonal convex hull of a point set]]
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| In [[geometry]], a set {{math|''K'' ⊂ [[real coordinate space|'''R'''<sup>''n''</sup>]]}} is defined to be '''orthogonally convex''' if, for every [[line (geometry)|line]] {{mvar|L}} that is parallel to one of [[standard basis]] vectors, the [[intersection (set theory)|intersection]] of {{mvar|K}} with {{mvar|L}} is empty, a point, or a single [[line segment|segment]]. The term "orthogonal" refers to corresponding [[Cartesian coordinate system|Cartesian]] basis and coordinates in [[Euclidean space]], where different basis vectors are [[perpendicularity|perpendicular]], as well as corresponding lines. Unlike ordinary [[convex set]]s, an orthogonally convex set is not necessarily [[connectedness|connected]].
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| The '''orthogonal convex hull''' of a set {{math|''S'' ⊂ '''R'''<sup>''n''</sup>}} is the intersection of all connected orthogonally convex supersets of {{mvar|S}}.
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| These definitions are made by analogy with the classical theory of convexity, in which {{mvar|K}} is [[convex set|convex]] if, for every line {{mvar|L}}, the intersection of {{mvar|K}} with {{mvar|L}} is empty, a point, or a single segment (interval). Orthogonal convexity restricts the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the [[convex hull]] of the same point set. A point {{mvar|p}} belongs to the orthogonal convex hull of {{mvar|S}} if and only if each of the closed axis-aligned [[orthant]]s having {{mvar|p}} as apex has a nonempty intersection with {{mvar|S}}.
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| The orthogonal convex hull is also known as the '''rectilinear convex hull''', or, in [[two-dimensional space|two dimensions]], the '''{{mvar|x}}-{{mvar|y}} convex hull'''.
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| ==Example==
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| The figure shows a set of 16 points in the plane and the orthogonal convex hull of these points. As can be seen in the figure, the orthogonal convex hull is a [[polygon]] with some degenerate edges connecting extreme vertices in each coordinate direction. For a discrete point set such as this one, all orthogonal convex hull edges are horizontal or vertical. In this example, the orthogonal convex hull is connected.
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| ==Algorithms==
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| Several authors have studied algorithms for constructing orthogonal convex hulls: {{harvtxt|Montuno|Fournier|1982}}; {{harvtxt|Nicholl|Lee|Liao|Wong|1983}}; {{harvtxt|Ottman|Soisalon-Soisinen|Wood|1984}}; {{harvtxt|Karlsson|Overmars|1988}}. By the results of these authors, the orthogonal convex hull of {{mvar|k}} points in the plane may be constructed in time {{math|[[big O notation|O]](''k'' [[logarithm|log]] ''k'')}}, or possibly faster using integer searching data structures for points with [[integer]] coordinates.
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| ==Related concepts==
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| It is natural to generalize orthogonal convexity to ''restricted-orientation convexity'', in which a set {{mvar|K}} is defined to be convex if all lines having one of a finite set of slopes must intersect {{mvar|K}} in connected subsets; see e.g. {{harvtxt|Rawlins|1987}}, {{harvs|last1=Rawlins|last2=Wood|year=1987|year2=1988|txt}}, or {{harvs|last1=Fink|last2=Wood|year=1996|year2=1998|txt}}.
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| In addition, the [[tight span]] of a finite metric space is closely related to the orthogonal convex hull. If a finite point set in the plane has a connected orthogonal convex hull, that hull is the tight span for the [[Manhattan distance]] on the point set. However, orthogonal hulls and tight spans differ for point sets with disconnected orthogonal hulls, or in higher dimensional [[Lp space|L<sup>''p''</sup> spaces]].
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| {{harvtxt|O'Rourke|1993}} describes several other results about orthogonal convexity and orthogonal [[visibility (geometry)|visibility]].
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| ==References==
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| *{{citation
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| | last1 = Fink | first1 = Eugene
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| | last2 = Wood | first2 = Derick
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| | doi = 10.1016/0020-0255(96)00056-4
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| | issue = 1–4
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| | journal = Information Sciences
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| | pages = 175–196
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| | title = Fundamentals of restricted-orientation convexity
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| | url = http://www.cs.cmu.edu/~eugene/research/full/restricted-convexity.pdf
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| | volume = 92
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| | year = 1996}}.
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| *{{citation
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| | last1 = Fink | first1 = Eugene
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| | last2 = Wood | first2 = Derick
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| | doi = 10.1007/BF01237603
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| | journal = Journal of Geometry
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| | pages = 99–120
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| | title = Generalized halfspaces in restricted-orientation convexity
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| | url = http://www.cs.cmu.edu/~eugene/research/full/restricted-halfspaces.pdf
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| | volume = 62
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| | year = 1998}}.
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| *{{citation
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| | last1 = Karlsson | first1 = Rolf G.
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| | last2 = Overmars | first2 = Mark H. | author2-link = Mark Overmars
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| | doi = 10.1007/BF01934088
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| | issue = 2
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| | journal = BIT
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| | pages = 227–241
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| | title = Scanline algorithms on a grid
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| | volume = 28
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| | year = 1988}}.
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| *{{citation
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| | last1 = Montuno | first1 = D. Y.
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| | last2 = Fournier | first2 = A.
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| | publisher = University of Toronto
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| | series = Technical Report 148
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| | title = Finding the {{mvar|x}}-{{mvar|y}} convex hull of a set of {{mvar|x}}-{{mvar|y}} polygons
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| | year = 1982}}.
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| *{{citation
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| | last1 = Nicholl | first1 = T. M.
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| | last2 = Lee | first2 = D. T. | author2-link = Der-Tsai Lee
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| | last3 = Liao | first3 = Y. Z.
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| | last4 = Wong | first4 = C. K.
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| | doi = 10.1007/BF01933620
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| | journal = BIT
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| | pages = 456–471
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| | title = On the X-Y convex hull of a set of X-Y polygons
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| | volume = 23
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| | year = 1983
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| | issue = 4}}.
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| *{{citation
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| | last = O'Rourke | first = Joseph | author-link = Joseph O'Rourke (professor)
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| | pages = 107–109
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| | publisher = Cambridge University Press
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| | title = Computational Geometry in C
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| | year = 1993}}.
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| *{{citation
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| | last1 = Ottman | first1 = T.
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| | last2 = Soisalon-Soisinen | first2 = E.
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| | last3 = Wood | first3 = Derick
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| | doi = 10.1016/0020-0255(84)90025-2
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| | journal = Information Sciences
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| | pages = 157–171
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| | title = On the definition and computation of rectilinear convex hulls
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| | volume = 33
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| | year = 1984
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| | issue = 3}}.
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| *{{citation
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| | last = Rawlins | first = G. J. E.
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| | publisher = University of Waterloo
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| | series = Ph.D. thesis and Tech. Rep. CS-87-57
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| | title = Explorations in Restricted-Orientation Geometry
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| | year = 1987}}.
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| *{{citation
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| | last1 = Rawlins | first1 = G. J. E.
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| | last2 = Wood | first2 = Derick
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| | doi = 10.1016/0890-5401(87)90045-9
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| | journal = Information and Computation
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| | pages = 150–166
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| | title = Optimal computation of finitely oriented convex hulls
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| | volume = 72
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| | year = 1987
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| | issue = 2}}.
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| *{{citation
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| | last1 = Rawlins | first1 = G. J. E.
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| | last2 = Wood | first2 = Derick
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| | contribution = Ortho-convexity and its generalizations
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| | editor-last = Toussaint | editor-first = Godfried T. | editor-link = Godfried Toussaint
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| | pages = 137–152
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| | publisher = Elsevier
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| | title = Computational Morphology
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| | year = 1988}}.
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| [[Category:Convex hulls]]
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| {{Computer science}}
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The writer is known by the name of Numbers Wunder. My working day job is a meter reader. Minnesota has always been his house but his spouse wants them to move. Body developing is 1 of the issues I love most.
my webpage ... epapyrus.com