Editing Circumscribed sphere

{{short description|Sphere touching all of a polyhedron's vertices}}
[[File:Вписанный куб.gif|right|thumb|Circumscribed sphere of a [[cube]]]]

In [[geometry]], a '''circumscribed sphere''' of a [[polyhedron]] is a [[sphere]] that contains the polyhedron and touches each of the polyhedron's [[Vertex (geometry)|vertices]].<ref>{{citation|title=The Mathematics Dictionary|first=R. C.|last=James |author-link=Robert C. James |publisher=Springer|year=1992|isbn=9780412990410|page=62|url=https://books.google.com/books?id=UyIfgBIwLMQC&pg=PA62}}.</ref>  The word '''circumsphere''' is sometimes used to mean the same thing, by analogy with the term ''[[circumcircle]]''.<ref>{{citation|title=Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere|first=Edward S.|last=Popko|publisher=CRC Press|year=2012|isbn=9781466504295|page=144|url=https://books.google.com/books?id=WLAFlr1_2S4C&pg=PA144}}.</ref> As in the case of two-dimensional circumscribed circles (circumcircles), the [[radius]] of a sphere circumscribed around a polyhedron {{mvar|P}} is called the [[circumradius]] of {{mvar|P}},<ref>{{citation|title=Methods of Geometry|first=James T.|last=Smith|publisher=John Wiley & Sons|year=2011|isbn=9781118031032|page=419|url=https://books.google.com/books?id=B0khWEZmOlwC&pg=PA419}}.</ref> and the center point of this sphere is called the [[circumcenter]] of {{mvar|P}}.<ref>{{citation|title=Modern pure solid geometry|first=Nathan|last=Altshiller-Court|edition=2nd|publisher=Chelsea Pub. Co.|year=1964|page=57}}.</ref>

==Existence and optimality==
When it exists, a circumscribed sphere need not be the [[Smallest-circle problem|smallest sphere containing the polyhedron]]; for instance, the tetrahedron formed by a vertex of a [[cube]] and its three neighbors has the same circumsphere as the cube itself, but can be contained within a smaller sphere having the three neighboring vertices on its equator. However, the smallest sphere containing a given polyhedron is always the circumsphere of the [[convex hull]] of a subset of the vertices of the polyhedron.<ref name="fgk">{{citation
 | last1 = Fischer | first1 = Kaspar
 | last2 = Gärtner | first2 = Bernd
 | last3 = Kutz | first3 = Martin
 | contribution = Fast smallest-enclosing-ball computation in high dimensions
 | doi = 10.1007/978-3-540-39658-1_57
 | pages = 630–641
 | publisher = Springer
 | series = [[Lecture Notes in Computer Science]]
 | title = Algorithms - ESA 2003: 11th Annual European Symposium, Budapest, Hungary, September 16-19, 2003, Proceedings
 | volume = 2832
 | year = 2003| isbn = 978-3-540-20064-2
 | url = http://www.mpi-inf.mpg.de/~mkutz/pubs/FiGaeKu_SmallEnclBalls.pdf
 }}.</ref>

In ''De solidorum elementis'' (circa 1630), [[René Descartes]] observed that, for a polyhedron with a circumscribed sphere, all faces have circumscribed circles, the circles where the plane of the face meets the circumscribed sphere. Descartes suggested that this necessary condition for the existence of a circumscribed sphere is sufficient, but it is not true: some [[bipyramid]]s, for instance, can have circumscribed circles for their faces (all of which are triangles) but still have no circumscribed sphere for the whole polyhedron. However, whenever a [[simple polyhedron]] has a circumscribed circle for each of its faces, it also has a circumscribed sphere.<ref>{{citation|title=Descartes on Polyhedra: A Study of the "De solidorum elementis"|title-link=Descartes on Polyhedra|first=Pasquale Joseph|last=Federico|authorlink=Pasquale Joseph Federico|series= Sources in the History of Mathematics and Physical Sciences|volume=4|publisher=Springer|year=1982|pages=52–53}}</ref>

==Related concepts==
The circumscribed sphere is the three-dimensional analogue of the [[circumscribed circle]].
All [[regular polyhedra]] have circumscribed spheres, but most irregular polyhedra do not have one, since in general not all vertices lie on a common sphere. The circumscribed sphere (when it exists) is an example of a [[bounding sphere]], a sphere that contains a given shape. It is possible to define the smallest bounding sphere for any polyhedron, and compute it in [[linear time]].<ref name="fgk"/>

Other spheres defined for some but not all polyhedra include a [[midsphere]], a sphere tangent to all edges of a polyhedron, and an [[inscribed sphere]], a sphere tangent to all faces of a polyhedron. In the [[regular polyhedra]], the inscribed sphere, midsphere, and circumscribed sphere all exist and are [[Concentric spheres|concentric]].<ref>{{citation|last=Coxeter|first=H. S. M.|authorlink=Harold Scott MacDonald Coxeter|title=[[Regular Polytopes (book)|Regular Polytopes]]|edition=3rd|year=1973|publisher=Dover|isbn=0-486-61480-8|pages=[https://archive.org/details/regularpolytopes0000coxe/page/16 16–17]|contribution=2.1 Regular polyhedra; 2.2 Reciprocation|contribution-url=https://books.google.com/books?id=iWvXsVInpgMC&pg=PA16}}.</ref>

When the circumscribed sphere is the set of infinite limiting points of [[hyperbolic space]], a polyhedron that it circumscribes is known as an [[ideal polyhedron]].

==Point on the circumscribed sphere==
There are five convex [[regular polyhedra]], known as the [[Platonic solids]]. All Platonic solids have circumscribed spheres. For an arbitrary point <math>M</math> on the circumscribed sphere of each Platonic solid with number of the vertices <math>n</math>, if  <math>MA_i</math> are the distances to 
the vertices <math>A_i</math>, then<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 1 November 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>
:<math>4(MA_1^{2}+MA_2^{2}+...+MA_n^{2})^2=3n(MA_1^{4}+MA_2^{4}+...+MA_n^{4}).</math>

==References==
{{reflist}}

==External links==
{{commonscat|Circumscribed spheres}}
* {{mathworld | urlname = Circumsphere | title = Circumsphere}}

[[Category:Elementary geometry]]
[[Category:Spheres]]