| 词条 | Circumscribed sphere |
| 释义 |
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices.[1] The word circumsphere is sometimes used to mean the same thing.[2] As in the case of two-dimensional circumscribed circles, the radius of a sphere circumscribed around a polyhedron P is called the circumradius of P,[3] and the center point of this sphere is called the circumcenter of P.[4] Existence and optimalityWhen it exists, a circumscribed sphere need not be the 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.[5] Related conceptsThe 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.[5] 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.[6] References1. ^{{citation|title=The Mathematics Dictionary|first=R. C.|last=James|publisher=Springer|year=1992|isbn=9780412990410|page=62|url=https://books.google.com/books?id=UyIfgBIwLMQC&pg=PA62}}. 2. ^{{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}}. 3. ^{{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}}. 4. ^{{citation|title=Modern pure solid geometry|first=Nathan|last=Altshiller-Court|edition=2nd|publisher=Chelsea Pub. Co.|year=1964|page=57}}. 5. ^1 {{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}}. 6. ^{{citation|last=Coxeter|first=H. S. M.|authorlink=Harold Scott MacDonald Coxeter|title=Regular Polytopes|edition=3rd|year=1973|publisher=Dover|isbn=0-486-61480-8|pages=16–17|contribution=2.1 Regular polyhedra; 2.2 Reciprocation|contribution-url=https://books.google.com/books?id=iWvXsVInpgMC&lpg=PP1&pg=PA16}}. External links{{commonscat|Circumscribed spheres}}
2 : Elementary geometry|Spheres |
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