Circumscribed sphere
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 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]
Contents
1 Existence and optimality
2 Related concepts
3 References
4 External links
Existence and optimality
When 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 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.[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]
References
^ James, R. C. (1992), The Mathematics Dictionary, Springer, p. 62, ISBN 9780412990410.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.
^ Popko, Edward S. (2012), Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, CRC Press, p. 144, ISBN 9781466504295.
^ Smith, James T. (2011), Methods of Geometry, John Wiley & Sons, p. 419, ISBN 9781118031032.
^ Altshiller-Court, Nathan (1964), Modern pure solid geometry (2nd ed.), Chelsea Pub. Co., p. 57.
^ ab Fischer, Kaspar; Gärtner, Bernd; Kutz, Martin (2003), "Fast smallest-enclosing-ball computation in high dimensions", Algorithms - ESA 2003: 11th Annual European Symposium, Budapest, Hungary, September 16-19, 2003, Proceedings, Lecture Notes in Computer Science, 2832, Springer, pp. 630–641, doi:10.1007/978-3-540-39658-1_57.
^ Coxeter, H. S. M. (1973), "2.1 Regular polyhedra; 2.2 Reciprocation", Regular Polytopes (3rd ed.), Dover, pp. 16–17, ISBN 0-486-61480-8.
External links
 | Wikimedia Commons has media related to Circumscribed spheres. |
- Weisstein, Eric W. "Circumsphere". MathWorld.
