In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing. 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, and the center point of this sphere is called the circumcenter of P.
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.
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.
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.
- James, R. C. (1992), The Mathematics Dictionary, Springer, p. 62, ISBN 9780412990410.
- 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.
- 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.
|Wikimedia Commons has media related to Circumscribed spheres.|