Circumscribed sphere

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, by analogy with the term circumcircle.[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]

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]

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 bipyramids, 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.[6]

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.[7]

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.