Circumsphere

A circumsphere, or circumscribed sphere, is the sphere that passes through all the vertices of a polyhedron. A polyhedron that admits such a sphere is called cyclic or circumscribable. In three dimensions, any tetrahedron (four non-coplanar vertices) has a unique circumsphere; for a general polyhedron a circumsphere exists if and only if all its vertices lie on a common sphere. If this condition holds, the circumsphere is unique.

To compute it for a tetrahedron with vertices v1, v2, v3, v4, the center c is the solution to the linear system obtained from ||vi − c||^2 = ||v1 − c||^2 for i = 2, 3, 4. This yields 2(vi − v1)·c = ||vi||^2 − ||v1||^2. Solving for c gives the circumcenter, and the radius is R = ||v1 − c||. For a larger set of vertices, the same idea applies, but a circumsphere exists only if all vertices lie on a common sphere.

In regular polyhedra the circumsphere is centered at the symmetry center, and all vertices are equidistant from it. The circumsphere is related to, but distinct from, the insphere, which is tangent to faces rather than passing through vertices.

Applications include mesh generation, computer graphics, and geometric modeling, where circumscribed spheres help define bounds, normalization, or quality measures for simplices and polyhedra.