circumspheres

A circumsphere is the unique sphere that passes through all the vertices of a simplex. In n-dimensional Euclidean space, an n-simplex has n+1 vertices, and its circumsphere is the sphere whose center is the circumcenter of the simplex and whose radius is the common distance to all vertices.

Existence and uniqueness follow from the geometry of a simplex: there is a single point equidistant from all its vertices, and hence a unique sphere with that common distance. In two dimensions this reduces to the circumscribed circle of a triangle. In higher dimensions the same principle generalizes to spheres through the vertices of the simplex. Not all polytopes admit a circumsphere; a polytope has a circumsphere only if all its vertices lie on a common sphere.

In three dimensions, every tetrahedron has a circumsphere. For a tetrahedron with vertices v1, v2, v3, v4, the circumsphere center x satisfies |x − vi| = R for all i. Equivalently, subtracting equations |x − vi|^2 = |x − v1|^2 gives a linear system: 2(v_i − v1)·x = ||v_i||^2 − ||v1||^2 for i = 2,3,4. Solving yields x, and R is the distance from x to any vertex.

In general, to compute the circumsphere of an n-simplex, solve for the point x that satisfies |x − v_i|^2 = |x − v_j|^2 for all i,j; the resulting x is the circumcenter and the radius is the distance to a vertex. Regular polytopes and many structured configurations possess circumspheres, while arbitrary polytopes may not.