TOPICS

# Circumsphere

The circumsphere of given set of points, commonly the vertices of a solid, is a sphere that passes through all the points. A circumsphere does not always exist, but when it does, its radius is called the circumradius and its center the circumcenter. The circumsphere is the 3-dimensional generalization of the circumcircle.

The figures above depict the circumspheres of the Platonic solids.

The circumsphere is implemented in the Wolfram Language as Circumsphere[pts], where pts is a list of points, or Circumsphere[poly], where poly is a Polygon (giving a two-dimensional circumcircle) or Polyhedron (giving a three-dimensional circumsphere) object.

By analogy with the equation of the circumcircle, the equation for the circumsphere of the tetrahedron with polygon vertices for , ..., 4 is

 (1)

Expanding the determinant,

 (2)

where

 (3)

is the determinant obtained from the matrix

 (4)

by discarding the column (and taking a plus sign) and similarly for (this time taking the minus sign) and (again taking the plus sign)

 (5) (6) (7)

and is given by

 (8)
 (9)

which is a sphere of the form

 (10)

with circumcenter

 (11) (12) (13)

 (14)