Circumsphere

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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 👁 R
is called the circumradius and its center the circumcenter. The circumsphere is the 3-dimensional generalization of the circumcircle.

👁 Circumsphere

The figures above depict the circumspheres of the Platonic solids.

The circumsphere is implemented in the Wolfram Language as [pts], where pts is a list of points, or [poly], where poly is a (giving a two-dimensional circumcircle) or (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 👁 (x_i,y_i,z_i)
for 👁 i=1
, ..., 4 is

👁 |x^2+y^2+z^2 x y z 1; x_1^2+y_1^2+z_1^2 x_1 y_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 z_4 1|=0.

(1)

Expanding the determinant,

👁 a(x^2+y^2+z^2)-(D_xx+D_yy+D_zz)+c=0,

(2)

where

👁 a=|x_1 y_1 z_1 1; x_2 y_2 z_2 1; x_3 y_3 z_3 1; x_4 y_4 z_4 1|,

(3)

👁 D_x
is the determinant obtained from the matrix

👁 D=[x_1^2+y_1^2+z_1^2 x_1 y_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 z_4 1]

(4)

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

👁 D_x	👁 =	👁 +\|x_1^2+y_1^2+z_1^2 y_1 z_1 1; x_2^2+y_2^2+z_2^2 y_2 z_2 1; x_3^2+y_3^2+z_3^2 y_3 z_3 1; x_4^2+y_4^2+z_4^2 y_4 z_4 1\|	(5)
👁 D_y	👁 =	👁 -\|x_1^2+y_1^2+z_1^2 x_1 z_1 1; x_2^2+y_2^2+z_2^2 x_2 z_2 1; x_3^2+y_3^2+z_3^2 x_3 z_3 1; x_4^2+y_4^2+z_4^2 x_4 z_4 1\|	(6)
👁 D_z	👁 =	👁 +\|x_1^2+y_1^2+z_1^2 x_1 y_1 1; x_2^2+y_2^2+z_2^2 x_2 y_2 1; x_3^2+y_3^2+z_3^2 x_3 y_3 1; x_4^2+y_4^2+z_4^2 x_4 y_4 1\|	(7)