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)
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Expanding the determinant,
(2)
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where
(3)
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is the determinant obtained from the matrix
(4)
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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)
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(6)
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(7)
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and is given by
(8)
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Completing the square gives
(9)
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which is a sphere of the form
(10)
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with circumcenter
(11)
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(12)
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(13)
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and circumradius
(14)
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