TOPICS
Circumsphere
The circumsphere of given set of points, commonly the vertices of a solid, is asphere that passes through all the points. A circumsphere does not always exist, but when it does, itsradius
is called thecircumradius and its center thecircumcenter. The circumsphere is the 3-dimensional generalization of thecircumcircle.
The figures above depict the circumspheres of thePlatonicsolids.
The circumsphere is implemented in theWolfram Language asCircumsphere[pts], wherepts is a list of points, orCircumsphere[poly], wherepoly is aPolygon (giving a two-dimensionalcircumcircle) orPolyhedron (giving a three-dimensional circumsphere) object.
By analogy with the equation of thecircumcircle, the equation for the circumsphere of thetetrahedron withpolygon vertices
for
, ..., 4 is
 | (1) |
Expanding thedeterminant,
 | (2) |
where
 | (3) |
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]](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fCircumsphere%2fNumberedEquation4.svg&f=jpg&w=240) | (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) |
Completing the square gives
 | (9) |
which is asphereof the form
 | (10) |
withcircumcenter
 |  |  | (11) |
 |  |  | (12) |
 |  |  | (13) |
andcircumradius
 | (14) |
See also
Circumcenter,Circumcircle,Circumradius,Insphere,MidsphereExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Circumsphere." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/Circumsphere.html