The equation of a sphere with centre in and radius is

or

that is

(1)

with

Writing (1) for four points on the sphere, , not belonging to the same plane, one gets a linear system of rank 4 in the four unknowns ; then of course

If more than four points on the sphere are given, and they don’t belong all to the same plane, one gets an overdetermined linear system of rank 4 whose solution, eg with the Singular Value Decomposition method (SVD), is an approximation of the sphere in the least squares sense.

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