Linear least squares fitting of a sphere

The equation of a sphere with centre in \left[ x_c, y_c, z_c \right] and radius r is

(x - x_c)^2 + (y - y_c)^2 + (z - z_c)^2 = r^2

or

x^2 - 2 x x_c + x_c^2 + y^2 - 2 y y_c + y_c^2 + z^2 - 2 z z_c + z_c^2 = r^2

that is

(1)  \alpha x + \beta y + \gamma z + \delta = \epsilon

with

\alpha = - 2 x_c

\beta = - 2 y_c

\gamma = - 2 z_c

\delta = x_c^2 + y_c^2 + z_c^2 - r^2

\epsilon = - x^2 - y^2 - z^2

Writing (1) for four points on the sphere, \left[ x_i \hspace{1 mm} y_i \hspace{1 mm} z_i \right], i=1 \cdots 4, not belonging to the same plane, one gets a linear system of rank 4 in the four unknowns \alpha, \beta, \gamma, \delta; then of course

x_c = - \dfrac{1}{2} \alpha

y_c = - \dfrac{1}{2} \beta

z_c = - \dfrac{1}{2} \gamma

r = \sqrt{x_c^2 + y_c^2 + z_c^2 - \delta}

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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