I would like to find the center of a sphere of fixed radius that best approximates a set of points.
I want to minimize:
$$E(C) = \sum (|C - X_i|^2 - r^2)^2 $$
Where $X_i$ is the point $i$. Deriving with respect to $C$ and setting to zero we get:
$$0 = 4 \sum (|C - X_i|^2 - r^2)(C - X_i)$$
If we call $\Delta_i = C - X_i$, we have
$$0 = \sum (\Delta_i^t \Delta_i - r^2) \Delta_i$$
How can I find $C \in \mathbb{R}^3$ such that that the above equation equals zero?
In the geometric tools library the approximation of a sphere without a fixed radius is explained.