Minimum bounding circle on a sphere

Use stereographic projection $P$ with center the North pole $N$ mapping any point $M$ of the unit sphere onto $P(M)=M'$ into its equatorial plane with property:

$$N,M,M' \ \text{aligned}\tag{1}$$

One of the main properties of this transformation is that circles on the sphere are mapped onto circles into the equatorial plane.

Therefore the principle is to transform the set of points $P_k$s on the sphere onto a set of points $P'_k$s on the equatorial plane, determine the smallest enclosing circle of these points, and then send it back on the sphere.

The transformation formula (1) can be given an analytical form : $$\begin{cases}x'&=&\frac{x}{1-z}\\y'&=&\frac{y}{1-z}\end{cases}$$

The inverse transform can be given the following form

$$\begin{cases}x&=&2x'/D\\y&=&2y'/D\\z&=&(-1+x^2+y^2)/D \end{cases} \ \text{with} \ D:=1+x^2+y^2$$

due to relationship:

$$\vec{NM'}=2 \dfrac{\vec{NM}}{\|NM\|^2}$$

Remark: the stereographic projection is a particular case of an inversion, i.e. , the inversion with center $N$ and power $2$.