I have $2$ arbitrary points in space, $p_1$ and $p_2$, that both lie on a sphere of known radius $r$.
The center of the sphere is such that if a line were drawn around the sphere through both points $p_1$ and $p_2$, the line would also cross through the bottom most point of the sphere and would create a slice of the sphere of the maximum size circle.
How can I find the center of the sphere $p$? And the equation that defines the circle that would represent the line drawn around it?