I have a set of points $P$ on the surface of a sphere of radius $R$. I wish to change the radius of the sphere to $R_2 = c*R_1$, where $c$ is some positive real number. Using Cartesian coordinates, how do I map the set of points $P$ to the surface of the new sphere such that the distance between the points remains proportionally the same?
Please assume that we know the point $p_s$ at the center of the sphere. We can set $p_s = (0,0,0)$.