# How do map a set of points on the surface of a sphere to the surface of a scaled sphere?

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 $p_s = (0,0,0)$.

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$(x,y,z) \mapsto (cx,cy,cz)$ (assuming the spheres are both centered at the origin)