Transform points on a sphere when center changes

Question

Given the center of a unit sphere at

\begin{equation} ( x_{c1},y_{c1},z_{c1} ) \end{equation}

we define a point on the surface of the sphere as

\begin{equation} ( x_{p1},y_{p1},z_{p1} ) \end{equation}

Now if the center of the sphere moves to

\begin{equation} ( x_{c2},y_{c2},z_{c2} ) \end{equation}

what's the formula to compute the new coordinates of the point

\begin{equation} ( x_{p2},y_{p2},z_{p2} ) \end{equation}

Answer 1

Let $x_{c_2} = x_{c_1} + u$, $y_{c_2} = y_{c_1} + v$, $z_{c_2} = z_{c_1} +w$.

Then, each point on the sphere gets shifted by the above units

So, $(x_{p_2},y_{p_2},z_{p_2}) = (x_{p_1}+u,y_{p_1}+v,z_{p_1}+w)$

Also, be careful while shifting the center. For example if $x$ is changed into $x-1$ and $y$ is changed into $y-2$, (in 2D for clarity) each point in the circle gets shifted to $x+1$ and $y+2$.

So in this case,

$x_{c_2} = x_{c_1} +1$ and $y_{c_2} = y_{c_1}+2$

Similarly a point

$P_1 \equiv(0,2)$ gets shifted to $P_2 \equiv(1,4)$

This is also true in 3 dimensions.