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Suppose there are two observational frames of reference with origins $O$ and $O'$, respectively, separated by some constant distance. A body located at point $P$ has Cartesian coordinates $\left(x,y,z\right)$ and $\left(x',y',z',\right)$ in $O$ and $O'$, respectively; and similarly, the spherical coordinates of $P$ in the unprimed and primed frames are $\left(r,\theta,\phi\right)$ and $\left(r',\theta',\phi'\right)$.

Say $O$ determines the Cartesian coordinates of $O'$ to be $\left(a,b,c\right)$. Then the Cartesian coordinates of $P$ in $O$ are simply related to the primed Cartesian coordinates: $$\left(x,y,z\right)=\left(x'+a,y'+b,z'+c\right)$$

My question, then, is how does one compute the affect of translation on spherical coordinates. That is, given $\left(a,b,c\right)$, how can one write $\left(r,\theta,\phi\right)$ as a function of $\left(r',\theta',\phi'\right)?$

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up vote 3 down vote accepted

There is kind of an answer over at Math. All you can do in spherical coordinates is to change the position of your "pole", i.e. you have $(1,0,0)$ in your first coordinate system, which is mapped to some $(r', \vartheta', \varphi')$ in the second coordinate system. The two angles represent a rotation, and the $r$ represents a scaling.

I think since we still deal with a vector space (for the angular part), you can simply add the origin of coordinate system two to vectors of the first coordinate system. The radius is a scaling, and needs to be multiplied. Hence for a point $p=(r, \vartheta, \varphi)$ in coordinate system 1 you get:

$$ p'=(r \cdot r', \vartheta+\vartheta', \varphi+\varphi') $$

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Reading this, I am not quite sure if maybe the $r'$ needs to be multiplied instead, since it represents a scaling... – Arne Dec 10 '13 at 10:04
I am rather sure now. Please correct me if I'm wrong -- edited the answer accordingly. – Arne Dec 10 '13 at 10:06

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