<p>There is kind of an answer <a href="https://math.stackexchange.com/questions/92301/transforming-from-one-spherical-coordinate-system-to-another">over at Math</a>. 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. </p>

<p>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>

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

CC BY-SA 3.0
2013-12-10T10:04:59.620Z
Reading this, I am not quite sure if maybe the $r'$ needs to be multiplied instead, since it represents a scaling...
Arne
CC BY-SA 3.0
2013-12-10T10:06:19.493Z
I am rather sure now. Please correct me if I'm wrong -- edited the answer accordingly.
Arne
CC BY-SA 3.0
2013-12-10T10:04:01.490Z
2013-12-10T10:04:01.490Z
2017-04-13T12:19:35.350Z
Arne
<p>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 $\left(r',\theta',\phi'\right)$.</p>

<p>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: $$\left(x,y,z\right)=\left(x'+a,y'+b,z'+c\right)$$</p>

<p>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 $\left(r',\theta',\phi'\right)?$</p>

<fundamental-astronomy><coordinate>
Coordinate transformations between reference frames in spherical astronomy
David H
# Coordinate transformations between reference frames in spherical astronomy

(Asked by: username_0 on 2013-12-10)

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 $\left(r',\theta',\phi'\right)?$

**Tags:** fundamental-astronomy, coordinate


---

## 1 answers

### Answer by username_1 on 2013-12-10. Score: 3 | Accepted answer

There is kind of an answer [over at Math](https://math.stackexchange.com/questions/92301/transforming-from-one-spherical-coordinate-system-to-another). 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:

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

> **Comments:**
> **username_1**: Reading this, I am not quite sure if maybe the $r'$ needs to be multiplied instead, since it represents a scaling...  
> **username_1**: I am rather sure now. Please correct me if I'm wrong -- edited the answer accordingly.  

---

