<p>Partial answer:</p>
<blockquote>
<p>Since a spherical surface (the celestial sphere) is projected onto a rectangular surface...</p>
</blockquote>
<p>There really isn't a spherical surface, but yes, a &quot;distortionless&quot; telescope maps directions on to a photographic plate or flat CCD sensor similarly to the <a href="https://en.wikipedia.org/wiki/Pinhole_camera_model" rel="nofollow noreferrer">pinhole camera model</a> which definitely does not convert RA and Dec into completely orthogonal, cartesian coordinates.</p>
<p><a href="https://i.sstatic.net/OBMT4.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OBMT4.png" alt="enter image description here" /></a></p>
<ul>
<li>also <a href="https://astronomy.stackexchange.com/q/38784/7982">Why was the Vera C. Rubin Observatory LSST Camera tested with a broccoli and a pinhole?</a></li>
</ul>
<blockquote>
<p>Why are pixel scales usually assumed to be constant?</p>
</blockquote>
<p><em>Who sez?</em> In what <em>astrometric contexts</em> are pixel scales are usually assumed to be constant? I think any instrument used for astrometry goes through an elaborate calibration, and the plate solutions used in astrometric exposures will include both a number of known objects and positions and certainly at least some polynomial model of order greater than one if not something even more elaborate and descriptive.</p>
<blockquote>
<p>For astrometry, how are these geometrical distortions corrected?</p>
</blockquote>
<p>Like I said, &quot;partial answer&quot; and I am sure an actual astronomer or astrometer will chime in with a thorough answer. Until then, I'll quote myself: &quot;...astrometric exposures will include both a number of known objects and positions and certainly at least some polynomial model of order greater than one if not something even more elaborate and descriptive.&quot;</p>
<blockquote>
<p>Is there any systematic way to read up on this topic? (of optics, imaging etc)</p>
</blockquote>
<p>A systematic approach may not be the best way, I'd say find the avenue of least resistance. If you like math, start with Wikipedia's:</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Projective_geometry" rel="nofollow noreferrer">Projective geometry</a></li>
<li><a href="https://en.wikipedia.org/wiki/Optical_aberration" rel="nofollow noreferrer">Optical aberration</a></li>
<li><a href="https://en.wikipedia.org/wiki/Zernike_polynomials" rel="nofollow noreferrer">Zernike polynomials</a></li>
</ul>
<p>and then pursue references therein or a book on optics you find suitable; they range from overly simplistic to absurdly obfuscative. Borne &amp; Wolf is about in the middle, and it will keep you busy for a long time.</p>
<p>You also might hunt around and choose a specific telescope/observatory that has a lot of supporting information and papers written about it. Or you can search for early papers including the search terms &quot;Schmidt&quot; &quot;plate&quot; &quot;solution&quot; or similar.</p>
<p>Or you can look at one of the newer, &quot;crazier&quot; optical systems that due to field distortion and especially field curvature (not-flat CCDs!) have solutions so interesting that they'll have several modern papers and perhaps even several thesises written about them.</p>
<ul>
<li><a href="https://astronomy.stackexchange.com/q/21478/7982">In a reflecting telescope (Cassegrain), is there a difference between using a concave or convex secondary mirror?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/15005/7982">What is a quaternary mirror and why does the E-ELT need one?</a></li>
<li><a href="https://astronomy.stackexchange.com/a/49247/7982">this answer to <em>Why aren't reflector telescopes built with an offset secondary mirror?</em></a></li>
</ul>
<p>Actually, a good thesis might be your best bet as it will strive to put an amazing amount of perspective -- from historical to state-of-the-art -- all in one place, since <a href="https://academia.stackexchange.com/a/187017/69206">most of us write way too much background</a>.</p>
<p>As for Van De Kamp's <em>Stellar Paths</em> and <em>Principles of Astrometry</em> and Smart's <em>Spherical Astronomy</em> yes they may address some issues but not in a systematic way (there's that word again).</p>

CC BY-SA 4.0
2022-07-25T06:17:45.047Z
2022-07-25T06:17:45.047Z
uhoh
<p>CCD's are widely used for astrometry. Since a spherical surface (the celestial sphere) is projected onto a rectangular surface, the plate scale or pixel scale (approximated with 1/focal length etc) is used to estimate the angular separation of stars in an image.</p>
<p>However, due to the actual plate scale being non-uniform across the whole sensor, different images with the same constellation of stars at different positions on the sensor will result in distorted images, compared to a centered image of the stars.</p>
<hr />
<p>Hence for accurate work e.g. astrometry,</p>
<p>Why are pixel scales usually assumed to be constant? Is it because other sources of error would usually dominate hence accounting for differences in pixel scale is unnecessary?</p>
<p>For astrometry, how are these geometrical distortions corrected?</p>
<p>Is there any systematic way to read up on this topic? (of optics, imaging etc)</p>
<hr />
<p>What I've found so far:</p>
<p>The series of articles:</p>
<p>Ground-based CCD astrometry with wide field imagers</p>
<p>(J. Anderson et al, 2006)
<a href="https://doi.org/10.1051/0004-6361:20065004" rel="nofollow noreferrer">https://doi.org/10.1051/0004-6361:20065004</a></p>
<p>(auto-calibration described in detail)</p>
<p>(A.Bellini and L.R.Bedin, 2010)
<a href="https://doi.org/10.1051/0004-6361/200913783" rel="nofollow noreferrer">https://doi.org/10.1051/0004-6361/200913783</a></p>
<p>(here the authors discuss auto-calibration briefly.)</p>
<blockquote>
<p>Since the scale is a free
parameter in deriving GD correction, choosing a particular scale
value will not invalidate the solution itself.</p>
</blockquote>
<p>Not sure whether the above remarks in the second paper means that non-uniform pixel-scale is not a concern, at least to the algorithm.</p>
<p>And a book
Stellar Paths Photographic Astrometry with Long-Focus Instruments, Peter Van De Kamp</p>
<p>Plate solving is also discussed in Spherical Astronomy, Smart</p>

CC BY-SA 4.0
2022-07-25T03:16:54.470Z
You need to keep in mind that even wide field imagers  usually cover less than a degree, so the change in scale when projecting is quite small.  Think of a map of the Earth that just covers a degree or so right at the Equator.  However, the mirrors are not perfect so there could be other distortions besides just projection issues.  If the telescope is used for astrometry, then usually there will be a series of calibration images taken to map out a low order polynomical fit of all the optical distortions.
eshaya
CC BY-SA 4.0
2022-07-25T03:24:10.570Z
@eshaya indeed; reading https://www.aanda.org/articles/aa/pdf/2008/16/aa8402-07.pdf

The pixel scale at the center is
0.2275" ± 0.0001 and the median value is 0.2254" ± 0.0001.

Yet refraction, light aberration is already on the order of several arcseconds right?
Cheng
CC BY-SA 4.0
2022-07-25T03:27:03.993Z
@eshaya It must be convenient to lump all distortions be it due to plate scale, optics, misalignments as "Geometrical Distortions" and deal with them in one go.
Cheng
CC BY-SA 4.0
2022-07-25T03:29:58.293Z
@eshaya Wide field imagers cover less than one degree, yet they already have the largest FOV, I guess this is why plate scale variations are neglected in all other telescope....
Cheng
<telescope><photography><coordinate><data-analysis><optics>
Astrometry and Geometric Distortions
Cheng
# Astrometry and Geometric Distortions

(Asked by: username_0 on 2022-07-25)

CCD's are widely used for astrometry. Since a spherical surface (the celestial sphere) is projected onto a rectangular surface, the plate scale or pixel scale (approximated with 1/focal length etc) is used to estimate the angular separation of stars in an image.

However, due to the actual plate scale being non-uniform across the whole sensor, different images with the same constellation of stars at different positions on the sensor will result in distorted images, compared to a centered image of the stars.

---

Hence for accurate work e.g. astrometry,

Why are pixel scales usually assumed to be constant? Is it because other sources of error would usually dominate hence accounting for differences in pixel scale is unnecessary?

For astrometry, how are these geometrical distortions corrected?

Is there any systematic way to read up on this topic? (of optics, imaging etc)

---

What I've found so far:

The series of articles:

Ground-based CCD astrometry with wide field imagers

(J. Anderson et al, 2006)
<https://doi.org/10.1051/0004-6361:20065004>

(auto-calibration described in detail)

(A.Bellini and L.R.Bedin, 2010)
<https://doi.org/10.1051/0004-6361/200913783>

(here the authors discuss auto-calibration briefly.)

> Since the scale is a free
> parameter in deriving GD correction, choosing a particular scale
> value will not invalidate the solution itself.

Not sure whether the above remarks in the second paper means that non-uniform pixel-scale is not a concern, at least to the algorithm.

And a book
Stellar Paths Photographic Astrometry with Long-Focus Instruments, Peter Van De Kamp

Plate solving is also discussed in Spherical Astronomy, Smart

**Tags:** telescope, photography, coordinate, data-analysis, optics

> **Comments:**
> **username_1**: You need to keep in mind that even wide field imagers  usually cover less than a degree, so the change in scale when projecting is quite small.  Think of a map of the Earth that just covers a degree or so right at the Equator.  However, the mirrors are not perfect so there could be other distortions besides just projection issues.  If the telescope is used for astrometry, then usually there will be a series of calibration images taken to map out a low order polynomical fit of all the optical distortions.  
> **username_0**: @username_1 indeed; reading https://www.aanda.org/articles/aa/pdf/2008/16/aa8402-07.pdf  The pixel scale at the center is 0.2275" ± 0.0001 and the median value is 0.2254" ± 0.0001.  Yet refraction, light aberration is already on the order of several arcseconds right?  
> **username_0**: @username_1 It must be convenient to lump all distortions be it due to plate scale, optics, misalignments as "Geometrical Distortions" and deal with them in one go.  
> **username_0**: @username_1 Wide field imagers cover less than one degree, yet they already have the largest FOV, I guess this is why plate scale variations are neglected in all other telescope....  

---

## 1 answers

### Answer by username_2 on 2022-07-25. Score: 1 | Accepted answer

Partial answer:

> Since a spherical surface (the celestial sphere) is projected onto a rectangular surface...

There really isn't a spherical surface, but yes, a "distortionless" telescope maps directions on to a photographic plate or flat CCD sensor similarly to the [pinhole camera model](https://en.wikipedia.org/wiki/Pinhole_camera_model) which definitely does not convert RA and Dec into completely orthogonal, cartesian coordinates.

[![enter image description here](https://i.sstatic.net/OBMT4.png)](https://i.sstatic.net/OBMT4.png)

* also [Why was the Vera C. Rubin Observatory LSST Camera tested with a broccoli and a pinhole?](https://astronomy.stackexchange.com/q/38784/7982)

> Why are pixel scales usually assumed to be constant?

*Who sez?* In what *astrometric contexts* are pixel scales are usually assumed to be constant? I think any instrument used for astrometry goes through an elaborate calibration, and the plate solutions used in astrometric exposures will include both a number of known objects and positions and certainly at least some polynomial model of order greater than one if not something even more elaborate and descriptive.

> For astrometry, how are these geometrical distortions corrected?

Like I said, "partial answer" and I am sure an actual astronomer or astrometer will chime in with a thorough answer. Until then, I'll quote myself: "...astrometric exposures will include both a number of known objects and positions and certainly at least some polynomial model of order greater than one if not something even more elaborate and descriptive."

> Is there any systematic way to read up on this topic? (of optics, imaging etc)

A systematic approach may not be the best way, I'd say find the avenue of least resistance. If you like math, start with Wikipedia's:

* [Projective geometry](https://en.wikipedia.org/wiki/Projective_geometry)
* [Optical aberration](https://en.wikipedia.org/wiki/Optical_aberration)
* [Zernike polynomials](https://en.wikipedia.org/wiki/Zernike_polynomials)

and then pursue references therein or a book on optics you find suitable; they range from overly simplistic to absurdly obfuscative. Borne & Wolf is about in the middle, and it will keep you busy for a long time.

You also might hunt around and choose a specific telescope/observatory that has a lot of supporting information and papers written about it. Or you can search for early papers including the search terms "Schmidt" "plate" "solution" or similar.

Or you can look at one of the newer, "crazier" optical systems that due to field distortion and especially field curvature (not-flat CCDs!) have solutions so interesting that they'll have several modern papers and perhaps even several thesises written about them.

* [In a reflecting telescope (Cassegrain), is there a difference between using a concave or convex secondary mirror?](https://astronomy.stackexchange.com/q/21478/7982)
* [What is a quaternary mirror and why does the E-ELT need one?](https://astronomy.stackexchange.com/q/15005/7982)
* [this answer to *Why aren't reflector telescopes built with an offset secondary mirror?*](https://astronomy.stackexchange.com/a/49247/7982)

Actually, a good thesis might be your best bet as it will strive to put an amazing amount of perspective -- from historical to state-of-the-art -- all in one place, since [most of us write way too much background](https://academia.stackexchange.com/a/187017/69206).

As for Van De Kamp's *Stellar Paths* and *Principles of Astrometry* and Smart's *Spherical Astronomy* yes they may address some issues but not in a systematic way (there's that word again).


---

