<p>I've found the solution to my problem, detailed neatly in <a href="https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1965JRASC..59..155M&amp;defaultprint=YES&amp;filetype=.pdf" rel="nofollow noreferrer">this paper</a> by the late astrophysicist Bruce A McIntosh.</p>
<p>Citation: McIntosh, B., 1965. The Stereographic Projection. <em>Journal of the Royal Astronomical Society of Canada</em>, 59, p.155.</p>

CC BY-SA 4.0
2022-06-07T22:38:20.323Z
As noted in the paper, this method is just an approximation for use when using a compass to draw the chart.  There is little to be gained from this method when drawing graphics with a computer.
Greg Miller
CC BY-SA 4.0
2022-06-08T07:44:22.677Z
Having actually now implemented this solution, I am 100% satisfied with it. It results in simpler code, and when rendered with vector graphics it is more performant. Although your recommendation would suit a situation where I am wanting to use other map projections, this question firmly defined the scope as concerning only the stereographic map.
h0dges
CC BY-SA 4.0
2022-06-10T19:11:04.010Z
For anyone else needing this I also found this paper: http://www.astro.ro/~roaj/26_3/19-dvasilca.pdf which outlines things quite comprehensively.
h0dges
CC BY-SA 4.0
2022-06-07T19:33:55.013Z
2022-06-07T19:33:55.013Z
h0dges
<p>I'm trying to find a set of formulae that allows me to plot equatorial grid circles (latitude and meridian sets) onto a stereographic projection of the night sky. I've already worked out how to project stars (i.e. points) onto the circular chart for an observer at a given time and location (thanks to this answer: <a href="https://astronomy.stackexchange.com/a/35888">https://astronomy.stackexchange.com/a/35888</a>), but I'm struggling to work out how to do the same for the equatorial grid.</p>
<p>I can plot the special case at the poles where the latitude circles are concentric, but I'm struggling to ascertain what the general formula looks like to plot non-concentric latitude circles. I can see that it must take the observer's latitude and translate the circles origin toward infinity on the y-axis (and tend the radius to infinity) in order, for example, to achieve a straight line bisecting the sky when the observer is at the equator.</p>
<p>Any help would be appreciated :)</p>

CC BY-SA 4.0
2022-06-06T21:15:24.810Z
Use the same code you have for stars to compute points along the circles, and draw lines between them.  Compute them as close together as necessary for the accuracy you need.
Greg Miller
CC BY-SA 4.0
2022-06-06T22:56:41.397Z
I second @GregMiller's suggestion; treat the coordinate grid as a set of synthetic stars. That avoids special cases (e.g. great circles through the zenith) and easily generalizes to other projections. Do you still want an analytic solution anyway?
Mike G
CC BY-SA 4.0
2022-06-07T07:20:38.530Z
Yes, I'm still seeking an analytic solution to this problem. I found some interesting material on the matter here https://apps.dtic.mil/sti/pdfs/ADA037381.pdf (pages 195-200), but I'm not sure it provides a general analytic solution with cartesian coordinates + radius as output.
h0dges
<projection><map>
Plotting equatorial grid circles onto stereographic projection
h0dges
# Plotting equatorial grid circles onto stereographic projection

(Asked by: username_0 on 2022-06-06)

I'm trying to find a set of formulae that allows me to plot equatorial grid circles (latitude and meridian sets) onto a stereographic projection of the night sky. I've already worked out how to project stars (i.e. points) onto the circular chart for an observer at a given time and location (thanks to this answer: <https://astronomy.stackexchange.com/a/35888>), but I'm struggling to work out how to do the same for the equatorial grid.

I can plot the special case at the poles where the latitude circles are concentric, but I'm struggling to ascertain what the general formula looks like to plot non-concentric latitude circles. I can see that it must take the observer's latitude and translate the circles origin toward infinity on the y-axis (and tend the radius to infinity) in order, for example, to achieve a straight line bisecting the sky when the observer is at the equator.

Any help would be appreciated :)

**Tags:** projection, map
> **Comments:**
> **username_1**: Use the same code you have for stars to compute points along the circles, and draw lines between them.  Compute them as close together as necessary for the accuracy you need.  
> **username_2**: I second @GregMiller's suggestion; treat the coordinate grid as a set of synthetic stars. That avoids special cases (e.g. great circles through the zenith) and easily generalizes to other projections. Do you still want an analytic solution anyway?  
> **username_0**: Yes, I'm still seeking an analytic solution to this problem. I found some interesting material on the matter here https://apps.dtic.mil/sti/pdfs/ADA037381.pdf (pages 195-200), but I'm not sure it provides a general analytic solution with cartesian coordinates + radius as output.  

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## 1 answers

### Answer by username_0 on 2022-06-07. Score: 1 | Accepted answer

I've found the solution to my problem, detailed neatly in [this paper](https://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1965JRASC..59..155M&defaultprint=YES&filetype=.pdf) by the late astrophysicist Bruce A McIntosh.

Citation: McIntosh, B., 1965. The Stereographic Projection. *Journal of the Royal Astronomical Society of Canada*, 59, p.155.

> **Comments:**
> **username_1**: As noted in the paper, this method is just an approximation for use when using a compass to draw the chart.  There is little to be gained from this method when drawing graphics with a computer.  
> **username_0**: Having actually now implemented this solution, I am 100% satisfied with it. It results in simpler code, and when rendered with vector graphics it is more performant. Although your recommendation would suit a situation where I am wanting to use other map projections, this question firmly defined the scope as concerning only the stereographic map.  
> **username_0**: For anyone else needing this I also found this paper: http://www.astro.ro/~roaj/26_3/19-dvasilca.pdf which outlines things quite comprehensively.  

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