<p>The example sky map appears to be a <a href="https://en.wikipedia.org/wiki/Stereographic_projection" rel="nofollow noreferrer">stereographic projection</a>; circles of right ascension and declination are projected as circular arcs, and their intersections are projected as right angles.
The example has a 195&deg; field of view, and the eastern and western horizons are mislabeled.</p>

<p>Let the celestial sphere be centered at the origin with radius 1, zenith at (0,&nbsp;0,&nbsp;1), northern horizon at (0,&nbsp;1,&nbsp;0), and eastern horizon at (1,&nbsp;0,&nbsp;0).
Assuming azimuth 0&deg; is north:
<span class="math-container">$$\begin{align}
x &amp;= \sin \mathrm{Az} \cos \mathrm{Alt} \\
y &amp;= \cos \mathrm{Az} \cos \mathrm{Alt} \\
z &amp;= \sin \mathrm{Alt}
\end{align}$$</span></p>

<p>To project from the nadir (0,&nbsp;0,&nbsp;-1) onto the plane <em>z</em>&nbsp;=&nbsp;0:
<span class="math-container">$$ (x', y') = \left( \frac{x}{z + 1}, \frac{y}{z + 1} \right) $$</span></p>

<p>To fit this between (0,&nbsp;0) and (2<i>R</i>,&nbsp;2<i>R</i>) on a canvas with northern horizon at (<em>R</em>,&nbsp;0) and eastern horizon at (0,&nbsp;<em>R</em>), scale by -<em>R</em> and translate by (<em>R</em>,&nbsp;<em>R</em>):
<span class="math-container">$$ (X, Y) = (R(1 - x'), R(1 - y')) $$</span></p>

<p>For a sky map centered overhead, if north is up, east should be on the left.</p>

CC BY-SA 4.0
2020-04-21T06:10:04.317Z
This is so cool! I'm gonna give it a try..
uhoh
CC BY-SA 4.0
2020-04-21T09:13:45.267Z
Thanks you and **uhoh** for helping me out with this project. I owe you big time!
maisteRR
CC BY-SA 4.0
2020-04-21T20:41:08.240Z
Can I ask you last question? how can i calculate the coordinates for the grid as in the screenshot?
maisteRR
CC BY-SA 4.0
2020-04-22T03:43:51.880Z
@maisteRR Those appear to be at intervals of 10° in declination and 1h (15°) in right ascension. Process your synthetic grid points the same way as the star data.
Mike G
CC BY-SA 4.0
2020-04-22T23:06:32.030Z
@MikeG, thanks, but I'm a little confused. This mean, that I must get alt and az for value "previous dec val + 10" and "previous RA + 15" every time when I getting coords the star? I know, most likely I'm wrong and should do it in another way. Can you give one more hint?
maisteRR
CC BY-SA 4.0
2020-04-22T23:19:10.937Z
@maisteRR For each circle to plot (RA=0h, RA=1h, RA=2h, ...; Dec=0°, Dec=10°, Dec=20°, ...), generate a set of fake stars at smaller intervals (RA circle: Dec=0°, Dec=1°, Dec=2°, ...; Dec circle: RA=0h, RA=0h5m, RA=0h10m, ...), find their projected coordinates on the canvas, and connect those with line segments.
Mike G
CC BY-SA 4.0
2020-04-21T03:58:56.867Z
2020-04-21T18:55:13.967Z
2020-04-21T18:55:13.967Z
Mike G
<p>Thanks to <a href="https://astronomy.stackexchange.com/users/7982/uhoh">@ohoh</a> for helped translate ra and dec into alt and az. Now that I have this data(altitude and azimuth), how can I plot it in something like <a href="https://i.sstatic.net/lXqvx.jpg" rel="nofollow noreferrer">this</a>, with the coordinate system going like <a href="https://i.sstatic.net/1DUHn.jpg" rel="nofollow noreferrer">this</a>?</p>

CC BY-SA 4.0
2020-09-11T18:23:55.287Z
Related: [How to make a star chart from entries in a Bright Star Catalogue?](/q/35864)
Mike G
<coordinate><night-sky>
How to make projection from altitude and azimuth to screen with screen coordinate system?
maisteRR
# How to make projection from altitude and azimuth to screen with screen coordinate system?

(Asked by: username_0 on 2020-04-20)

Thanks to [@ohoh](https://astronomy.stackexchange.com/users/7982/username_2) for helped translate ra and dec into alt and az. Now that I have this data(altitude and azimuth), how can I plot it in something like [this](https://i.sstatic.net/lXqvx.jpg), with the coordinate system going like [this](https://i.sstatic.net/1DUHn.jpg)?

**Tags:** coordinate, night-sky

> **Comments:**
> **username_1**: Related: [How to make a star chart from entries in a Bright Star Catalogue?](/q/35864)  

---

## 1 answers

### Answer by username_1 on 2020-04-21. Score: 3 | Accepted answer

The example sky map appears to be a [stereographic projection](https://en.wikipedia.org/wiki/Stereographic_projection); circles of right ascension and declination are projected as circular arcs, and their intersections are projected as right angles.
The example has a 195° field of view, and the eastern and western horizons are mislabeled.

Let the celestial sphere be centered at the origin with radius 1, zenith at (0, 0, 1), northern horizon at (0, 1, 0), and eastern horizon at (1, 0, 0).
Assuming azimuth 0° is north:
$$\begin{align}
x &= \sin \mathrm{Az} \cos \mathrm{Alt} \\
y &= \cos \mathrm{Az} \cos \mathrm{Alt} \\
z &= \sin \mathrm{Alt}
\end{align}$$

To project from the nadir (0, 0, -1) onto the plane *z* = 0:
$$ (x', y') = \left( \frac{x}{z + 1}, \frac{y}{z + 1} \right) $$

To fit this between (0, 0) and (2*R*, 2*R*) on a canvas with northern horizon at (*R*, 0) and eastern horizon at (0, *R*), scale by -*R* and translate by (*R*, *R*):
$$ (X, Y) = (R(1 - x'), R(1 - y')) $$

For a sky map centered overhead, if north is up, east should be on the left.

> **Comments:**
> **username_2**: This is so cool! I'm gonna give it a try..  
> **username_0**: Thanks you and **username_2** for helping me out with this project. I owe you big time!  
> **username_0**: Can I ask you last question? how can i calculate the coordinates for the grid as in the screenshot?  
> **username_1**: @username_0 Those appear to be at intervals of 10° in declination and 1h (15°) in right ascension. Process your synthetic grid points the same way as the star data.  
> **username_0**: @username_1, thanks, but I'm a little confused. This mean, that I must get alt and az for value "previous dec val + 10" and "previous RA + 15" every time when I getting coords the star? I know, most likely I'm wrong and should do it in another way. Can you give one more hint?  
> **username_1**: @username_0 For each circle to plot (RA=0h, RA=1h, RA=2h, ...; Dec=0°, Dec=10°, Dec=20°, ...), generate a set of fake stars at smaller intervals (RA circle: Dec=0°, Dec=1°, Dec=2°, ...; Dec circle: RA=0h, RA=0h5m, RA=0h10m, ...), find their projected coordinates on the canvas, and connect those with line segments.  

---

