How to make projection from altitude and azimuth to screen with screen coordinate system?

Question

Thanks to @ohoh 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, with the coordinate system going like this?

Answer 1

The example sky map appears to be a 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 (2R, 2R) 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.