Repost from StackOverflow, where I was advised to seek help here :D
I'm neither a geometry student or a native speaker, so apologies if my question isn't clear enough.
As part of my master's thesis, I have to plot bounded regions of the night sky onto a 2D plane. My current solution consists of a rectangular mapping where (ra, dec) values are plotted to (x,y) coordinates. While this approach works well enough for small regions in relatively low ascension values, the resulting plots get progressively distorted for higher ||dec|| values, as expected.
At some point I'll have to change this to a more versatile approach. Thing is, I'm not exactly clear on what to search for. I guess I have to be able to map angular coordinates to a square (or hexagon) subgrid, but most search results I get are concerned with full-surface mapping.
I know I won't be able to achieve a perfect, distortion-free plotting, but I don't require perfect solutions; only a more general projection that will work well near the poles. Something like this, where I put my Photoshop skills to work and try to simulate a 20º region under my current approach and the one I'm looking for:
TL;DR: how do I convert between coordinates on a sphere (ra/dec) to cartesian coordinates on a locally-defined grid?
I think that near the poles this may work:
The complement of the declination $d$, so $90^\circ - d$, is proportional to the great circle distance from the pole. Scale that distance to whatever units you need; call the answer $r$.
The right ascension measures the longitude, as an angle $\theta$.
Then use $(r, \theta)$ as the polar coordinates of a point on your plane. If you need rectangular coordinates, just do the standard polar to rectangular conversions.
You'll have to decide whether the distortions are too bad $20^\circ$ from the pole. If they are, you need a more sophisticated projection. Consult a cartographer.