Creating a Star Map?

Question

I have a set of data on all the stars (well, to a magnitude of 9 or so) with the values of the following properties: - magnitude, - right ascension and, - declination.Now I'd like to create a planar (rectangle in shape) map of the entire sky (both hemispheres), without stretching anything (keeping all constellations etc., as seen in the sky.)I know this is more of a math question, but I guess there's some fairly known function or ratio to achieve this; if so, how?

Answer 1

This is fundamentally impossible, unfortunately, since the constellations are drawn on the celestial sphere, you can't squash them down to a rectangular shape. Think about if you drew a constellation on a fully blown-up balloon; if you deflated the balloon and cut the piece with the constellation, the proportions would be all messed up; some areas would shrink more than others. This is true for one constellation; now imagine drawing all of them on a balloon and trying this same thing. This is because a sphere has intrinsic curvature, which means there's no way you can represent a sphere as a non-curved object without distortion. A rectangle is a non-curved object. There's a very interesting theorem called the ball of hair theorem; essentially, it concludes that if you have a ball covered with hair, you cannot comb all the hair on it straight, in the same direction; there will always be some hair pointing in a different direction. However, there are plenty of functions that map images on spheres onto a rectangular area, like the famous Mercator projection for rectangular maps in classrooms. Same with Mollweide or stereographic projections; Mercator is probably what you want, though. With all of these, however, there is distortion. All you would need to do is plug your RA and declination into the projection formula, and it will tell you the x and y coordinates on the rectangle you're mapping onto.