I have a set of data on all the stars (well, to a magnitude of 9 or so) with the

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? I haven't found it on Google, though) to achieve this; if so, how?

Explanation / Answer

I'd like to create a planar (rectangle shaped) map of the entire (both hemispheres) sky, with stretching anything (keeping all constellations etc. as seen in the sky).

This is a physical impossibility. You simply cannot map a spherical entity (the celestial sphere) onto a plane without introducing some distortion. Cartographers have developed many different projections in their efforts to solve this problem, but none is perfect. All of them are forced to introduce distortion at some point.

The Mercator projection suggested in another answer is notoriously inaccurate as you get closer to the poles, making Greenland as large as South America, and stretching Antarctica into an impossible shape.

The best solution is to forget about mapping the whole sky, but to concentrate on smaller areas where the distortions are not as severe. This is what most star atlases and planetarium software programs do.

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