How do I turn a 3D Star Map into a 2D Grid Map?

I don't know if this is the correct forum to post this in (I think the problem is related to Geometry), but here goes anyway. Ok, so I've been trying to create a kind of 2-Dimensional grid of the Stars which are closest to the Sun (see list here, if you don't know what I'm talking about). I'm doing it so that I can possibly create a mod for a space game which would use the *real* Stars in our Astronomical vicinity (rather then fictional ones), but of course the problem is that space is 3-Dimensional (some Stars would appear above others from a perspective from the Sun's North Pole) and I need to somehow make that 3D arrangement into a 2D map.

When I started drawing it out in MS Paint I thought it would be easy; I just started with a square grid layout with each square representing 0.1 of a Light-year in distance (with the Sun in the Centre square), so that Alpha Centauri (the closest Star to the Sun, at 4.37 Light-years away) would be 44 grid squares away from the central square. That's all fine, but then when you add another Star, Barnard's Star (5.98 LYs away, or 60 grid squares), you then also have to make sure that it is not only the proper distance away the Sun, but also from Alpha Centauri. It turns out Barnard's Star is actually 6.49 LYs away from Alpha Centauri (I know this because I have Celestia, which can tell you all that kind of detail), I could work out where to put these two Stars so they would be correct, but I realized that if I was going to put in 50 to 100 Stars it would become practically impossible to place them so that the grid square distances between them corresponded to the real astronomical distances.

So basically, I need to convert a 3D map (like the ones here and here) into a 2D map of grid squares. So far, I have a drawing with 200 x 200 grid squares (each representing 0.1 LYs, so 20 LYs in all direction, that's 40,000 squares in total) the Central square represents the Sun and our Solar System, and each of the Stars near the Sun must be represented as being in another square somewhere else on the grid (with the appropriate number of squares in-between them). But each Star must also be the correct distance from *its* neighbors.

Sorry if this sounds overly complicated, but essentially I need some way of converting 3D distances and positions into 2D distances and positions, I figure there may be a mathematical solution. So can anyone tell me what to do?

Re: How do I turn a 3D Star Map into a 2D Grid Map?

Any help at all with this? Though I tried experimenting more and I'm starting to think it's not possible to do.

Re: How do I turn a 3D Star Map into a 2D Grid Map?

It's trivial to see that this is impossible. Just consider four stars equidistant from each other. In 3d this is a tetrahedral. In 2d no such arrangement is possible.