I'm aware of the projections of the hyperbolic plane at http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane, but how would I project the hyperbolic plane onto a flat piece of paper to preserve area? Obviously, what happens is that circles get stretched out as you go away from the center, but how would I work out the actual shapes? What do straight lines look like?
How would you do it so that shapes were not distorted along a straight line? I'm interested in drawing a lattice of regular octagons and moving them around with the mouse. It would be nice if there's a projection where there's an axis which is basically an infinitely long line of equal-sized octagons, and the rest of the octagons get distorted progressively as you move away from that line. Of course, as you move away from the center point, the octagons begin to look pretty un-octagonal. But that's ok.
How would I work out the symmetries? More to the point - has someone already done this?
(FYI: I'm thinking of drawing out a map of the astral plane for D&D. The cells are octagonal to correspond with the cardinal compass directions. Players will find it … odd to navigate :D )