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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 )

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  • $\begingroup$ You know - with the octagons-along-a-horizonatl-line model, it's clear that each octagon will have a column of octagons above and below it (with a large nuber of other octagons fitted into the gaps). We could treat this as a repeating patter of (say) 360 octagons along that base, and glue the whole structure into a cylinder. That might be nice for game - each octagon is a day, the Sun travels along that circuit once a year. $\endgroup$ Nov 20, 2014 at 15:14
  • $\begingroup$ Your question is probably a little too vague for this forum. The main issue is there's no concrete shape for you to aim for -- if you had a precise idea of what it is you wanted, your question would be a better fit for the forum. Have you looked up Taimina and Henderson's crochet models of hyperbolic space? cabinetmagazine.org/issues/16/crocheting.php That would be a good place to start. $\endgroup$ Nov 20, 2014 at 18:54
  • $\begingroup$ It's ok - I think I know what I want to do. The mapping will be: For a point A, find the point B on the horizontal axis such that A is above or below it (i.e., AB is at right angles to OB). The x coordinate is simply the hyperbolic distance between OB, the ycoordinate the hyperbolic distance BA. This will not be equal-area, but will line p the octagons along the axes nicely. So delete the question, I'm good. Ta. $\endgroup$ Nov 26, 2014 at 5:11

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I think this question is intrinsically interesting, but not research-level. Let me still give you some hints.

There are many, many ways to achieve what you want; this might be why you have trouble, there is no way to "compute the solution" to your problem, you have to choose a suitable solution among many. The simplest way to go would be to impose more conditions. For example, in the case where you ask a line to be undistorted, you could choose to project this hyperbolic line $L$ to any given Euclidean line $M$, then to projects the hyperbolic lines crossing $L$ orthogonally to Euclidean lines orthogonal to $M$. Then you have no choice left if you want areas to be preserved, you have to solve in the natural coordinates given by your choice the equation that says the Jacobian of your map is $1$.

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