# Map projection for a pseudospherical world

A pseudosphere (or antisphere or tractricoid) has a surface of constant negative Gaussian curvature, in contrast to a sphere, which has constant positive Gaussian curvature.

Source: http://xahlee.info/surface/pseudosphere/pseudosphere.html

Handwaving the model of physics and spacetime that would allow the formation of bodies closely approximating this top-like shape—pseudosphere planets, pseudosphere stars, etc—how would you optimally (i.e. with least distortion) display the surface as a 2D map?

My instinct says it would be comparable to the Sinusoidal projection, but with the maximum distortion at the equator instead of poles. But I have a hard time resolving the equator as the widest and largest portion of the surface having the most distortion.

Source: https://en.wikipedia.org/wiki/Sinusoidal_projection

• The surface of body shown in the first picture does not have constant negative curvature. There is an infinite discountinuity at what you call the equator. This infinite discontinuity is discarded on the map; that's the massive distortion which you ignored. Commented Jul 30, 2020 at 21:34
• I suspect the short answer is "use a projection with minimal distortion". Sounds tautological, but the way you achieve that is by cutting the map into pieces, a la Dymaxion or Waterman. Commented Jul 30, 2020 at 21:48
• @AlexP 1) By definition is does; the 'equator' is asymptotic but the surface is finite. 2) I clearly said the planet "closely approximates" the shape.
– rek
Commented Jul 30, 2020 at 22:40
• @Matthew OK, so what does that look like? Bear in mind Dymaxion maps can be recut to split landforms to keep bodies of water continuous, so it doesn't matter what the arrangement of continents might be.
– rek
Commented Jul 30, 2020 at 22:43
• I'm not sure, but you might actually try asking on a gamedev site; if you're willing to add splits, you're essentially dealing with a texture skinning problem. Commented Jul 30, 2020 at 23:00

Something similar to the "polar projections" of spheroid planets like Earth would probably work well.

Your planet has a natural and rather extreme barrier at the equator, so separating the map into 2 "polar" maps, which are separated by this equatorial barrier, just makes practical sense.

You could even look at the history of the planet, how they began cartography long before they were able to discover the existence of the opposite end of the planet, and this means the first maps would be a single polar projection. Call this "tradition" and "culture", and say it's too hard and impractical to change.

• Perhaps I'm misunderstanding, but wouldn't this have a great deal of distortion at the poles (the middle of each map section)? The land there would be approaching side-on from the top-down perspective. Having the least distorted 2D surface would let me map the world objectively before layering on the cultural distortions etc.
– rek
Commented Jul 31, 2020 at 3:38
• You could split this into a ring and a more traditional cylindrical projection, and could further add a seam to the ring and 'unroll' it into a rectangle (adding some distortion in the process). Or you could go really overboard and just split the thing into many such unrolled cylinders, which you could then fold, spindle and mutilate into split and stacked sections or even triangles. Commented Jul 31, 2020 at 13:37

After further research I think the best way to minimize or regulate, but not eliminate, distortion would be to take inspiration from how pseudospheres are represented on Poincaré's disk:

Source: https://www.cs.unm.edu/~joel/NonEuclid/pseudosphere.html

Or shown another way to illustrate the equivalent of Tissot's indicatrix:

Source: http://web1.kcn.jp/hp28ah77/us20_pseu.htm

In both examples the circumference of the outer circle is an infinite distance from the middle. As the planet (anti-planet?) is meant to be a physical approximation of these properties, we can ignore that aspect and focus on the shape of the flattened surface in this non-Euclidian space, pointilized in the first figure and light purple in the second.

This approach also allows for considerable flexibility in fitting landforms, as the 2D shape needn't be symmetrical to describe the 3D. Imagine the pink and orange is a continuous landmass:

As a result we get what I'd call a Gingko Leaf Projection.

• I will leave the question open to encourage other approaches.
– rek
Commented Nov 17, 2020 at 17:03