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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.

Diagram of a pseudosphere, resembling a toy top

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.

Sinusoidal projection with Tissot's indicatrix visible

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

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    $\begingroup$ 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. $\endgroup$ – AlexP Jul 30 '20 at 21:34
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    $\begingroup$ 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. $\endgroup$ – Matthew Jul 30 '20 at 21:48
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    $\begingroup$ @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. $\endgroup$ – rek Jul 30 '20 at 22:40
  • $\begingroup$ @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. $\endgroup$ – rek Jul 30 '20 at 22:43
  • $\begingroup$ 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. $\endgroup$ – Matthew Jul 30 '20 at 23:00
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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.

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  • $\begingroup$ 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. $\endgroup$ – rek Jul 31 '20 at 3:38
  • $\begingroup$ 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. $\endgroup$ – Matthew Jul 31 '20 at 13:37
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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:

A Sector of the Pseudosphere, cut to the "Infinitely Distant Boundary" – Rucker

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

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

Concentric red circles and white circles at regular intervals

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:

A pink and orange circle on an asymmetrical flat rendering forms a 'saddle' when rolled in three dimensions

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

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  • $\begingroup$ I will leave the question open to encourage other approaches. $\endgroup$ – rek Nov 17 '20 at 17:03
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Normalize the Radius

The way i see it, the best way is to declare a Radius r and stretch every point of the Surface to that Radius. This gives you the very hard to map surface of a Cylinder. The truth is that for the most part, only the Equator region will be massivily out of Proportion as most of the Plant has a "similar" Radius already. And there just is not that much Curviture from y 100 to y 20000.

You can go all fancy with those maps too and maybe add a Graph or i guess the Crossection of the Planet on the side of that map to show how much everything is distorted.

But yeah, dont overthink it. Nobody on your world would. A Normalized Radius is easy to understand and it is also easy to get why stuff is distorted.

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  • $\begingroup$ It sounds like you're saying equirectangular, but that can't be the least distorted projection (as asked for) for an antisphere because it isn't for a regular sphere. $\endgroup$ – rek Nov 17 '20 at 20:47

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