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I'm interested in illustrating a hyperbolic plane for a report I'm writing. Here's the metric that I'm using:

$$ds^2=dr^2+\sqrt{\lvert k\rvert}^{-1}\sinh\left(r\sqrt{\lvert k\rvert}\right)\left( d\theta^2+\sin^2\theta\space d\phi^2 \right)$$

And I'd like it to look like this:

surface

Marco's answer below is a good start:

Plot3D[Sinh[x] Sinh[y], {x, y} \[Element] Disk[], BoxRatios -> {1, 1, 1}, Boxed -> False, Axes -> False]

I'd like to do basically this, but using radial coordinates (like in the picture).

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    $\begingroup$ ... and what have you tried? It would seem to me that a Plot3D with an appropriate RegionFunction (a disk?), or using the newer Element[{x, y}, region] to specify the support of your lot could help. $\endgroup$
    – MarcoB
    May 21 '20 at 4:44
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    $\begingroup$ I was thinking of something like Plot3D[Sinh[x] Sinh[y], {x, y} \[Element] Disk[], BoxRatios -> {1, 1, 1}, Boxed -> False, Axes -> False] plus appropriate formatting modifications. $\endgroup$
    – MarcoB
    May 21 '20 at 4:52
  • $\begingroup$ @MarcoB He may mean to use differential geometry to solve this problem. For example, using FrenetSerretSystem function. $\endgroup$ May 21 '20 at 7:39
  • $\begingroup$ @MarcoB - That looks like a good start. Is it possible to do the same thing with radial coordinates (like in the plot)? Write it up as an answer and I'll up-vote it. $\endgroup$
    – Quarkly
    May 23 '20 at 13:30
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    $\begingroup$ @Quarkly I'd be happy to if the question gets reopened. In any case, the special sauce to get the polar-style gridlines would be MeshFunctions -> {Norm[{#1, #2}] &, ArcTan[#1, #2] &}. Give me a ping if the question gets reopened and I'll write a proper answer. $\endgroup$
    – MarcoB
    May 23 '20 at 21:25
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Here is the approach I teased in comments:

Plot3D[
  Sinh[x] Sinh[y], {x, y} ∈ Disk[],
  MeshFunctions -> {Norm[{#1, #2}] &, ArcTan[#1, #2] &}, 
  Mesh -> {15, 40},
  MeshStyle -> Black,
  PlotPoints -> 75,
  PlotTheme -> {"Classic"},
  Background -> Black,
  BoxRatios -> {1, 1, 1}, Boxed -> False, Axes -> False,
  ViewPoint -> {3, -1, 2}, ViewVertical -> {-0.5, 0.2, 1}
]

3D plot with black background no axes no frames

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  • $\begingroup$ Exactly what I needed. $\endgroup$
    – Quarkly
    May 24 '20 at 20:48

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