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There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane:

enter image description here

where geodesics are represented by straight lines. The following image, on the other hand, depicts the Poincare model of the same hyperbolic plane:

enter image description here

where geodesics are represented by segments of circles intersecting the boundary of the disk orthogonally. Both of these models capture the entire $n$-dimensional hyperbolic space in a disk (or more generally, a Euclidean $n$-ball).

I was thinking about what hyperbolic space would "really look like" from the perspective of an observer within the space. What came to mind was the exponential map, which maps an element of the tangent space $\mathrm{T}_PM$ of a point $p$ on a manifold $M$ to another point on the manifold:

enter image description here

Intuitively, the exponential map follows the geodesic over the manifold that "departs" from the specified direction belonging to the tangent space. For example, the exponential map of the Earth as viewed from the north pole is the polar azimuthal equidistant projection in cartography:

enter image description here

This seems to be what elliptic space would "really look like" from the perspective of an observer within the space, since light reaches our eyes by traveling along geodesics.

This page and this page have demos. You may click around to get a feeling of what moving through an elliptic space would look like, though of course geodesics can go beyond the boundary of the disk in the demo by repeatedly wrapping around the sphere of the Earth.

Given this background, my question is as follows:

What does the exponential map from some point on a hyperbolic space look like, assuming that the contents of the space are depicted by the models above? Are there any demos or examples?

Edit: This question seems to be related.

Edit 2: The last section of this video shows the azimuthal equidistant projection of a hyperbolic plane to be

enter image description here

Edit 3: See also Riemann normal coordinates.

Last edit: Someone made a virtual reality demo of hyperbolic space. This is a definitive answer!

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    $\begingroup$ suggest reading original Bolyai in translation; it's in Bonola's book. Martin's book follows up pretty well, intrinsic viewpoint. $\endgroup$ – Will Jagy Aug 24 '15 at 3:55
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    $\begingroup$ I wouldn't want to live in hyperbolic space. As you walk, the stuff you see in the horizon moves quickly to get behind you as new hyperbolic lands unfold in front of you. Wait no. You wouldn't be able to see very far away (a candle gets dimmer exponentially as you walk away from it) and if you get drunk for 5 minutes you could be lost forever... $\endgroup$ – mercio Aug 24 '15 at 18:08
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    $\begingroup$ Whoah, I just posted the exact same tiling on 9gag last night (odd place to post it, I know). Were you inspired by that post? $\endgroup$ – Akiva Weinberger Aug 24 '15 at 18:10
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    $\begingroup$ I think you're looking for #10 of this YouTube video. (I recommend watching the whole thing.) $\endgroup$ – Akiva Weinberger Aug 24 '15 at 18:36
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    $\begingroup$ There is a math movie called "Not knot", made in 1990s, which describes ``what is it like to live in a hyperbolic manifold." Maybe you can find the movie online. $\endgroup$ – Moishe Kohan Aug 25 '15 at 20:59
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One common way to visualize the "intrinsic appearance" of a simply-connected universe of constant curvature $\pm 1$ is to give the angular size of an object modeled as a geodesic arc of (sufficiently small) length $\ell$ placed at distance $d$ from one's eye. (I didn't try to run the linked applets, and am not sure if either implements this strategy.)

On a sphere of unit curvature, a circle of geodesic radius $d$ has circumference $2\pi \sin d$; an object of length $\ell$ (placed "orthogonal to the line of sight") therefore subtends an angle $\theta \approx \ell/\sin d$.

The angle subtended by an object on a sphere

Playing the same game in the hyperbolic plane, a circle of geodesic radius $d$ has circumference $2\pi \sinh d$; an object of length $\ell$ (placed orthogonal to the line of sight) therefore subtends an angle $\theta \approx \ell/\sinh d$. As in mercio's comment, this angle decreases exponentially with $d$.

If the characteristic length is one meter (i.e., a circle of radius $d$ meters has circumference $C = 2\pi \sinh d$ meters), then an object at hyperbolic distance $d$ meters appears (to our Euclidean intuition) to lie at distance $d' = \sinh d$ meters: $$ \begin{array}{l|ccccccc} d = & 1 & 2 & 3 & 4 & 5 & 10 & 100 \\ \hline % C \approx & 7.384 & 22.79 & 62.944 & 171.468 & 466.233 & 69198.183 & 8.445 \times 10^{43} \\ d' \approx & 1.175 & 3.627 & 10.018 & 27.29 & 74.203 & 11013.233 & 1.344 \times 10^{43} \\ \end{array} $$ Particularly, an object ten meters away in hyperbolic space appears to be over eleven kilometers distant, and an object one hundred meters away subtends an angle too small to be cosmologically meaningful (a formal distance of about $1.4 \times 10^{27}$ light-years).

Analogous conclusions hold in a three-dimensional sphere or three-dimensional hyperbolic space. The main qualitative point is, it's easy to hide (or to become irretrievably lost) in hyperbolic space.

Jeffrey Weeks' geometry software seems likely to be of interest. His book The Shape of Space (q.v.) is an excellent read.

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The Beltrami-Klein model is an accurate depiction of what it would look like in hyperbolic space. To be a bit more precise, if you live in 3-dimensional hyperbolic space $\mathbb{H}^3$, and if $P \subset \mathbb{H}^3$ is a 2-dimensional hyperbolic plane tiled in red and white triangles with angles $\pi/2,\pi/3,\pi/7$ as in the picture shown in the question, and if your eye is situated at a point $Q$ a certain distance from $P$, then what you would see is exactly that picture.

The intuitive reason for this is that geodesics are straight lines, and that is how they appear to your eye.

In a bit more detail, one can prove analytically that that if you take the straight line projection of a geodesic in $X$ onto the unit tangent sphere $T^1_Q (\mathbb{H}^3)$ of $\mathbb{H}^3$ at the point $Q$, then the result is a great circle segment in $T^1_Q (\mathbb{H}^3)$.

One interesting feature of this fact is that from $Q$ one can "see" the circle at infinity of $P$, just as the picture shows.

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  • $\begingroup$ Are you sure this is correct? I think the direction of points in the hyperbolic plane from the origin is the same as that in the Klein model, but I presume the distance is not, since points on the edge of the disk really are infinitely far away (whereas the exponential map geodesic covers a finite distance). $\endgroup$ – user76284 Aug 30 '15 at 17:24
  • $\begingroup$ Yes, it is really true. The question of what hyperbolic space "looks like" is equivalent to the question of how things project to the unit tangent bundle at the obseration point. So for instance if you take any geodesic line $L$ (e.g. one of the lines in your picture of the 2,3,7 tiling), and then form the unique plane $\overline{QL}$ passing through both $Q$ and $L$, then that plane intersects the unit tangent bundle at $Q$ in a great circle, and the rays in that plane that pass through actual points of $L$ form an arc of that great circle. $\endgroup$ – Lee Mosher Aug 31 '15 at 3:21
  • $\begingroup$ "The question of what hyperbolic space "looks like" is equivalent to the question of how things project to the unit tangent bundle at the obseration point." I think it's not just the unit tangent bundle, but the entire tangent bundle, where the length of the tangent vector denotes the distance traversed over the geodesic heading in that direction. I believe what you say is true if you only want the direction, though. $\endgroup$ – user76284 Aug 31 '15 at 3:47
  • $\begingroup$ Furthermore -- consider an equidistant surface in $\mathbb{H}^3$, which is of course also internally hyperbolic. If this equidistant surface is $d$ units below a plane in $\mathbb{H}^3$, and our eye is $d$ units above this plane, we see it in the Poincaré model. (This is the 3D model used in HyperRogue) $\endgroup$ – Zeno Rogue Jan 5 '18 at 19:46

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