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:

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

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:

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:

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

**Edit 3:** See also Riemann normal coordinates.

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