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I am searching for formulae to convert Euclidean distance into hyperbolic distance.

The problem that I am confronting is that I want to calculate if a bug travels $x$ units in Euclidean space, how much distance would that be in hyperbolic space. I am aware of the following formula

$$d_h\ =\frac{\sinh ^{-1}\left(\frac{\sqrt{\left| k\right| } d_e}{2}\right)}{\sqrt{\left| k\right| }}$$ where $k$: Gaussian Curvature, $d_e$ : Euclidean distance and $d_h$: hyperbolic distance.

[there is a formula from Poincaré disk

$$d_h\ =2 \tanh ^{-1}(r)$$ where $r$ is distance from the center of the disk.

But it is not very helpful since Euclidean distance inside the disk is limited by the size of the disk]

Are there other formulae, which can convert any Euclidean distance into hyperbolic distance?

Thank you!


Thanks Andrey. I have a thought about comparing Euclidean world with hyperbolic world at least from the point of view of distance covered by the bug. See if this makes sense. Please tell me if I am completely off the mark.

In any unit disk model of hyperbolic world (Poincaré, Klein etc.) we have no comparison of Euclidean world since it is constrained by the size of the disk. How about if we create a unit disk model of Euclidean world [like A Euclidean Model for Euclidean Geometry by Mader ].

In this Euclidean model if we assume the distance from the origin of this disk along the diameter to be $r$ where $r<1$ and Euclidean distance in Euclidean space to be $d_e$. Now $d_e$ and $r$ are related by $d_e=\frac{r}{\sqrt{1-r^2}}$which is $r=\frac{d_e}{\sqrt{1+d_e^2}}$

If we now superimpose this disk on Poincaré disk and again assume $r$ to be the distance from the origin along the diameter in Poincaré disk. We know $d_h=2 \tanh ^{-1}(r)$. If we substitute $r$ with $r=\frac{d_e}{\sqrt{1+d_e^2}}$ we get $$d_h=2 \tanh ^{-1}\left(\frac{d_e}{\sqrt{d_e^2+1}}\right)$$.

Will this be a reasonable way to compare Euclidean distance covered by the bug with hyperbolic distance?

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1 Answer 1

up vote 1 down vote accepted

In Beltrami-Klein model (assuming the curvature is $-1$), for any two points $p=(x_p,y_p)$ and $q=(x_q,y_q)$ within the unit disk the hyperbolic distance $d_h$ between the points satisfies

$$\sinh d_h = \frac{\sqrt{|D_{pq}^2-A_{pq}^2|}}{\sqrt{1-D_p^2}\sqrt{1-D_q^2}},$$

where $D_{pq}=\sqrt{(x_p-x_q)^2+(y_p-y_q)^2}$ is the Euclidean distance between the points, $D_p=\sqrt{x_p^2+y_p^2}$ and $D_q=\sqrt{x_q^2+y_q^2}$ are the Euclidean distances from the origin, and $A=x_py_q-x_qy_p$.

Distances to or between points outside the unit disk are not defined in hyperbolic geometry.

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Thanks Andrey! Please see my edits. –  Sid Sep 20 '13 at 5:07
    
In Beltrami-Klein model, a straight line provides the shortest distance between two points. So, as long as your bug travels along a straight line, you can compare Euclidean and hyperbolic distances directly, provided the bug is within the unit circle. On the other hand, the shortest distance is provided by half-ellipses in Mader's model, and by arc of circles in Poincare model. So, you cannot compare the distances unless you integrate along the path of motion. Think of your formula for distance as a result of the integration along a "straight" line in a given model. –  Andrey Sokolov Sep 20 '13 at 6:15
    
I agree, Andrey. I was thinking of comparing bug’s movements only along the diameter of the Poincare and Mader’s model. Since movements along the diameter provide shortest distance both in the Poincare’s and in the Mader’s model, shouldn’t we be able to compare hyperbolic and Euclidean distances covered by the bug? –  Sid Sep 20 '13 at 15:53
    
Mader's model seems like an arbitrary choice. His model maps the plane $\mathbb E^2$ to the unit disk using a half-sphere. It might be useful if you are particularly interested in ellipses. There are other options though. For instance, you could use $z=x^2+y^2$ instead of a half-sphere. Then, following the same recipe as in Mader's model you will get $r=\frac{2d_e}{1+\sqrt{1+4d_e^2}}$. In any case, your conversion formula will not be particularly meaningful. The Poincare model is convenient for measuring angle between lines, but the Beltrami-Klein model is more natural for distances. –  Andrey Sokolov Sep 21 '13 at 6:01
    
If you allow your bug to move from the origin along a diameter only, you might as well consider 1D spaces. You can parametrise the coordinate $x$, $|x|<1$, by $\phi$ according to $x=\tanh\phi$. Here, $|\phi|$ is the hyperbolic distance from the origin to $x$. So, you have $d_h=|\phi|$, $d_e=|x|$, and the comparison formula is then $d_e=\tanh d_h$, but it only applies if $d_e<1$. –  Andrey Sokolov Sep 21 '13 at 6:02

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