I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows:(adsbygoogle = window.adsbygoogle || []).push({});

... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2:

[tex]ds^2 = \frac{4}{(1-r^2)^2} dx^2[/tex]

I don't understand how what ds^2 means or how to use this formula to compute distances and arc lengths. A naive guess is that the arc length should be given by

[tex]s = \int_a^b \sqrt{ds^2}[/tex]

but that doesn't seem to give me the correct answer. For example, take a point with Euclidean distance r from the origin. What is its distance in the hyperbolic metric?

I know that the distance should be the arc length of the straight line connecting 0 to x, since the straight line through the origin is a geodesic in the hyperbolic metric. My guess would give me an arc length of

[tex]s = \int_0^r \frac{2}{1-x^2} dx = 2 arctanh(r)[/tex]

However, another website claims that the answer should be log(1+r)/log(1-r).

Can someone help?

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# Computing arc length in Poincare disk model of hyperbolic space

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