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I'm wondering how to trisect a line/segment in $\mathbb{H}^2$ (using the Poincaré Disk model). Bisection of a hyperbolic line seems rather straightforward (e.g. as described in the paper Compass and Straightedge in the Poincaré Disk by Goodman-Strauss) ( link: http://comp.uark.edu/~strauss/papers/hypcomp.pdf ) as a construction similar to the Euclidean one can be used. However, the approach for trisection in the Euclidean plane I'm familiar with uses parallel lines and can therefore probably not be extended to a hyperbolic setting. Any ideas?

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It is not generally possible to trisect a line segment in the hyperbolic plane.

There is a description of lengths in the hyperbolic plane that can be constructed with compass and straightedge. It begins with the field $E,$ which is the lengths (and their negatives) that are constructible in the ordinary plane. As you probably know, the main rule is that if $x \in E$ and $x > 0,$ we then have $\sqrt x \in E.$

The lengths constructible in the hyperbolic plane are those $x$ for which $e^x \in E,$ or $\sinh x \in E,$ or $\cosh x \in E,$ or $\tanh x \in E,$ these conditions being equivalent.

Probably easiest to see with the exponential function. We can construct the length $x$ that has $e^x = 2.$ About $x/3,$ we then have $e^{x/3} = \sqrt[3] 2,$ which is not in $E.$

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