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In the rough sketch four concurrent lines are drawn in the Poincaré disk model and in the Euclidean model.

If same angles $ (\alpha,\beta,\gamma,\delta) $ are enclosed at respective points of concurrency in either model then does the same trig definition of Cross Ratio hold good?

enter image description here

If that is so, and if the same three adjacent angles are given then is it correct to say they have the same Cross Ratio in euclidean and hyperbolic geometries?

I need your help, appreciate your comments.

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  • $\begingroup$ I don't understand your picture. Does it omit the boundary of the disc model? Your "hyperbolic lines" appear to be nearly full circles, whereas they should intersect the boundary at right angles. $\endgroup$
    – HJRW
    Dec 6, 2020 at 9:50
  • $\begingroup$ I'm also confused by your question. The definition of the cross-ratio in the disc model is the same as the definition in ordinary complex geometry. What do you mean by an "inversion"? There are no inversions in the isometry group of the disc. On the other hand, if you want to know how the cross-ratio relates to hyperbolic distances, you can look at formula (3) in these notes on a course by Dylan Thurston: math.berkeley.edu/~qchu/Notes/274/Lecture11.pdf .(Hat tip to Matt Stover, from whom I learned this.) $\endgroup$
    – HJRW
    Dec 6, 2020 at 9:53
  • $\begingroup$ Thank you. I edited it somewhat leaving out the inversions. My question does not involve hyperbolic distances but angles only at the concurrent point. Thanks also for the reference. Also, btw, can one, by studying this article start to gain an understanding of Thruston geometrization Conjecture? $\endgroup$
    – Narasimham
    Dec 6, 2020 at 14:08

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Yes. The Poincaré disk model preserves angles everywhere, so we are free to (hyperbolically) translate the point of concurrency to the (Euclidean) center of the disk. Then we can swap out the Poincaré disk model for the Beltrami–Klein model, which preserves projective invariants everywhere including the cross-ratio; the Beltrami–Klein model does not preserve angles everywhere, but it does preserve them at the center.

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A convenient definition of the cross-ratio, in hyperbolic geometry, is as follows.

We work in the upper half plane model; see here, for example. Suppose that $a$, $b$, $c$, and $d$ are points on the boundary of the upper half plane; that is, on the $x$-axis. Then the cross-ratio of these four points (in order) is defined in the usual way; see here, for example.

The relevance for hyperbolic geometry is as follows. The isometry group (a copy of $\textrm{PSL}(2, \mathbb{R})$) acts three-transitively on the the ideal boundary of hyperbolic space. Thus for four-tuples of points of the boundary there is only one invariant - this is the cross ratio.

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  • $\begingroup$ Thank you, apologies for rough sketch. My query is whether the same Cross Ratio definition of given angles in hyperbolic trigonometry and euclidean trigonometries holds as valid. $\endgroup$
    – Narasimham
    Dec 6, 2020 at 15:47

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