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You can uniquely specify any point in 2D Euclidian space using 2 numbers: the distance from the infinitely long X-axis, and the distance from the infinitely long Y-axis.

How do you uniquely specify a point in 2D hyperbolic space? Can you do it with just 2 numbers? Can you do it in a "uniform" way? How would that work?

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Do you know the upper half space or disc model of hyperbolic space? In these, you can of course specify a point with 2 coordinates.

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    Caroline Series's notes on the subject are particularly good. http://homepages.warwick.ac.uk/~masbb/Papers/MA448.pdf2012-06-01