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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?

MathematicalOrchid
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1 Answers1

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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.

  • OK. But then how would you add or subtract coordinates? Clearly the usual Euclidian rules won't work... – MathematicalOrchid Jun 01 '12 at 13:21
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    @MathematicalOrchid: How would you add and subtract points in the hyperbolic plane to begin with? It's not a vector space. – Hans Lundmark Jun 01 '12 at 13:29
  • You did not ask about adding coordinates in your OP. If you need an analogy to the distance description, then this can be achieved -- eg in the disc model -- using the inteersection of the disc with the Euclidean $x$ and $y$ axis and then use geodesic distance. ($x$ and $y$ axis are geodesics in that model). –  Jun 01 '12 at 13:29
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    (Oh and if you think of adding coordinates as translations -- the group of isometries of hyperbolic space is well known and easy to describe). –  Jun 01 '12 at 13:33
  • Check out any introduction to hyperbolic geometry. Lectures on Hyperbolic Geometry by Benedetti and Petronio, Three Dimensional Geometry and Topology by Thurston are two books that come to mind. – Neal Jun 01 '12 at 15:39
  • Caroline Series's notes on the subject are particularly good. http://homepages.warwick.ac.uk/~masbb/Papers/MA448.pdf – Sean Jun 01 '12 at 18:31