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July 28

Hyperbolic reflection

Hyperbolic reflection is defined by projecting the hyperbolic plane into a Poincaré disk and inverting the point about the circle which formed the hyperbolic line. How can I reflect a point across a hyperbolic line while keeping all calculations in hyperbolic coordinates, without projecting the hyperbolic plane into the Euclidean plane? --138.110.25.31 (talk) 19:47, 28 July 2010 (UTC)[reply]

Probably the most transparent way of seeing this is in the upperhalf plane model, where the hyperbolic reflection about the y-axis is simply the map (x,y) -> (-x,y). Since every hyperbolic line can be moved by an isometry to the y-axis, this is in fact the general picture. Tkuvho (talk) 20:01, 28 July 2010 (UTC)[reply]

I have to be in the Poincaré disk, though, so that doesn't help at all. --138.110.25.31 (talk) 20:05, 28 July 2010 (UTC)[reply]

I don't understand what is meant by "hyperbolic coordinates" so presumably you can't assert the hyperbolic line in the Poincaré disc is a diameter by some isometry and go from there? 95.150.19.83 (talk) 22:52, 28 July 2010 (UTC)[reply]