Unit Disc and Hyperbolic length

Lengths from the centre O of our coordinate system imposed on the disc to some point P run from 0 to 1 in Euclidean space and from 0 to infinity in hyperbolic terms.

The hyperbolic tan function tanh gives us means of switching between measurements of d-line segments from O to P (which look like straight line segments to us as observers) as follows;

If we denote the Euclidean distance as E(OP) and the hyperbolic distance as nonE(OP) (non-Euclidean was the name Gauss gave to hyperbolic space) then

nonE(OP) = 2 arctanh(E(OP))

E(OP) = tanh(nonE(OP)/2)

Length Hyperbolic circles

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