Length

Reflection in the d-line is inversion and if we drop a perpendicular d-line from M to the d-line that acts as a mirror and cutting it at the point P, the image point is seen to be on this line as well. For the distances MP and PImage to be the same, when measured in hyperbolic terms, the length from the point z1 to the point z2 (points corresponding to complex numbers to make the algebra easier) is given by
nonE(z1, z2) =
2 arctanh( | (z2-z1)/(1-bar(z1)z2|)

If we measure from the point O, the centre O of Euclidean plane coordinate system, to the point z,
then hyperbolic length is given by the somewhat simpler formula
nonE(O,z1) = 2 arctanh(|z|)

Inversion Length continued

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