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The Metric for the Poincaré Hyperbolic Disk, a model for Hyperbolic Geometry. The hyperbolic metric is invariant under conformal maps of the disk onto itself.
See also Hyperbolic Geometry, Poincaré Hyperbolic Disk
References
Bear, H. S. ``Part Metric and Hyperbolic Metric.'' Amer. Math. Monthly 98, 109-123, 1991.