D = {z = x + i y : ∣ z ∣ < 1}

equipped with the Riemannian metric

d s^{2} = \frac{4 ( d x ^{2} + d y ^{2} )}{( 1 - x ^{2} - y ^{2} ) ^{2}} = \frac{4∣ d z ∣ ^{2}}{( 1 - ∣ z ∣ ^{2} ) ^{2}} .

This normalization gives Gaussian curvature

- 1

. In particular, a Euclidean curve has hyperbolic length

\int 2∣ d z ∣/ (1 - ∣ z ∣^{2})

; the boundary circle is infinitely far away in hyperbolic distance.

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