The Poincare metric and distance on the unit disc

On the unit disc $\mathbb{D}$ of The unit disc, the upper half-plane, and Blaschke factors fix the Poincare metric, also called the hyperbolic metric of the disc,

$d{s}_{\mathbb{D}}:=\frac{2\mathrm{\mid }dz\mathrm{\mid }}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}\mathrm{.}$

For a piecewise $C^1$ curve $\gamma :[a,b]\to \mathbb{D}$ its Poincare length is

${\mathrm{\ell }}_{\mathbb{D}}(\gamma ):={\int }_{\gamma }\frac{2\mathrm{\mid }dz\mathrm{\mid }}{1-\mathrm{\mid }z{\mathrm{\mid }}^2}={\int }_a^b\frac{2\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }}{1-\mathrm{\mid }\gamma (t){\mathrm{\mid }}^2}dt,$

and the Poincare distance between $z,w\in \mathbb{D}$ is

$d_{\mathbb{D}}(z,w):=\inf \{{\mathrm{\ell }}_{\mathbb{D}}(\gamma ):\gamma \text{ a piecewise }{C}^1\text{ curve in }\mathbb{D}\text{ from }z\text{ to }w\}\mathrm{.}$

The image of $\gamma$ is a compact subset of $\mathbb{D}$, so $1-\mathrm{\mid }\gamma (t){\mathrm{\mid }}^2$ is bounded away from $0$. On each of the finitely many $C^1$ pieces, $\mathrm{\mid }{\gamma }^{\mathrm{\prime }}(t)\mathrm{\mid }$ is continuous and bounded; hence the displayed integrand is piecewise continuous and integrable, and each ${\mathrm{\ell }}_{\mathbb{D}}(\gamma )$ is a finite real number. The infimum is taken over a nonempty set, because $\mathbb{D}$ is convex and the segment from $z$ to $w$ lies in $\mathbb{D}$. This normalisation — factor $2$, curvature $-1$ — is fixed for every later surface page. The explicit formula $d_{\mathbb{D}}(z,w)=2\mathrm{artanh}\mathrm{\mid }{\phi }_z(w)\mathrm{\mid }$ and the metric axioms are established on this page by the Poincare-distance formula theorem; the present definition records the intrinsic path-metric construction it evaluates.