Solution
= Solution
The <Poincare disc model> is the open <unit disc> $D=\{z=x+iy:|z|<1\}$ equipped with the <Riemannian metric>
$$
\boxed{ds^2=\frac{4(dx^2+dy^2)}{(1-x^2-y^2)^2}=\frac{4|dz|^2}{(1-|z|^2)^2}.}
$$
This normalization gives <Gaussian curvature> $-1$. In particular, a Euclidean curve has <hyperbolic length> $\int 2|dz|/(1-|z|^2)$; the boundary circle is infinitely far away in <hyperbolic distance>.