Poincaré disk model

The Poincaré disk model is the open unit disk $\mathbb D=\{z=x+iy\in\mathbb C:|z|<1\}$ equipped with the Riemannian metric

ds^2=\frac{4(dx^2+dy^2)}{(1-|z|^2)^2}.

Properties

This metric has constant Gaussian curvature $-1$ , so $\mathbb D$ is a model of the hyperbolic plane.

Geodesics and boundary

The hyperbolic geodesics are the Euclidean circles and straight lines that meet the unit generalized circle orthogonally. The ideal boundary is the unit circle $S^1=\partial\mathbb D$ .

Relation to the half-plane model

A Möbius transformation from the upper half-plane to $\mathbb D$ is an isometry between the corresponding models of the hyperbolic plane.

References

Benson Farb and Dan Margalit, A Primer on Mapping Class Groups, Princeton University Press, 2011, Chapter 1, §1.1.2, pp. 19–20. Chapter excerpt.