the upper half-plane model
The upper half-plane model is the disk's twin sibling: a second, equally faithful drawing of the same hyperbolic plane, but on a different canvas. Instead of an infinite world squeezed into a round disk, here the hyperbolic plane is drawn on the region above a horizontal line — all complex numbers z = x + i y with y > 0. The real axis along the bottom plays the role of the 'horizon at infinity'. For many purposes, especially in number theory, this canvas is far more convenient than the disk.
On the upper half-plane H, the hyperbolic line element is ds = |dz| / y, where y = Im(z) is the height above the real axis. The density 1/y is large near the bottom edge (small y) and small high up, so just as in the disk, distances near the boundary (here the real axis) are stretched to infinity. The geodesics take an especially clean form: they are the vertical half-lines (rays perpendicular to the real axis) and the semicircles centered on the real axis. The model is connected to the Poincare disk by the Cayley transform w = (z - i)/(z + i), which maps the upper half-plane conformally and isometrically onto the unit disk, sending i to the center 0 and the real axis to the boundary circle. So the two models are the same hyperbolic geometry in different coordinates — anything true in one is true in the other.
The upper half-plane model is the natural home of the automorphism group PSL(2,R) acting by real Mobius maps, and through its discrete subgroup the modular group SL(2,Z) it is the stage for modular forms, the modular j-function, elliptic curves, and vast tracts of number theory. Its tidy vertical-and-semicircle geodesics also make hyperbolic trigonometry computations cleaner than in the disk. A caveat worth flagging: the real axis (and the point at infinity above it) is the boundary of H, the horizon — it is NOT part of the hyperbolic plane, just as the unit circle is not part of the disk model. Points on the real axis are 'cusps' or 'points at infinity', infinitely far from any interior point.
The geodesic between i and 2i is the vertical segment of the imaginary axis joining them; its hyperbolic length is the integral of dy/y from 1 to 2, namely log 2. The geodesic between -1 + i and 1 + i, however, is NOT the horizontal segment between them but the semicircle of radius sqrt(2) centered at the origin on the real axis, arcing up and over — because horizontal paths run parallel to the costly boundary.
Geodesics in H are vertical rays and real-axis-centered semicircles; ds = |dz|/y.
The half-plane and disk models are isometric, so they describe the SAME geometry — never claim a hyperbolic fact holds in one but not the other. Just remember the density changes form: 1/y on the half-plane versus 2/(1 - |z|^2) on the disk.