the upper half-plane model
The upper half-plane model is a second portrait of the very same hyperbolic plane, drawn not in a disk but on the whole region of the ordinary plane above a horizontal line. The points are all (x, y) with y strictly greater than 0 — everything above the x-axis. The x-axis itself, together with a single 'point at infinity' high above, plays the role of the boundary at infinity and is not part of the plane. As in the disk, the ruler warps: heights are measured by ds = (Euclidean length)/y, so the same Euclidean step counts for more and more hyperbolic distance the closer you get to the x-axis.
Its dictionary is famously clean. A hyperbolic 'line' is either a vertical ray shooting straight up from the x-axis, or a semicircle whose centre sits ON the x-axis (so it meets the axis at right angles). Like the disk, this model is CONFORMAL — it gets every angle exactly right — but it is often preferred because its formulas are gorgeous. The rigid motions of the hyperbolic plane become the Mobius transformations z -> (az + b)/(cz + d) with real coefficients and ad - bc > 0, acting on the complex coordinate z = x + iy. That marriage with complex analysis is why number theorists and physicists reach for this model first.
Concretely, to find the hyperbolic distance between two points you find the unique vertical line or x-axis-centred semicircle through both, and integrate ds = (path length)/y along it. Two vertical lines are limiting-parallel (they 'meet' only at the shared point at infinity); a vertical line and a semicircle ending at the same x-axis point are likewise limiting-parallel. It is the exact same geometry as the disk — there is an explicit Mobius map carrying one model onto the other — just a different, often more convenient, chart.
The hyperbolic distance between (0, 1) and (0, e) on the same vertical line is the integral of dy/y from 1 to e, which equals ln(e) - ln(1) = 1. Notice the points (0, 1) and (0, 100) are also distance ln(100), about 4.6, apart — and as a point slides down toward the x-axis its hyperbolic distance from any fixed point grows without bound, because the axis is at infinity.
Lines are vertical rays or semicircles centred on the x-axis; distances use ds = (length)/y, and the x-axis is infinity.
This is the same hyperbolic plane as the Poincare disk, not a different geometry — an explicit Mobius transformation maps one to the other. The x-axis is the unreachable boundary at infinity, never a place you can stand.