the hyperboloid model
The disk and half-plane models are flat maps that distort the hyperbolic plane to fit it on paper. The hyperboloid model takes the opposite tack: it lifts the hyperbolic plane out of the flat page and realises it as a genuine curved surface sitting in 3D space, where nothing is squashed and the geometry sits there undistorted — at the cost of using an unusual way to measure distance. The surface is one sheet of a two-sheeted hyperboloid, the bowl-shaped graph of x^2 + y^2 - z^2 = -1 with z > 0, hovering above the origin.
The twist is the measuring rule. Instead of ordinary Euclidean distance, you use the Minkowski form, the same algebra Einstein's relativity uses for spacetime: the 'inner product' of vectors (x1, y1, z1) and (x2, y2, z2) is x1 x2 + y1 y2 - z1 z2, with a MINUS on the last term. Under this Lorentzian rule the hyperboloid's points are exactly the hyperbolic plane's points, and a hyperbolic 'straight line' (geodesic) is the curve where the hyperboloid is sliced by a flat plane through the origin — clean, just as great circles are the geodesics on a sphere. The rigid motions of hyperbolic geometry become the Lorentz transformations of special relativity.
Why bother with a third model? Because it is the unifying one. The Klein disk is literally the shadow of the hyperboloid projected from the origin onto the plane z = 1, and the Poincare disk is its projection from the point (0, 0, -1) — so the hyperboloid is the master picture from which both flat models drop out as different shadows. It also makes the kinship between hyperbolic geometry and relativity explicit: the hyperbolic plane is, in a precise sense, the unit sphere of velocity space. The honest note: the strange minus sign is not optional decoration — it IS the source of the negative curvature, and reading it as ordinary Euclidean distance gives nonsense.
The apex point (0, 0, 1) sits on the hyperboloid, since 0 + 0 - 1 = -1. Slice the surface with the vertical plane y = 0 through the origin: the cut is the curve x^2 - z^2 = -1, a hyperbola, and that curve is one hyperbolic straight line. Projecting the whole bowl down from the origin onto z = 1 flattens it into the Klein disk, turning each such geodesic into a straight chord.
The hyperbolic plane as a curved sheet measured with the Minkowski minus sign; the disk models are its projected shadows.
The minus sign in x1 x2 + y1 y2 - z1 z2 is the whole point — it is the Minkowski (Lorentzian) form of relativity, not ordinary Euclidean distance. Read it as Euclidean and the model collapses into meaninglessness.