the Klein model
/ kline /
The Klein model (also called the Beltrami-Klein model, after Eugenio Beltrami who introduced it and Felix Klein who studied it) is yet another faithful map of the hyperbolic plane inside an open disk — but with one delightful simplification over the Poincare disk. Its points are again all the points strictly inside the boundary circle. The payoff: a hyperbolic 'straight line' really IS a straight Euclidean line here — specifically, an open chord of the disk (a straight segment joining two boundary points, endpoints excluded). No arcs to draw, no perpendicularity conditions to check; straight is straight.
That makes parallelism beautifully visual. Through a point P inside the disk, draw the chord L it does not meet; every other chord through P that stays inside without crossing L is a parallel, and you can see at a glance the whole fan of them between the two chords that share an endpoint with L on the rim — those two are the limiting parallels, kissing L only on the boundary. So the infinitely-many-parallels phenomenon is laid bare with a ruler alone. This straightness is exactly why the model is so handy for questions about incidence, betweenness, and convexity in hyperbolic geometry.
Every model pays a price, and here it is angles. The Klein model is NOT conformal: the angle between two chords as measured with a protractor is generally NOT the true hyperbolic angle. A hyperbolic right angle can look acute or obtuse on the page. This is the mirror-image trade-off to the Poincare disk: Poincare keeps angles and bends lines; Klein keeps lines straight and distorts angles. Choose the chart to fit the job — there is no single picture that gets everything right at once, which is the standing honesty about models of a curved geometry.
In a Klein disk, a hyperbolic line through the centre and a hyperbolic line off-centre are both just straight chords. Take a chord L and a point P beside it: the two chords through P that hit the same boundary points as the ends of L are its limiting parallels, and any chord through P landing strictly between those endpoints is an ultraparallel — all visible with a straightedge, though their angles read 'wrong'.
Hyperbolic lines are straight chords here; the model keeps straightness but, unlike the Poincare disk, does not preserve angles.
Do not measure angles on the Klein model with a protractor and trust them — it is not conformal. Use the Poincare disk when angles matter and the Klein model when straightness and incidence matter; both depict the identical geometry.