The hyperbolic upper half-plane

Try

Travel vertically from i to 4i in the upper half-plane H, the complex numbers z=x+iy with y>0. Hyperbolic length is measured by ds=√(dx²+dy²)/y. Along the vertical path it is ∫₁⁴dy/y=log 4, not the Euclidean length three.

The height-dependent scale makes motion near the real boundary expensive. The boundary y=0 is excluded from the point set. It is not a row of ordinary points whose distances happen to be large.

Predict

Predict whether doubling both coordinates of both endpoints changes their hyperbolic distance. The numerator and height denominator both double, so every path length remains the same. Euclidean lengths double instead.

Predict which complete paths replace straight lines. Vertical lines and semicircles orthogonal to the real axis are the hyperbolic geodesics. We will justify their minimizing property rather than infer it from the appearance of the arc.

Experiment

The live experiment uses its own endpoint choices; the fixed worked example remains i to 4i. Change one height, compare the geodesic with the Euclidean straight path measured in the hyperbolic metric, and then scale both endpoints together.

Live experiment: distance depends on height

In the upper half-plane y>0, hyperbolic length is the integral of Euclidean arc length divided by height y. Predict how moving the same horizontal separation upward changes its distance. The boundary y=0 is not part of this space.

First x -2-1012First height 0.512Second x -2-1012Second height 0.512Scale both points 12Reset experiment

Actual points: P=(-1,1), Q=(1,1). Axes have equal, fixed coordinate scales. A curve is drawn from 81 sampled positions; the distance below comes from the exact model formula, not measuring the pixels.

Hyperbolic distance: 1.76275. Euclidean straight-line distance: 2. Hyperbolic length of that straight path: 2.

cosh(d)=1+[(x₂−x₁)²+(y₂−y₁)²]/(2y₁y₂)=3. Applying inverse hyperbolic cosine gives d. The geodesic lies on a circle centered at (0,0) with radius 1.41421. Such circles meet the excluded boundary orthogonally.

The dashed path has hyperbolic length ℓ/y₁ for equal heights, or ℓ log(y₂/y₁)/(y₂−y₁) otherwise, where ℓ is its Euclidean length. The geodesic minimizes hyperbolic length, so its value is no larger. This comparison checks these two paths; the general minimizing property needs the proof in the lesson.

Compare equal heights 0.5 and 2 with fixed x coordinates, then double both coordinates using the scale control. Scaling multiplies ordinary coordinate distances but leaves hyperbolic distance unchanged: both numerator and denominator in cosh(d) scale by the same square. The picture uses Euclidean screen coordinates and does not make a Euclidean straight segment a hyperbolic geodesic. Settings are temporary.

Fixed endpoints	Euclidean distance	Hyperbolic distance
i and 4i	3	log 4
2i and 8i	6	log 4
i and 1+i	1	arcosh(3/2)

The graph illustrates exact formulas. A sampled or rendered curve does not prove that every competing path is longer. Check the distance calculation and the argument below separately.

Understand

For any piecewise smooth path between iy₁ and iy₂, its length is at least ∫|y’|/y, which is at least |∫y’/y|=|log(y₂/y₁)|. The vertical monotone segment attains this bound, proving vertical distance.

Now take T(z)=(az+b)/(cz+d) with real coefficients and ad−bc=1. Direct algebra gives Im T(z)=Im z/|cz+d|² and |T’(z)|=1/|cz+d|². Thus |dT(z)|/Im T(z)=|dz|/Im z: T preserves every path length.

A semicircle with distinct real endpoints can be taken by such a real Möbius map to a vertical line, sending its endpoints to zero and infinity. The minimizing property transfers back. The resulting distance formula is cosh d(z,w)=1+|z−w|²/(2 Im z Im w). It can be derived by this normalization and checked on vertical endpoints, where cosh(log(y₂/y₁))=(y₂/y₁+y₁/y₂)/2. Equal endpoints give d=0.

Apply

Repair “A horizontal segment is shortest because it is Euclidean straight.” For endpoints at equal positive height, the hyperbolic geodesic rises along a circle. The increased height reduces the metric cost of horizontal movement. The comparison must use the same hyperbolic length functional on both candidate paths.

As y approaches zero vertically, distance from i to iy is |log y| and tends to infinity. No finite step reaches the ideal boundary. For optional depth, completeness and full verification of the synthetic neutral axioms require further arguments beyond one distance formula. The model calculations establish concrete metric facts and parallel counterexamples; they do not by themselves prove every axiom scheme.

Continue

Record endpoint coordinates, the metric, the selected geodesic and the exact distance before interpreting a picture. Continue to the Poincaré disk to represent the same geometry in a bounded region while retaining its infinite metric extent.