In the unit disk D, the radial hyperbolic distance from zero to r=1/2 is log 3. From zero to r=3/4 it is log 7. The second point is only one quarter farther in Euclidean radius, but the added hyperbolic distance is substantial.
The Poincaré disk metric is ds=2|dw|/(1−|w|²). Points satisfy |w|<1 strictly. The boundary circle is not part of the geometry. Bounded coordinates do not imply bounded metric diameter.
Predict
Predict whether Euclidean circles near the boundary have the same geometric size as equal-looking circles near the center. The scale factor grows near the boundary, so screen size alone is misleading.
Predict whether the disk picture preserves angles. Its metric is a positive scalar multiple of the Euclidean metric at each interior point, so it preserves local unsigned angles between tangent directions. It does not preserve distances or the Euclidean straightness of geodesics.
Experiment
Compare radial positions with their exact metric values.
Disk radius r
exp of distance from zero
Distance
0
1
0
1/2
3
log 3
3/4
7
log 7
Integrate 2/(1−r²) by partial fractions to reproduce the last column. Then solve the inverse problem r=(eᵈ−1)/(eᵈ+1) for a chosen distance d. Equal increments in d produce shrinking visible gaps as the boundary approaches.
Understand
The Cayley map C(z)=(z−i)/(z+i) takes H to D. Indeed |z+i|²−|z−i|²=4 Im z>0, so |C(z)|<1. Solving for z gives z=i(1+w)/(1−w), whose imaginary part is (1−|w|²)/|1−w|²>0. Thus the map is a bijection.
Substitute this inverse and its derivative into |dz|/Im z to obtain 2|dw|/(1−|w|²). This proves the two models are isometric when each uses its specified metric. Vertical lines and real-axis-orthogonal semicircles become disk diameters and circle arcs orthogonal to the boundary.
For radial distance from zero, every path has Euclidean speed at least the absolute rate of change of its radius. Consequently hyperbolic length is at least the net change of ∫2dr/(1−r²). A monotone radial segment attains the bound, giving d(0,r)=log((1+r)/(1−r)).
Apply
Angle preservation follows from a local calculation. If the metric inner product is λ² times the Euclidean dot product, then both the angle numerator and the norm-product denominator acquire λ². The normalized cosine is unchanged. This requires nonzero tangent vectors at the same point; it says nothing about finite Euclidean distances.
Repair “The boundary circle is another complete geodesic.” Its points are absent from D and radial distance diverges as r approaches one. Complete interior geodesics have ideal endpoints on that circle, but do not contain those endpoints.
For optional depth, disk automorphisms can move any interior point to zero, making the radial formula a route to the general distance formula. The coordinate formulas must carry the disk metric along; otherwise they are merely Euclidean transformations of a drawing.
Continue
Distinguish conformality, which concerns angles, from isometry, which concerns the chosen metric. Continue to Klein and hyperboloid models to trade angle preservation for straight-looking geodesics and useful linear algebra.
Apply · explain · revise
Practice and reasoning
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For disk points 0 and r=1/3, compute exp of their hyperbolic distance.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
What does the Poincaré disk preserve visually?
Show C(z)=(z−i)/(z+i) maps the upper half-plane bijectively to the unit disk and find its inverse.
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Integrate 2dr/(1−r²) from 0 to r, 0≤r<1, to derive radial distance.
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Why does a metric of the form λ(x,y)²(dx²+dy²), λ>0, preserve the Euclidean angle between tangent vectors?
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Explain which Euclidean circles or lines represent complete disk geodesics and why the boundary circle itself is excluded.
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Reflect and continue
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