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Geodesics of the Poincare disc

The geodesics are diameters and arcs of circles orthogonal to the unit circle. A radial segment is length minimizing because

\int \frac{2 r ˙ ^{, 2} + r ^{2} θ ˙ ^{, 2}}{1 - r ^{2}} d t \geq 2 artanh r,

with equality for constant angle and monotone radius. Hence the distance from the origin to

z

is

2 artanh ∣ z ∣

.

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