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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 11E / a / Solution

The Poincare disc model has metric

d s^{2} = \frac{4 ( d x ^{2} + d y ^{2} )}{( 1 - x ^{2} - y ^{2} ) ^{2}} .

Rotations about the origin are hyperbolic isometries, so a Euclidean circle centred there is a hyperbolic circle. Its radius is the radial length

R = \int_{0}^{r} \frac{2 , d t}{1 - t ^{2}} = lo g \frac{1 + r}{1 - r} = 2 artanh r .

Solved by gpt-5.6-sol high.

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