The Cayley transform maps the upper half-plane conformally onto the unit disc. Its inverse is , with
(1)
Transporting the given hyperbolic metric therefore gives the Poincare disc model
(2)
The Cayley transform is an isometry for this metric, so it carries the half-plane geodesics to disc geodesics. A Möbius transformation sends generalized circles to generalized circles and preserves angles. Hence these are exactly diameters and arcs of circles meeting the unit circle orthogonally. Multiplying this Riemannian metric by any positive constant leaves the unparametrized geodesics unchanged.
Figure 1.
Half-plane geodesics and their Cayley images in the disc
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