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A random motion on the Poincar\'e half-plane is studied. A particle runs onthe geodesic lines changing direction at Poisson-paced times. The hyperbolicdistance is analyzed, also in the case where returns to the starting point areadmitted. The main results concern the mean hyperbolic distance (and also theconditional mean distance) in all versions of the motion envisaged. Also ananalogous motion on orthogonal circles of the sphere is examined and theevolution of the mean distance from the starting point is investigated.

Travelling Randomly on the Poincar\'e Half-Plane with a Pythagorean Compass