  We show that all geodesic rays in the uniform infinite half-planar
quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering
thereby a recent question of Curien. However, the possible intersection points
are sparsely distributed along the boundary. As an intermediate step, we show
that geodesic rays in the UIHPQ are proper, a fact that was recently
established by Caraceni and Curien (2015) by a reasoning different from ours.
Finally, we argue that geodesic rays in the uniform infinite half-planar
triangulation behave in a very similar manner, even in a strong quantitative
sense.
