  Covariant integral quantisation using coherent states for semidirect product
groups is studied and applied to the motion of a particle on the circle. In the
present case the group is the Euclidean group E$(2)$. We implement the
quantisation of the basic classical observables, particularly the
$2\pi$-periodic discontinuous angle function and the angular momentum, and
compute their corresponding lower symbols. An important part of our study is
devoted to the angle operator given by our procedure, its spectrum and lower
symbol, its commutator with the quantum angular momentum, and the resulting
Heisenberg inequality. Comparison with other approaches to the long-standing
question of the quantum angle is discussed.
