  This article is an introductory presentation of the quantization of the
half-plane based on affine coherent states (ACS). The half-plane is viewed as
the phase space for the dynamics of a positive physical quantity evolving with
time, and its affine symmetry is preserved due to the covariance of this type
of quantization. We promote the interest of such a procedure for transforming a
classical model into a quantum one, since the singularity at the origin is
systematically removed, and the arbitrariness of boundary conditions can be
easily overcome. We explain some important mathematical aspects of the method.
Three elementary examples of applications are presented, the quantum breathing
of a massive sphere, the quantum smooth bouncing of a charged sphere, and a
smooth bouncing "dust" sphere as a simple model of quantum Newtonian cosmology.
