  The coefficients in the confluent hypergeometric equation specify the Regge
trajectories and the degeneracy of the angular momentum states. Bound states
are associated with real angular momenta while resonances are characterized by
complex angular momenta. With a centrifugal potential, the half-plane is
tessellated by crescents. The addition of an electrostatic potential converts
it into a hydrogen atom, and the crescents into triangles which may have
complex conjugate angles; the angle through which a rotation takes place is
accompanied by a stretching. Rather than studying the properties of the wave
functions themselves, we study their symmetry groups. A complex angle indicates
that the group contains loxodromic elements. Since the domain of such groups is
not the disc, hyperbolic plane geometry cannot be used. Rather, the theory of
the isometric circle is adapted since it treats all groups symmetrically. The
pairing of circles and their inverses is likened to pairing particles with
their antiparticles which then go one to produce nested circles, or a
proliferation of particles. A corollary to Laguerre's theorem, which states
that the euclidean angle is represented by a pure imaginary projective
invariant, represents the imaginary angle in the form of a real projective
invariant.
