  Recently, the existence of an Amplituhedron for tree level amplitudes in the
bi-adjoint scalar field theory has been proved by Arkhani-Hamed et al. We argue
that hyperbolic geometry constitutes a natural framework to address the study
of positive geometries in moduli spaces of Riemann surfaces, and thus to try to
extend this achievement beyond tree level. In this paper we begin an
exploration of these ideas starting from the simplest example of hyperbolic
geometry, the hyperbolic plane. The hyperboloid model naturally guides us to
re-discover the moduli space Associahedron, and a new version of its
kinematical avatar. As a by-product we obtain a solution to the scattering
equations which can be interpreted as a special case of the two well known
solutions in terms of spinor-helicity formalism. The construction is done in
$1+2$ dimensions and this makes harder to understand how to extract the
amplitude from the dlog of the space time Associahedron. Nevertheless, we
continue the investigation accommodating a loop momentum in the picture. By
doing this we are led to another polytope called Halohedron, which was already
known to mathematicians. We argue that the Halohedron fulfils many criteria
that make it plausible to be understood as a 1-loop Amplituhedron for the cubic
theory. Furthermore, the hyperboloid model again allows to understand that a
kinematical version of the Halohedron exists and is related to the one living
in moduli space by a simple generalisation of the tree level map.
