  Building on the prior work in [1] we locate a family of positive geometries
in the kinematic space which are a specific class of convex realisations of the
associahedron. These realisations are obtained by scaling and translating the
kinematic space associahedron discovered by by Arkani-Hamed, Bai, He and Yan
(ABHY). We call the resulting polytopes, deformed realisations of the
associahedron. The deformed realisations shed new light on the CHY formula. One
of the striking discoveries in [2] was the fact that the CHY scattering
equations generate diffeomorphism between the (compactified) CHY moduli space
and the ABHY associahedron. As we argue, the deformed realisation of the
associahedron can also be interpreted as an diffeomorphic image of the CHY
moduli space under scattering equations that we call deformed scattering
equations. The canonical form in the kinematic space is thus once again the
push-forward of the Parke-Taylor form . A natural off-shoot of our analysis is
the universality of the Parke-Taylor form as a CHY Integrand for a class of
(tree-level and planar) multi-scalar field amplitudes. These ideas help us in
proving the existence of positive geometries for certain specific multi-scalar
interactions. We prove that in a field theory with a massless and a massive
bi-adjoint scalar fields which interact via cubic interaction, the tree-level
S-matrix with massless external states and at most one massive propagator is a
weighted sum over the canonical forms defined by certain deformed realisations
of the associahedron. Finally, we show that these ideas admit an extension to
one-loop. In particular, the one loop S-matrix integrand with at most one
massive propagator is a weighted sum over canonical forms of a family of
deformed realisations of the type-D cluster polytope, discovered in [3,4].
