  In this note we make a field-theoretical derivation of a series of new
recursion relations by a one-parameter deformation of kinematic variables for
tree and one-loop amplitudes of bi-adjoint $\phi^3$ theory. Tree amplitudes are
given by canonical forms/functions of associahedra realized in kinematic space
by Arkani-Hamed, Bai, He and Yan (ABHY); the construction has been extended to
generalized associahedra, where type $\mathcal{B}$/$\mathcal{C}$ polytopes
compute tadpole diagrams and type $\mathcal{D}$ polytopes compute one-loop
planar $\phi^3$ amplitudes. The new recursions are natural generalizations of
the formula we found in 1810.08508, and are shown to work for all
"C-independent" ABHY polytopes. Geometrically, the formula indicates
triangulation of the generalized associahedron by projecting the whole polytope
onto its boundary determined by the deformation. When projecting onto one
facet, our recursion gives, e.g. "soft-limit triangulation" and "forward-limit
triangulation" for tree and one-loop level. But we also find a lot of new
formulae from our recursion relation, by projecting onto lower dimensional
facets.
