  The biadjoint scalar theory has cubic interactions and fields transforming in
the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde
N}\big)$. Amplitudes are "color" decomposed in terms of partial amplitudes
computed using Feynman diagrams which are simultaneously planar with respect to
two orderings. In 2019, a generalization of biadjoint scalar amplitudes based
on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of
Feynman diagrams derived by incorporating an additional constraint of "local
planarity" into the construction of the arrangements of metric trees in
combinatorics. In this work, we propose a natural generalization of color
orderings which leads to color-dressed amplitudes. A generalized color ordering
(GCO) is defined as a collection of standard color orderings that is induced,
in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$.
We present results for $n\leq 9$ generalized color orderings and GFDs,
uncovering new phenomena in each case. We discover generalized decoupling
identities and propose a definition of the "colorless" generalized scalar
amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$,
discuss some of their properties, and comment on their GFDs. In a companion
paper, we explore the definition of partial amplitudes using CEGM integral
formulas.
