  We study properties of "hyperbolic directions" in groups acting cocompactly
on properly convex domains in real projective space, from three different
perspectives simultaneously: the (coarse) metric geometry of the Hilbert
metric, the projective geometry of the boundary of the domain, and the singular
value gaps of projective automorphisms. We describe the relationship between
different definitions of "Morse" and "regular" quasi-geodesics arising in these
three different contexts. This generalizes several results of Benoist and
Guichard to the non-Gromov hyperbolic setting.
