  Relying on rays, we search for submodules of a module V over a supertropical
semiring on which a given anisotropic quadratic form is quasilinear. Rays are
classes of a certain equivalence relation on V, that carry a notion of
convexity, which is consistent with quasilinearity. A criterion for
quasilinearity is specified by a Cauchy-Schwartz ratio which paves the way to a
convex geometry on Ray(V), supported by a "supertropical trigonometry".
Employing a (partial) quasiordering on Ray(V), this approach allows for
producing convex quasilinear sets of rays, as well as paths, containing a given
quasilinear set in a systematic way. Minimal paths are endowed with a
surprisingly rich combinatorial structure, delivered to the graph determined by
pairs of quasilinear rays -- apparently a fundamental object in the theory of
supertropical quadratic forms.
