  Our previous exploration of the $\cE_g^{PD}$-geometry has shown that the
field is promising. Namely, the $\cE_g^{PD}$-approach is amenable to
development of novel trends in relativistic and metric differential geometry
and can particularly be effective in context of the Finslerian or Minkowskian
Geometries. The main point of the present paper is the tenet that the
$\cE_g^{PD}$-space-associated one-vector Finslerian metric function admits in
quite a natural way an attractive two-vector extension, thereby giving rise to
angle and scalar product. The underlying idea is to derive the angular measure
from the solutions to the geodesic equation, which prove to be obtainable in an
explicit simple form. The respective investigation is presented in Part I. Part
II serves as an extended Addendum enclosing the material which is primary for
the $\cE_g^{PD}$-space. The Finsleroid, instead of the unit sphere, is taken
now as carrier proper of the spherical image. The indicatrix is, of course, our
primary tool.
