  We present a generalized Poincar\'e sphere (G sphere) and generalized Stokes
parameters (G parameters), as a geometric representation, which unifies the
descriptors of a variety of vector fields. Unlike the standard Poincar\'e
sphere, the radial dimension in the G sphere is not used to describe the
partially polarized field. The G sphere is constructed by extending the basic
Jones vector basis to the general vector basis with the continuously changeable
ellipticity (spin angular momentum, SAM) and the higher dimensional orbital
angular momentum (OAM). The north and south poles of different spherical shell
in the G sphere represent the pair of different orthogonal vector basis with
different ellipticity (SAM) and the opposite OAM. The higher-order Poincar\'e
spheres are just the two special spherical shells of the G sphere. We present a
quite flexible scheme, which can generate all the vector fields described in
the G sphere. The salient properties of this representation are able to treat
the complicated geometric phase of arbitrary vector fields in the future.
