  The construction of parametric curve and surface plays important role in
computer aided geometric design (CAGD), computer aided design (CAD), and
geometric modeling. In this paper, we define a new kind of blending functions
associated with a real points set, called generalized toric-Bernstein
(GT-Bernstein) basis functions. Then the generalized toric-Bezier (GT-B\'ezier)
curves and surfaces are constructed based on the GT-Bernstein basis functions,
which are the projections of the (irrational) toric varieties in fact and the
generalizations of the classical rational B\'ezier curves and surfaces and
toric surface patches. Furthermore, we also study the properties of the
presented curves and surfaces, including the limiting properties of weights and
knots. Some representative examples verify the properties and results.
