| The non-polynomial B-splines in the new schemes of parameter curves and surfaces modeling are the research focus in Computer aided Geometry design (CAGD). In application of CAGD, B-spline curves and surfaces have been the kernel of geometric modeling. Since trigonometric splines introduced by Schoenberg in 1964; it has been investigated by other researchers because of its properties which are similar to classical B-spline curves. Later analogous results have also been established for hyperbolic B-splines, but there isn't a unified form for B-splines. In 2001, M.K.Jena introduced GBB-curves and generalized the subdivision algorithm for Bezier curves to GBB-curves, which is the new research field of the splines.D-polynomial space is introduced in this paper by considering the null space of a second order constant coefficient differential operator and the unique solution to an initial-value problem, and generalized B-spline basis and generalized B-spline curves are defined on the basic of the space. Then a new representation to B-splines and the concept of generalized B-spline are presented, whose particular forms are classical B-spline curves, trigonometric B-splines curves and hyperbolic B-splines curves. They can be used as an efficient new model for geometric design in the fields of CAD/CAM. The convex-hull property and variation-diminishing result are also analysised. The evaluation algorithm and knot insertion algorithm for generalized B-spline curves are also presented; the curve can be obtained from the control polygon via subdivision, and this help to accelerate the rendering of the curve. The generation of tensor product surfaces by this new spline is straightforward. On the other hand, we can use different kinds of the splines to construct the tensor product surfaces and it is easy to construct some special surfaces. At last, numerical examples show that the algorithms are valid to both curves and surfaces.
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