Abstract-In this paper a new kind of splines, called quadratic non-uniform algebraic trigonometric B-splines (quadratic NUAT Bsplines) with multiple shape parameters are constructed over the space spanned by { } . As each piece of the curves is generated by three consecutive control points, they possess many properties of the quadratic B-spline curves and quadratic trigonometric B-spline curves. These curves have continuity with non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves and quadratic trigonometric B-spline curves when choosing special shape parameters. The shape parameters serve to local control in the curves. The changes of one shape parameter will only affect two curve segments. Taking different values of the shape parameters, one can globally or locally adjust the shapes of the curves. The generation of tensor product surfaces by these new splines is straightforward. Keywords-Quadratic NUAT B-spline; shape parameter; continuity; spline surface. IntroductionThe trigonometric B-splines were presented in [12]. The recurrence relation for the trigonometric B-splines of arbitrary order was established in [8]. The construction of exponential tension B-splines of arbitrary order was given in [7]. It was further shown in [13] that the trigonometric Bsplines of odd order form a partition of a constant in the case of equidistant knots. In recent years, several bases in new spaces other than the polynomial space have been proposed for geometric modeling in CAGD. For instance, in [10] a basis is constructed for Cm=span { }. In [11] a basis for the space of trigonometric polynomials { } is constructed. Some bases are constructed in [9] for the spaces { }, { } and { }. In [1] C-Bézier curves are constructed in the space spanned by . Non-uniform algebraic trigonometric B-splines (NUAT B-splines), are generated in [14] over the space spanned by in which k is an arbitrary integer larger than or equal to 3. But all these curves do not have any shape parameter. In [15,16] C-B-splines are presented in the space spanned by { } with parameter .

A C 2 rational quadratic trigonometric spline interpolation has been studied using two kinds of rational quadratic trigonometric splines. It is shown that under some natural conditions the solution of the problem exists and is unique. The necessary and sufficient condition that constrain the interpolant curves to be convex in the interpolating interval or subinterval are derived. approximation properties has been discussed and confirms the expected approximation order is h 2 .

In Computer Aided Geometric Design it is often needed to produce a positivity preserving curve according to the given positive data. The main focus of this work is to visualize the positive data in such a way that its display looks smooth and pleasant. A rational quadratic trigonometric spline function with three shape parameters has been developed. In the description of the rational quadratic trigonometric spline interpolant, positivity is preserved everywhere. Constraints are derived for shape parameters to preserve the positivity thorough positive data. The curves scheme under discussion is attained C 1 continuity.

In the present paper a new method is developed for smooth rational cubic trigonometric interpolation based on values of function which is being interpolated. This rational cubic trigonometric spline is used to constrain the shape of the interpolant such as to force it to be in the given region by selecting suitable parameters. The more important achievement mathematically of this method is that the uniqueness of the interpolating function for the given data would be replaced by uniqueness of the interpolating curve for the given data and selected parameters. Approximation properties have been discussed and confirms that the expected approximation order is ) (

In this paper a new kind of splines, called cubic trigonometric polynomial B-spline (cubic TP B-spline) curves with a shape parameter, are constructed over the space spanned by As each piece of the curve is generated by three consecutive control points, they posses many properties of the quadratic B-spline curves. These trigonometric curves with a non-uniform knot vector are C1 and G2 continuous. They are C2 continuous when choosing special shape parameter for non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves when choosing special shape parameters. With the increase of the shape parameter, the trigonometric spline curves approximate to the control polygon. The given curves posses many properties of the quadratic B-spline curves. The generation of tensor product surfaces by these new splines is straightforward.