Conical representation of Rational Quartic Trigonometric Bèzier curve with two shape parameters

I. Introduction Rational spline is a commonly used spline function. In many cases the rational spline curves better approximating functions than the usual spline functions. It has been observed that many simple shapes including conic section and quadric surfaces can not be represented exactly by piecewise polynomials, whereas rational polynomials can exactly represent all conic sections and quadric surfaces in an easy manner (see Sarfraz, M. Recently, many papers investigate the trigonometric B´ezier-like polynomial, trigonometric spline and their applications. Many kinds of methods based on trigonometric polynomials were also established for free form curves and surfaces modeling (see [10] presented a study on class of TC-Bèzier curve with shape] have discussed the rational quadratic trigonometric Bèzier curve with two shape parameters. Base on this idea, we have constructed the Rational Quartic Trigonometric Bèzier curve with two shape parameters. In this paper, rational quartic trigonometric Bèzier curve with two shape parameters is presented. The purposed curve enjoys all the geometric properties of the traditional rational quartic Bèzier curve. The local control on the shape of the curve can be attained by altering the values of the shape parameters as well as the weight. The curve exactly represents some quartic trigonometric curves such as the arc of an ellipse and the arc of a circle and best approximates the ordinary rational quadratic Bèzier curve. The paper is organized as follows. In section 2, the asis functions of the quartic trigonometric Bèzier curve with two shape parameters are established and the properties of the basis function has been described. In section 3, rational quartic trigonometric Bèzier curves and their properties are discussed. In section 4, By using shape parameter, shape control of the curves is studied and explained by using figures. In section 5, the representation of ellipse and circle are given. In section 6, the approximation of the rational quartic trigonometric Bèzier curve to the ordinary rational quadratic Bèzier curve is presented. II. Quartic Trigonometric Bèzier Basis Functions In this section, definition and some properties of quartic trigonometric Bèzier basis functions with two shape parameters are given as follows: