C1 rational quadratic trigonometric spline

C1Rational Quadratic Trigonometric Interpolation Spline for Data Visualization

“…The data in Table 4 lies above line = 0.06 + 0.02, whereas the data given in Table 5 Table 1, such as (a) method [4], (b) method [5], (c) method [6], (d) method [7], and (e) our method. Their corresponding parameters for all segments are set as ( Table 2, such as (a) method [4], (b) method [5], (c) method [6], (d) method [7], and (e) our method. Their corresponding parameters for all segments are set as ( = 3.6, = 2.9), ( 0 = 1.3, Table 4.…”

“…Random values are assigned to the shape parameters and it is clear that the resulting curves do not preserve the positivity. The positivity 4 Mathematical Problems in Engineering Figures 2 and 3 are the curves generated by methods [4][5][6][7] with a set of appropriate parameters, and the last curves are generated by our scheme. From the results, it can be seen that our 1 piecewise rational quadratic trigonometric spline describes the positive data set more fairly than [4][5][6][7].…”

“…Recently, some approaches in shape preserving interpolation, using the trigonometric splines, are developed. Hussain and Saleem [4] introduced a 1 piecewise rational quadratic trigonometric polynomial spline which defines two shape parameters in each subinterval, and they extended the spline to piecewise rational biquadratic function with four shape parameters in each rectangular patch. Ibraheem et al [5] constructed a 1 piecewise rational cubic trigonometric function with four shape parameters.…”

Total Positivity of the Cubic Trigonometric Bézier Basis

“…The expressions (16) and (18) describe a corner cutting algorithm for computing the cubic T-Bézier curve (14). See Figure 1 for an illustration of this new developed algorithm.…”

“…Recently, based on the theory of envelop and topological mapping, shape analysis of the cubic T-Bézier curve with shape parameters was given in [14,15]. For the problems of shape preserving interpolation, the cubic and quadratic T-Bézier bases show great potential applications; see [16][17][18][19]. Blossom is a powerful tool for studying Bézier-like bases and B-spline-like bases.…”