Cubic Trigonometric Nonuniform Spline Curves and Surfaces

Abstract

A class of cubic trigonometric nonuniform spline basis functions with a local shape parameter is constructed. Their totally positive property is proved. The associated spline curves inherit most properties of usual polynomialB-spline curves and enjoy some other advantageous properties for engineering design. They haveC2continuity at single knots. For equidistant knots, they haveC3continuity andC5continuity for particular choice of shape parameter. They can express freeform curves as well as ellipses. The associated spline surfaces can exactly represent the surfaces of revolution. Thus the curve and surface representation scheme unifies the representation of freeform shape and some analytical shapes, which is popular in engineering.

Application of piecewise­cubic functions for constructing a Bezier type curve of C1 smoothness

“…Paper [14] studied the dependence of geometrical properties of the proposed cubic trigonometric curves and the Bezier surfaces on the shape parameters. Authors of [15] also considered cubic trigonometric basis functions of a spline with a local parameter of the shape. Thus, the introduction of parameters makes it possible to build a class of functions among which one can choose the one that is most suitable for a given data set.…”

The Cubic α-Catmull-Rom Spline

“…For different n (where n + 1 is the number of the given data points), the optimal value of the shape parameter α solved by Equation (24), the interpolation errors of the optimum cubic α-Catmull-Rom spline interpolation function and the interpolation error of the standard cubic Catmull-Rom spline interpolation function calculated by Equation (21) are shown in Table 1. Table 1 shows that the interpolation result of the optimum cubic α-Catmull-Rom spline interpolation function is better than the standard cubic Catmull-Rom spline interpolation function, which is due to the fact that the shape parameter α is taken as the optimal value.…”

“…Hence, the curves with shape parameters have been paid more and more attention by many researchers in geometric modeling. Some examples are the Bézier-like curves with shape parameters [1][2][3][4][5], the B-spline-like curves with shape parameters [6][7][8][9][10][11][12][13][14], the trigonometric curves with shape parameters [15][16][17][18][19][20][21], and so on. Curves with shape parameters not only inherit similar or the same properties as the corresponding classical curves, but also have better performance ability because of the shape parameters.…”

A Class of Trigonometric Bernstein‐Type Basis Functions with Four Shape Parameters

“…Later, in [10], the totally positive property of the cubic trigonometric Bernstein-type basis functions with two shape parameters given in [8] was proved, which implies that the cubic trigonometric Bernstein-type basis with two shape parameters is suitable for conformal design. Recently, a class of cubic trigonometric nonuniform B-spline basis functions having a local shape parameter was proposed in [11], which is an extension of the cubic trigonometric nonuniform spline basis functions with a global shape parameter given in [6]. In [12], a class of C-Bézier basis of the space span{1, , sin , cos } was constructed, where the length of the interval serves as shape parameter.…”