Mathematical Problems in Engineering

Volume 2016, Article ID 7067408, 9 pages

http://dx.doi.org/10.1155/2016/7067408

## Cubic Trigonometric Nonuniform Spline Curves and Surfaces

^{1}School of Mathematics and Statistics, Central South University, Changsha 410083, China^{2}School of Science, East China University of Technology, Nanchang 330013, China

Received 29 September 2015; Revised 9 January 2016; Accepted 11 January 2016

Academic Editor: Ofer Hadar

Copyright © 2016 Lanlan Yan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 polynomial -spline curves and enjoy some other advantageous properties for engineering design. They have continuity at single knots. For equidistant knots, they have continuity and continuity 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.

#### 1. Introduction

As a unified mathematical model with many desirable properties, -splines are widely applied to the modeling of freeform curves and surfaces. However, there are several limitations of the -spline model, which restrict its applications. For example, once the knot vectors are specified, the positions of -spline curves are relatively fixed to their control points. On the other hand, -spline curves fail to represent conic curves except the parabolas, as well as some transcendental curves such as the helix and the catenary, which are often used in engineering.

Although the nonuniform rational -spline (NURBS) can overcome the first shortcoming of -splines to a certain extent, it fails to model transcendental curves. The NURBS model has several other potential limitations due to the relative complexity of rational basis functions. For example, rational form may be unstable, and derivatives and integrals are hard to compute. Consequently, in order to overcome the drawbacks of -splines, it is necessary to explore new models.

To enhance the flexibility of -spline models, some researchers have suggested many types of curves with shape parameters incorporated into the basis functions. For instance, Xu and Wang [1] proposed three kinds of extensions of cubic uniform -spline. The advantage of the extensions is that they have shape parameters, which can be used to adjust the shape of the curves without shifting the control points. Costantini et al. [2] presented a method for the construction of cubic like -splines with multiple knots. The proposed -splines are equipped with tension parameters, associated to the knots, which permit a modification of their shape. Han [3] constructed a kind of piecewise quartic polynomial curves with a local shape parameter. Han [4] defined a class of piecewise quartic spline curves with three local shape parameters. Hu et al. [5] constructed a kind of -spline curves with two local shape parameters. To expand the scope of shape representation of -spline models, some researchers have suggested many types of curves defined on nonpolynomial space. For instance, Han presented the quadratic [6] and cubic [7] trigonometric polynomial curves with a shape parameter. Han and Zhu [8] defined trigonometric -spline curves with three local shape parameters and a global shape parameter. Dube and Sharma [9] constructed the cubic trigonometric polynomial -spline curves with a shape parameter. Wang et al. [10] presented nonuniform algebraic-trigonometric -splines. The curves in [6–10] not only enjoy adjustable shape, but also can exactly represent ellipses. Yan and Liang [11] constructed a class of algebraic-trigonometric blending spline curves with two shape parameters. Except for shape adjustability, the curves admit exact representations for several remarkable curves. Lü et al. [12] defined uniform algebraic-hyperbolic blending -spline curves, which possess adjustable shape and can represent exactly the hyperbola and the catenary.

Totally positive property is one of the important properties of basis functions. Curves defined by totally positive basis must have variation diminishing and convexity preserving property. Goodman and Said [13] proved that the generalized Ball basis given in [14] is normalized totally positive and hence it possesses the same kind of shape preserving properties as the Bernstein basis [15]. Han and Zhu [16] proved that the cubic trigonometric Bézier basis given in [17] forms an optimal normalized totally positive basis. Zhu et al. [18] constructed four Bernstein-like basis functions, which form an optimal normalized totally positive basis. Based on the Bernstein-like basis, a class of totally positive -spline-like basis functions is constructed. The associated -spline curves have continuity at single knots and can be () continuous for particular choice of shape parameters.

The curves given in [1–10, 12, 18] have adjustable shape. In addition, the curves given in [6–10, 12] can represent exactly some conic curves and transcendental curves. However, whether the blending functions in [1, 3–10, 12] have total positivity is unknown, so whether the associated curves have variation diminishing is unknown. The curves in [2, 18] have variation diminishing, but they cannot represent conic curve or transcendental curve. The purpose of this paper is to define a kind of -spline-like curves and surfaces, which have adjustable shape and can represent some elementary analytic curves and surfaces, and the curves have variation diminishing thus having a good shape control.

The research topics of this paper and some existing documents, such as [19, 20], are similar. There are also some differences, though. By extending the global parameter to local parameter, Wu and Chen [19] presented cubic nonuniform trigonometric polynomial curves with multiple shape parameters. The curves are continuity for a nonuniform knot vector and continuity for a uniform knot vector, respectively. Han [20] presented quadratic trigonometric polynomial curves with local basis. The curves have continuity with a nonuniform knot vector and any value of the bias. Compared to [19, 20], novelty of this paper is listed as follows. First, it discusses the total positivity of the basis functions. This property makes the corresponding curves have variation diminishing property, which is one of the important properties of dominant Bézier curves and -spline curves. Second, it provides the representation method of surface of revolution. In surface modeling, the construction of the rotational surface is a common problem. Third, it provides a class of higher order continuous curves, which can meet most of the needs in engineering.

The rest of the paper is organized as follows. Section 2 gives the definition and properties of the basis functions. Section 3 defines the associated curves and gives the representation of the ellipses and parabolas. Section 4 defines the associated surfaces and gives the representation of the rotating surfaces. Section 5 concludes the paper.

#### 2. Basis Functions

##### 2.1. The Construction of the Basis Functions

In [7], cubic trigonometric splines are presented for a nonuniform knot vector. These splines are used to define trigonometric spline curves. As special cases, the author also introduces a class of cubic trigonometric polynomial basis functions used to construct trigonometric Bézier curves. The original expression of the basis functions is as follows:This set of basis functions contains only one shape parameter. In [17], it was further extended to possess two shape parameters.

By changing the in the last two functions in (1) to , a class of cubic trigonometric Bézier basis functions with two shape parameters is defined in [17] as follows. Let , for ; the following four functions are defined to be the cubic trigonometric Bézier basis functions (-Bézier basis for short), with two shape parameters and :

In [16], the -Bézier basis was proved to be the optimal normalized totally positive basis of the space for . Hence the corresponding cubic trigonometric Bézier curves are suited for conformal curve design. However, the Bézier curve is a single curve segment. When using Bézier curve to describe complex shapes, the problem of joining curve segments smoothly needs to be solved. The Bézier form is the special case of -spline. -spline curves consist of many polynomial pieces, offering much more versatility than Bézier curves. Considering the -spline is more suitable for expressing complex curve and surface, this paper will discuss more generally the -spline form.

Next we will construct a kind of cubic trigonometric spline basis function based on the -Bézier basis.

Given knots , we refer to as a knot vector. Let , and , ; we want to construct the associated spline basis functions as follows:for . Here in which for are the -Bézier basis given in (2) and for are undetermined coefficients.

To determine the coefficient values, we impose two conditions on , which have continuity at each knot and form a partition of unity on the interval . Then, we can compute the coefficients as follows:where , , , and , .

For , , we have for , and We can easily check thatwhere

Theorem 1. *When , is a nonsingular stochastic and totally positive matrix.*

*Proof. *When , it is obvious that are all positive for . In addition, from (5) we can verify that . This means that is stochastic. With direct computation we havewhere , . From these, we can easily deduce that is nonsingular and all its remaining minors are nonnegative. Therefore, is a nonsingular stochastic and totally positive matrix.

*Definition 2. *Given a knot vector and an array , where all , , by using the -Bézier basis and the coefficients given in (5), functions (3) are defined to be the associated cubic trigonometric -spline basis functions (--spline basis for short).

It is easy to know the shape of is related to . We refer to the array as the shape parameter. For equidistant knots, we refer to the as a uniform --spline basis and refer to the knot vector as a uniform knot vector. For nonequidistant knots, and are called a nonuniform --spline basis and a nonuniform knot vector, respectively.

Figure 1 shows some graphs of uniform --spline basis functions with different shape parameters. As can be seen from the figure, with the increase of the parameter , the maximum of decreases and the highest point of the curve moves toward the right. With the increase of , the maximum of increases correspondingly. With the increase of , the maximum of decreases and the highest point of the curve moves toward the left. When , the graph of is symmetry.