Cubic Trigonometric Nonuniform Spline Curves and Surfaces

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.


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 algebraictrigonometric -splines.The curves in [6][7][8][9][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.

Mathematical Problems in Engineering
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  2 continuity at single knots and can be  2 ∩  +3 ( ∈  + ) 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][7][8][9][10]12] can represent exactly some conic curves and transcendental curves.However, whether the blending functions in [1,[3][4][5][6][7][8][9][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  2 continuity for a nonuniform knot vector and  3 continuity for a uniform knot vector, respectively.Han [20] presented quadratic trigonometric polynomial curves with local basis.The curves have  2 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.

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 ,  ∈ [−2, 1], for  ∈ [0, /2]; 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  , := Span{1, sin 2 , (1−sin ) 2 (1− sin ), (1−cos ) 2 (1− cos )} for ,  ∈ (−2, 1].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.
It is easy to know the shape of   () is related to  +1 ,  +2 ,  +3 .We refer to the array   = ( +1 ,  +2 ,  +3 ) 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  +1 , the maximum of   () decreases and the highest point of the curve moves toward the right.With the increase of  +2 , the maximum of   () increases correspondingly.With the increase of  +2 , the maximum of   () decreases and the highest point of the curve moves toward the left.When  +1 =  +3 , the graph of   () is symmetry.
So far in the discussion of the --spline basis, we have assumed that each knot is single.On the other hand, the --spline basis also makes sense when knots are considered with multiplicity  ≤ 4. For multiple knots, we shrink the corresponding intervals to zero and drop the corresponding pieces of the basis function.For example, if   =  +1 is a double knot, then we define As a direct application of functions ( 3) and (e) in Theorem 3, we have the following corollary which shows the geometric meaning of multiple knots.
For open curves, we choose the knot vector This ensures that the points  0 and   are the end points of the curves.An example is given in Figure 2 (left).
Based on the properties of the --spline basis, we can know that the curve () has the following properties: The corresponding --spline curve with all   = 0 is a segment of an elliptic arc, whose equation is According to the method of constructing closed curves given in Section 3.1, adding three control points  4 =  0 ,  5 =  1 , and  6 =  2 , we can obtain the entire ellipse.When | 1 −  2 | = | 2 − 1 |, we obtain an entire circle.An example is given in Figure 3.

The Representation of a Surface of Revolution. A surface of revolution is given by
For fixed V, an isoparametric line V = const traces out a circle of radius (V), called a meridian.Since a circle may be exactly represented by --spline curve, we may find an exact representation of a surface of revolution provided we can represent (V), (V) in --spline curve form.The most convenient way to define a surface of revolution is to prescribe the (planar) generating curve, or generatrix, given by (V) = ((V), 0, (V)) T (take the curve in  plane as an example) and by the axis of revolution, in the same plane as .

Conclusion
With total positivity, the --spline basis is suitable for conformal design.By using the --spline curves, we can represent ellipses and parabolas exactly.The --spline curves can be  3 or  5 continuous by taking equidistant knots.By using the --spline surfaces, we can represent rotating surfaces exactly.The --spline surfaces can also reach  3 or  5 continuity.One of our future works is to apply the --spline basis to generate shape preserving interpolation curves.