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.
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  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.  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  constructed a kind of piecewise quartic polynomial curves with a local shape parameter. Han  defined a class of piecewise quartic spline curves with three local shape parameters. Hu et al.  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  and cubic  trigonometric polynomial curves with a shape parameter. Han and Zhu  defined trigonometric -spline curves with three local shape parameters and a global shape parameter. Dube and Sharma  constructed the cubic trigonometric polynomial -spline curves with a shape parameter. Wang et al.  presented nonuniform algebraic-trigonometric -splines. The curves in [6–10] not only enjoy adjustable shape, but also can exactly represent ellipses. Yan and Liang  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.  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  proved that the generalized Ball basis given in  is normalized totally positive and hence it possesses the same kind of shape preserving properties as the Bernstein basis . Han and Zhu  proved that the cubic trigonometric Bézier basis given in  forms an optimal normalized totally positive basis. Zhu et al.  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  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  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 , 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 , 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  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 , 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.
2.2. The Property of the Basis Functions
Theorem 3. The --spline basis has the following properties:(a)Partition of unity. for .(b)Nonnegativity. If , then for .(c)Linear independence. For , , the set is linearly independent on .(d)Normalized total positivity. For , , , the system is a normalized totally positive basis of the space .(e)Continuity. With a nonuniform knot vector, the --spline basis has continuity for at each of the knots. With a uniform knot vector, has continuity for and continuity for at each of the knots.
Proof. (a) Notice that . For , , we have for ; then (b) When , we can see from (5) that (). Thus, from the positivity of the -Bézier basis, we can conclude that for .
(c) For any (), , we define For , , direct computation gives thatThus, we can obtain the following linear systems of equations with respect to :Since , we have the determinant of the coefficient matrix given by the above linear systems of equations as follows: Therefore, we can conclude that for .
(d) Notice that the system is the optimal normalized totally positive basis of the space . Thus, by relation (7) and Theorem 1, we can see that the system is a totally positive basis.
(e) For any direct computation we have When , for any , we have Further, when and , we have This means that the conclusion is correct.
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 . For multiple knots, we shrink the corresponding intervals to zero and drop the corresponding pieces of the basis function. For example, if 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.
Corollary 4. Suppose that a --spline basis has a knot with multiplicity () at a parameter value ; then at this point the continuity of the --spline basis is reduced from to ( means discontinuous) for any . Moreover, the support interval of the basis is reduced from segments to segments.
3. Spline Curves
3.1. The Construction of the Curves
Definition 5. Given control points () and a knot vector and an array , where and all (), the curve is called a --spline curve.
Obviously, for , , the curve can be represented by curve segment
Analogous to the cubic -spline curves, since curve is generated on the interval , the choice of the first and last three knots is free, and these knots can be adjusted to give the desired boundary behavior of the curve; see the following descriptions.
For open curves, we choose the knot vectorThis ensures that the points and are the end points of the curves. An example is given in Figure 2 (left).
To construct closed curves, we can extend the given points by setting , , and and letting (). Thus the parametric formula for a closed curve is where () are given by expanding (3). An example is given in Figure 2 (right).
Based on the properties of the --spline basis, we can know that the curve has the following properties:(a)Curve lies in the convex hull of the points for .(b)Curve has variation diminishing property.(c)If a knot has multiplicity (), then the curve has continuity at this point. With a uniform knot vector, the curve has continuity for any and continuity if all .
3.2. The Representation of an Ellipse
Given a uniform knot vector, and given control points , , , and , where , . The corresponding --spline curve with all 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 , , and , we can obtain the entire ellipse. When , we obtain an entire circle. An example is given in Figure 3.
3.3. The Representation of a Parabola
Given a uniform knot vector and given control points , , , and , where , , the corresponding --spline curve with all is a segment of a parabola, whose equation is
Adding a control point , we obtain a symmetric parabolic segment composed of two curve segments. An example is given in Figure 4.
4. Spline Surfaces
4.1. The Construction of the Surfaces
Exactly as in the construction of -spline tensor product surfaces from -spline curves, we can construct --spline surface from --spline curve. For --spline surface, the shape parameters in -direction and -direction can be different.
Definition 6. Given control points (), two knot vectors , , and two arrays , , where and all (), then we can define a --spline surface as follows:
For and , , , the surface can be represented by surface patch
Most properties of the --spline surfaces follow in a straightforward way from those of the --spline curves. For example, if and are single knot vectors, the --spline surfaces have continuity in both directions. If and are uniform knot vectors, the --spline surfaces have continuity in general and continuity when all shape parameters are equal to .
4.2. The Representation of a Surface of Revolution
A surface of revolution is given byFor fixed , an isoparametric line traces out a circle of radius , 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 , 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 (take the curve in plane as an example) and by the axis of revolution, in the same plane as .
Suppose the axis of revolution is -axis. Assume that is given by a planar --spline curve with control points (), knot vector , and shape parameter array . Denote , . Here and are row vectors formed by the abscissa and ordinate of the control points ().
By the method given in Section 3.2, the --spline curve with control points , , , , , , and , uniform knot vector , and shape parameter array of all elements are zero which is a unit circle whose center is . Denote , , and . Here and are row vectors formed by the abscissa and ordinate of the control points (). is a row vector with all elements being equal to , and it has the same dimension with and .
Let , , and be matrixes defined by , , and , where , denotes the th row of the matrix and denotes the th element of the vector . Suppose , , and are the coordinate component matrices of the control points (). Then the --spline surface with control points , knot vectors and , and shape parameter arrays and is a surface of revolution. Its generatrix is , and its axis of revolution is -axis.
In this way, we represent exactly “classical” surfaces such as cylinders, spheres, or tori. For example, taking and , we obtain a cylinder. Taking and , we obtain a circular cone. Taking and , we obtain a sphere. Taking and , we obtain an ellipsoid of revolution. Taking and , we obtain a paraboloid of revolution. Taking and , we obtain a tori. In each case we take uniform knot vector and shape parameter array with all elements which are equal to . Figure 5 shows the results.
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 or continuous by taking equidistant knots. By using the --spline surfaces, we can represent rotating surfaces exactly. The --spline surfaces can also reach or continuity. One of our future works is to apply the --spline basis to generate shape preserving interpolation curves.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (no. 11261003).
H. Wu and X. Chen, “Cubic non-uniform trigonometric polynomial curves with multiple shape parameters,” Journal of Computer-Aided Design & Computer Graphics, vol. 18, no. 10, pp. 1599–1606, 2006.View at: Google Scholar