Quartic Rational Said-Ball-Like Basis with Tension Shape Parameters and Its Application

Four new quartic rational Said-Ball-like basis functions, which include the cubic Said-Ball basis functions as a special case, are constructed in this paper. The new basis is applied to generate a class of C1 continuous quartic rational Hermite interpolation splines with local tension shape parameters. The error estimate expression of the proposed interpolant is given and the sufficient conditions are derived for constructing a C1 positivityor monotonicitypreserving interpolation spline. In addition, we extend the quartic rational Said-Ball-like basis to a triangular domain which has three tension shape parameters and includes the cubic triangular Said-Ball basis as a special case. In order to compute the corresponding patch stably and efficiently, a new de Casteljautype algorithm is developed. Moreover, the G1 continuous conditions are deduced for the joining of two patches.


Introduction
Constructing practical basis functions to generate free form curves and surfaces is an important topic of CAGD (computer aided geometric design) and computer graphics.Generally speaking, the basis function with good properties plays a vital role in curve and surface design.In [1][2][3], Ball introduced a kind of cubic rational polynomial to be the basis of CONSURF surface lofting program.In [4], Said extended the cubic basis proposed by Ball into arbitrary odd degree basis to get a type of generalized Ball basis.After that, Goodman and Said proved that the generalized Ball basis is normalized totally positive and hence it possesses the same kind of shape preserving properties as the Bernstein basis [5].In [6], Hu and his colleagues suggested an extension of the generalized Ball basis to arbitrary even degree, usually known as Said-Ball basis.In [7], based on the necessary and sufficient conditions for the conic representation in a rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis, Hu and Wang deduced the representation theory of rational cubic and quartic Said-Ball conics.Because surfaces modeling over a triangular domain have great potential of constructing complex shapes, there are some researchers who have put many efforts on the establishments of triangular Said-Ball bases.In [8], Goodman and Said defined a bivariate Said-Ball basis on a triangle and designed a triangular Said-Ball surface based on the research of Said-Ball curve.In [9], Hu et al. extended the univariate Wang-Ball basis to the bivariate case on a triangle, designed a triangular Wang-Ball surface, and pointed out its virtues.Chen and Wang [10] constructed a class of triangular DP surfaces.Recently, Zhu and Han [11] constructed a class of -Bernstein-Bézier basis possessing three-exponentialshape parameters over a triangular domain, which includes the cubic triangular Said-Ball basis and the cubic triangular Bernstein-Bézier basis as special cases.
In industrial design and scientific data visualization, constructing shape preserving interpolation splines for the interpolation of positive or monotonic data is an essential problem and has attracted widespread interest.In [12], Schmidt and Heß derived a necessary and sufficient criterion under which a kind of  1 quadratic rational positivitypreserving interpolation spline was developed.Sakai and Schmidt [13] proposed a method for the construction of positive cubic rational splines of continuity class  2 .In [14,15], two kinds of  1 rational cubic interpolation splines were constructed by Hussain and Sarfraz to visualize the positive data.In [16], Fritsch and Carlson gave necessary and sufficient conditions for a cubic spline to be monotone on an interval.Manni and Sablonnière [17] developed an explicit expression of a  1 cubic rational spline which can be applied to monotonic data.In [18,19], two kinds of  2 cubic rational interpolation splines were proposed by Sarfraz for the visualization of monotonic data.In [20], Hussain and Sarfraz presented a  1 cubic rational interpolating scheme to deal with the problem of monotonicity.In [21], by using weighted  1 quadratic splines, Kvasov proposed two algorithms with automatic selection of the shape parameters to construct a shape preserving interpolation spline for monotonic or convex data.Recently, the quadratic and cubic rational trigonometric bases are also considered by some scholars to construct positivity-preserving and monotonicity-preserving interpolation splines; see [22][23][24].
The purpose of this paper is to present four new quartic rational Said-Ball-like basis functions with two-tensionshape parameters, which include the cubic Said-Ball basis functions.By using the new basis, a class of  1 continuous quartic rational Hermite interpolation spline with local tension shape parameters is also constructed.Its error estimate expression is given and sufficient conditions for constructing a  1 positivity-or monotonicity-preserving interpolation spline are derived.The quartic rational Said-Ball-like basis functions are also extended to a triangular domain.In order to compute the corresponding patch stably and efficiently, a new de Casteljau-type algorithm is developed.
The rest of this paper is organized as follows.Section 2 gives the definition and properties of the new quartic rational Said-Ball-like basis and discusses the properties of the corresponding quartic rational Said-Ball-like curve.In Section 3, based on the new proposed quartic rational Said-Ball-like basis, a new class of  1 quartic rational Hermite interpolation spline is developed.Sufficient conditions are derived for constructing a  1 positivity-or monotonicitypreserving interpolation spline.Section 4 gives the construction and properties of the quartic rational Said-Ball-like basis functions over a triangular domain.The conditions for  1 continuously and smoothly joining of two quartic rational triangular Said-Ball-like patches are deduced.Conclusion is given in Section 5.

Quartic Rational Said-Ball-Like Basis
2.1.Construction of the Basis.We construct four new quartic rational Said-Ball-like basis functions as follows.
Definition 1.For any ,  ∈ [0, +∞),  ∈ [0, 1], the new quartic rational Said-Ball-like basis functions are given by . ( Notice that we can see immediately that the quartic rational Said-Balllike basis functions (1) are constructed in the function space span {1, 3 It is easy to check that, for  =  = 0, the quartic rational Said-Ball-like basis is precisely the cubic Said-Ball basis; see [1][2][3].
From the definition of the quartic rational Said-Ball-like basis given in (1), we have the following important properties of the new basis.Proof.It is easy to check that the basis (1) has the properties of partition of unity, nonnegativity, and symmetry.In addition, for any ,  ∈ [0, +∞),   ∈ R( = 0, 1, 2, 3), we consider a linear combination Differentiating with respect to the variable  on both sides, we have For  = 0, from (3) and (4), we can obtain the following system of equations with respect to  0 and  1 : Then, we have  0 =  1 = 0. Similarly, for  = 1, we can get  2 =  3 = 0. Thus, the basis functions (1) are linearly independent.

Definition and Properties of the Quartic Rational Said-
Ball-Like Curve Definition 3. Given control points   ( = 0, 1, 2, 3) in R 2 or R 3 , then is called a quartic rational Said-Ball-like curve with two shape parameters  and .
Since the quartic rational Said-Ball-like basis (1) has the properties of partition of unity and nonnegativity, we can see that the corresponding curve (6) defined by it has the properties of affine invariance and convex hull, which are crucial properties in curve design.Direct computation gives the following end-point property of the quartic rational Said-Ball-like curve: These imply that for any ,  ∈ [0, +∞) the curve has the end-point interpolation property and  0  1 ,  2  3 are the tangent lines of the curve at the points  0 ,  3 , respectively.From these, we can find that the quartic rational Said-Balllike curve has some properties analogous to that of the cubic Said-Ball curve.For  =  = 0, the quartic rational Said-Balllike curve, specially, is just the cubic Said-Ball curve.
For the analysis of the effect of the shape parameters on the shape of the obtained quartic rational Said-Ball-like curve, we rewrite (6) as follows: From (8), it is obvious that the shape parameters  and  only affect the curve on the control edges  0 −  1 and Figure 2: The effect of shape parameters on the quartic rational Said-Ball-like curves. 3 −  2 , respectively.Moreover,  0 (; ) decreases with the increase of  for any fixed  ∈ (0, 1), which indicates that the curve moves in the same direction of the edge  0 −  1 as  increases.Inversely, as  decreases, the curve moves in the opposite direction to the edge  0 −  1 .The shape parameter  has similar effects on the edge  3 − 2 .When the shape parameters satisfy  = , the curve moves in the same direction or the opposite direction to the edge  2 − 1 when  increases or decreases, respectively.And when ,  increase at the same time, the curve tends to the edge  2 −  1 .Thus we can see that the shape parameters  and  serve as local tension parameters.Figure 2 shows the effect of the shape parameters on the quartic rational Said-Ball-like curves.
Corner cutting algorithm is an efficient and stable process for computing the quartic rational Said-Ball-like curve.In order to develop such an algorithm, we rewrite the quartic rational Said-Ball-like curve (6) in the following matrix form: From ( 9), we can immediately obtain a corner cutting algorithm for computing the quartic rational Said-Ball-like curve.See Figure 3 for an illustration of this algorithm.
Control polygons provide an important tool in geometric modeling.Here, we adopt the method given in [25] to show some relations of the quartic rational Said-Ball-like curves and the classical cubic Bézier curves corresponding to the same control polygons.
Figure 4 shows the comparisons between the quartic rational Said-Ball-like curves and the cubic Bézier curves.It is clear that, by changing the shape parameters, the quartic rational Said-Ball-like curves can yield tight envelopes for the cubic Bézier curves.Figure 5 shows the comparisons between the quartic rational Said-Ball-like curves (solid lines and dash-dotted lines) and the cubic trigonometric Bézier curves with two shape parameters  and  given in [25] (dashed lines and dotted lines).Clearly, the quartic rational Said-Ball-like curves can be closer to the control polygon than the cubic trigonometric Bézier curves. =  = 0.5 Figure 4: Comparisons between the quartic rational Said-Ball-like curves and the cubic Bézier curves.
Figure 5: Comparisons between the quartic rational Said-Ball-like curves and the cubic trigonometric Bézier curves with two shape parameters  and  given in [25].

𝐶 1 Quartic Rational Shape Preserving Interpolation Spline
In this section, based on the quartic rational Said-Ball-like basis given in (1), we will construct a kind of  1 quartic rational Hermite interpolation spline.Meanwhile, sufficient conditions are also derived for constructing the interpolation spline which can preserve the shape of positive or monotonic set of data.
The spline given in ( 11) is a  1 Hermite interpolation spline as it satisfies the following interpolatory properties: And it can be easily checked that, for   =  +1 = 0, the quartic rational Hermite interpolation spline is exactly the classical cubic Hermite interpolation spline.
Here, all tangent information   is computed from the given data (  ,   ),  = 1, 2, . . ., , as follows: which is used usually for generating Hermite interpolation curves.
In practical applications, the corner cutting algorithm given in ( 9) can be used to compute the quartic rational Hermite interpolation spline () effectively and stably.From (11), it is obvious that the shape parameter   only affects two associated curve segments.As   increases, the curve () is pulled towards the point (  ) in the neighborhood of the knot position   .In other words, the shape parameter   serves as a local tension shape parameter.
Smooth shape preserving interpolation splines are of great significance in the area of computer graphics and scientific data visualization.In the next two subsections, we will show that a  1 shape preserving interpolation spline for positive and/or monotonic data sets can be easily constructed by constraining the shape parameters   in the quartic rational Hermite spline () given in (11).
Figure 6 shows our positivity-preserving interpolation spline (on the left) for the positive dataset given in Table 1 and the graphics of their first derivatives (on the right).It can be seen that the spline curves preserve the positive shape of the data well and they all achieve  1 continuity.The visually pleasing positivity-preserving interpolation spline is generated by applying the sufficient conditions in (23) with different free shape parameters   .Figure 7 shows the comparison between our positivity-preserving interpolation spline curves ((on the left), with the choice of free parameters  = (0, 0, 4, 4, 0, 0, 0, 0)) and the positivity-preserving interpolation spline curves given in [26] for the same positive dataset given in Table 2. From the results, it is obvious that our positivity-preserving interpolation spline curves and the scheme given in [26] can construct similar curves to describe the positive dataset.
Without loss of generality, we just consider the function () constrained on the subinterval [  ,  +1 ].From (11), direct computation gives that where the necessary conditions   > 0,  +1 > 0 are assumed.From ( 27), we can easily obtain the following sufficient conditions for constructing a  1 monotonicity-preserving interpolation spline () where the free parameter   ≥ 0,  = 1, 2, . . ., , can be specified by users to achieve a monotonicity-preserving interpolation spline.Similarly, we can deal with a monotonic decreasing data.It should be noted that, if  +1 =   , it is necessary to set   =  +1 = 0. Thus for any   ,  +1 ∈ [0, +∞), () =   =  +1 , is a constant on [  ,  +1 ].For the case where the data is monotonic but not strictly monotonic (i.e., when some  +1 =   ) it would be necessary to divide the data into several strictly monotonic parts.And we should set   =  +1 = 0, whenever  +1 =   .
Figure 8 shows our monotonicity-preserving interpolation spline (on the left) for the dataset in Table 3 and the graphics of their first derivatives (on the right).Obviously, all of the interpolation spline curves have the properties of monotonicity-preserving and they are all  1 continuous.The visually pleasing monotonicity-preserving interpolation spline is generated by applying the sufficient conditions in (28) with different free shape parameters   .Figure 9 shows the comparison between our monotonicity-preserving interpolation spline curves (on the left, with the choice of the free parameters  = (0, 0, 0, 0)) and the monotonicitypreserving interpolation spline curves given in [26] for the monotone dataset given in Table 4. From the results, it can be seen that our monotonicity-preserving interpolation spline curves describe the monotone dataset as well as the scheme given in [26].

Quartic Rational Said-Ball-Like Basis over a Triangular Domain
Based on the proposed quartic rational Said-Ball-like basis with two shape parameters given in (1), by using the method of tensor product, we can easily construct a kind of biquartic rational Said-Ball-like patch over a rectangle domain with four shape parameters.However, the patch over the triangular  3 and the graphics of their first derivatives.
domain is not a tensor product patch exactly.These imply that we cannot extend the basis (1) to a triangular domain by using the method of tensor product.In this section, we will construct a new class of quartic rational Said-Ball-like basis over a triangular domain with three shape parameters, which is an extension over a triangular domain of the basis (1).

Construction and Properties of the Basis.
Here, we give the definition of the new quartic rational Said-Ball-like basis functions over a triangular domain as follows.Figure 9: Comparison between our monotonicity-preserving interpolation spline curves (on the left) and the monotonicity-preserving interpolation spline curves given in [26].defined as new quartic rational Said-Ball-like basis functions, with three shape parameters , , and , over the triangular domain : Remark 6.In the following discussion, we will denote the new quartic rational Said-Ball-like basis functions as  3 ,, (, , ; , , ), , ,  ∈ N,  +  +  = 3.It is easy to check that for  =  =  = 0, the basis functions given in (29) are precisely the cubic Said-Ball basis functions over a triangular domain; see [8].
(e) Boundary property: when one of the three variables  is taken as zero, the ten quartic rational basis functions  3 ,, (, , ) degenerate to the corresponding quartic rational Said-Ball-like basis functions   (; , ) (notice  = 1 − ) given in (1).
Proof.We will prove (b) and (f).Simple and direct computation can give out the remaining properties.
These imply the theorem.
Figure 10 shows some plots of quartic rational Said-Balllike basis functions over a triangular domain.The three shape parameters take values  = 0,  = 1,  = 2.

Definition and Properties of the
the quartic rational triangular Said-Ball-like patch with three shape parameters , , .
From the properties of the quartic rational Said-Ball-like basis functions over a triangular domain, some properties of the quartic rational triangular Said-Ball-liek patch (37) can be obtained easily as follows.
These indicate that the patch (37) interpolates at the corner points.
(e) Shape adjustable property: without changing the control net, we can adjust the shape of the quartic rational triangular Said-Ball-like patch conveniently by using the three shape parameters , , and .As the three shape parameters increase at the same time, the patch will be made close to the control net.Thus the three shape parameters , ,  serve as tension parameters.Furthermore, from the boundary property of the patch, we can see that the three shape parameters , , and  have nothing to do with the boundary curves (0, , ), (, 0, ), and (, , 0), respectively.These imply that altering the value of a single shape parameter, one corresponding boundary curve of the generated patch will not change.
Figure 11 shows the quartic rational triangular patches and the effect on the patches by altering the values of the shape parameters without changing the control points.
From Theorem 9, we can know that the conditions for smoothly joining two quartic rational triangular Said-Balllike patches are similar to the ones necessary for joining two triangular Bernstein-Bézier cubic patches; see [28].However, we can adjust the shape of the obtained  1 continuous surface conveniently by changing the tension shape parameters in the quartic rational triangular Said-Ball-like patch.Figure 12 shows the  1 continuous smooth surfaces generated by the joining of two quartic rational triangular Said-Ball-like patches with different shape parameters.The parameters take values  = 1,  = −1.

Conclusion
The four quartic rational Said-Ball-like basis functions have the properties of partition of unity, nonnegative property, and linear independence and include the cubic Said-Ball basis functions as a special case.The new proposed  1 quartic rational Hermite interpolation spline can be used to construct  1 positivity-or monotonicity-preserving interpolation splines by constraining the shape parameters in the spline.The quartic rational Said-Ball-like basis with three-shape parameters over a triangular domain is a new construction for geometric design and computing, which has the properties of partition of unity, nonnegativity, and linear independence, and includes the classical cubic Said-Ball basis over a triangular domain as a special case.Without changing the control net, the shape of the obtained patch can be adjusted conveniently by using the shape parameters.Future work will concentrate on applying the new proposed basis to construct shape preserving interpolation spline surfaces.

Figure 1 :
Figure 1: Some plots of the new quartic rational Said-Ball-like basis functions.

Figure 7 :
Figure 7: Comparison between our positivity-preserving interpolation spline curves (on the left) and the positivity-preserving interpolation spline curves given in[26].

Figure 8 :
Figure 8: Monotonicity-preserving interpolation spline curves for the Akima's data given in Table3and the graphics of their first derivatives.

Theorem 7 .
The basis functions over a triangular domain given in (29) have the following properties.

Figure 10 :
Figure 10: Some plots of quartic rational Said-Ball-like basis functions over a triangular domain.