Abstract
Four new trigonometric Bernstein-like bases with two denominator shape parameters (DTB-like basis) are constructed, based on which a kind of trigonometric Bézier-like curve with two denominator shape parameters (DTB-like curves) that are analogous to the cubic Bézier curves is proposed. The corner cutting algorithm for computing the DTB-like curves is given. Any arc of an ellipse or a parabola can be exactly represented by using the DTB-like curves. A new class of trigonometric B-spline-like basis function with two local denominator shape parameters (DT B-spline-like basis) is constructed according to the proposed DTB-like basis. The totally positive property of the DT B-spline-like basis is supported. For different shape parameter values, the associated trigonometric B-spline-like curves with two denominator shape parameters (DT B-spline-like curves) can be continuous for a non-uniform knot vector. For a special value, the generated curves can be continuous for a uniform knot vector. A kind of trigonometric B-spline-like surfaces with four denominator shape parameters (DT B-spline-like surface) is shown by using the tensor product method, and the associated DT B-spline-like surfaces can be continuous for a nonuniform knot vector. When given a special value, the related surfaces can be continuous for a uniform knot vector. A new class of trigonometric Bernstein–Bézier-like basis function with three denominator shape parameters (DT BB-like basis) over a triangular domain is also constructed. A de Casteljau-type algorithm is developed for computing the associated trigonometric Bernstein–Bézier-like patch with three denominator shape parameters (DT BB-like patch). The condition for continuous jointing two DT BB-like patches over the triangular domain is deduced.
1. Introduction
The construction of basis functions has always been a difficulty of computer-aided geometric design (CAGD). A class of practical basis functions often plays a decisive role in the geometric industry. Conventional cubic B-spline curves and surfaces are widely applied for CAGD due to their remarkable local adjustment properties. However, given control points keep a generated conventional cubic B-spline curve over a single location. Although cubic rational B-spline curves and surfaces can adjust positions and shapes by changing the weighting factor [1–3], their adjustment effect is difficult to predict due to its own defects. In recent years, trigonometric polynomials and splines with one or more shape parameters have been widely used with CAGD, especially in the design of curves and surfaces. Details can be found in [4–7] and the corresponding references therein. For example, researchers have used shape parameters to propose quadratic and cubic trigonometric polynomial splines [8, 9]. In [10], the extended cubic trigonometric spline curve of [8] was given. In [11], a class of C-Bézier curves was constructed in the space span , where the length of the interval serves as shape parameter. The sine and ellipse curves can be represented by the C-Bézier curves.
Many basis functions with tension effect, which possess one or more parameters, have been proposed to predictably adjust the positions and shapes of generated curves. For example, Nieson [12] proposed a cubic rational spline curve that can generate an interactive curve. The proposed curve can be continuous with different tension parameters. Barsky [13, 14] provided a class of beta-splines that can locally modify the shape of the generated curves. In a special case, the curve proposed by Barsky contains a kind of uniform B-spline. In [15], Gregory proposed a class of cubic continuous rational spline, which possesses tension shape parameters so that the shape of the generated curve can be modified. A class of spline curves was proposed in [16] by using tension shape parameters. According to [17] and the related references therein, curve design has attracted widespread interest due to the appearance of plenty of exponential splines. In the space , Costantini constructed a kind of continuous polynomial splines [18]. In [19, 20], this space was reported to be a quasi-extended Chebyshev space. In [21–25], different points of view about variable degree spaces were proposed. In [26], a class of -Bernstein-like basis was proposed in the space span . In the space span , four trigonometric Bernstein-like basis functions were constructed [27]. In the space , researchers in [28–30] discussed the subdivision scheme with respect to a constructed class of basis functions. In the space span , a cubic rational basis was constructed [31]. In [32, 33], variable degree polynomial splines exhibited enormous potential in the geometric industry in view of the problem of shape preservation and approximation.
Although many improved methods are available, they are rarely applied in solving practical problems. In the final analysis, these techniques increase the flexibility of the curve by adding shape parameters compared with the traditional Bézier and B-spline methods. However, the technique itself cannot replace the traditional method, and several aspects still need improvement. For example, the majority of these methods discuss only basic properties, such as nonnegativity, partition of unity, symmetry, and linear independence. Shape preservation, total positivity, and variation diminishing, which are important properties for curve design, are often overlooked. However, the basis function, which has total positivity, ensures that the related curve contains variation diminishing and shape preservation. Therefore, possessing total positivity is highly important for basis functions. In addition, constructing cubic curves and surfaces remains the main method among the improved techniques. In general, these improved methods have continuity, thereby meeting engineering requirements. However, in many practical applications, if the requirement for continuity is high, then these methods are slightly insufficient and often need to increase the number of times the curve is constructed. The B-spline curve and surface are regarded as examples. Notably, the continuity and locality of the curved surface are directly related to the number of times. The more times the curve is constructed, the higher the continuous order, but the locality is poor, and the computational complexity is high. Therefore, sacrificing the local property of its dominant position is necessary to achieve the special requirements of high-order continuity. Therefore, in the construction of curves and surfaces, the importance of meeting high-order continuity without increasing the computational complexity and without affecting its local properties is highlighted.
The traditional surface over rectangular domains, which possesses research and application value, has been widely used in CAGD. Obtaining a surface over a rectangular domain is easy because the traditional surface over such domain is a direct extension of the traditional Bézier curve by the tensor product method. However, by using the tensor product method, we fail to extend the patch over a triangular domain because it is not a tensor product surface. In many practical applications, surface modeling based on patch construction over triangular domains is important. Thus, the study of patches over triangular domains is of considerable interest. Therefore, the construction of a practical method that generates patches is important. For this reason, researchers have conducted numerous works. In [34], Cao showed a class of basis functions over a triangular domain. The related patch can be rendered flexible by an adjustment of the values. In [35], Han proposed a patch over a triangular domain, which can construct boundaries that can exactly represent elliptic arcs. A kind of quasi-Bernstein–Bézier polynomials over a triangular domain was proposed in [36]. Recently, Zhu constructed the -Bernstein–Bézier-like basis, which possesses 10 functions; the related exponential parameter has tension effect.
This study proposes a class of DTB-like bases with tension effects that is based on previous studies. The proposed basis has two denominator shape parameters constructed in the space and can form an optimal normal normalized totally positive basis (B-basis) and a new class of DT BB-like basis functions over a triangular domain with three denominator shape parameters. The presented DT B-spline-like curves and surfaces are continuous with respect to a nonuniform knot vector. The corresponding curves and surfaces are continuous for the shape parameters, which select a special value with respect to a uniform knot vector. The denominator parameter introduced in the basis function has a tension effect, and the parameters can be used to predictably adjust the corresponding curves and surfaces generated.
The remainder of this work is organized as follows. Section 2 provides the definition and properties of the DTB-like basis functions and shows the corresponding curves. Section 3 presents a class of DT B-spline-like basis with two denominator shape parameters. The properties of the proposed basis are analyzed, and the associated DT B-like curves are shown. Section 4 proposes a class of DT BB-like basis over a trigonometric domain with three denominator shape parameters. We provided the definition and properties of related DT BB-like patches on the basis of the presented basis functions. Then, we developed a de Casteljau algorithm to calculate the proposed patch. Finally, connecting conditions of the two proposed patches are given. Section 5 presents the conclusion.
2. Trigonometric Bernstein-Like Basis Functions
2.1. Preliminaries
For a good understanding of this study, related background knowledge about the extended Chebyshev (EC) space and the extended completed Chebyshev (ECC) space is provided in this subsection. Additional details are available in [37–39].
for any closed bounded interval , which can be denoted by . The function space is called -dimension ECC-space, which is generated by the positive weight function in canonical form. The weight function shows that
The necessary and sufficient condition of -dimension function space is called an ECC-space on that is for arbitrary , , and an arbitrary nontrivial linear combination of the elements of the subspace with the most zeros (counting multiplicities).
If the collocation matrix related to the basis for an arbitrary sequence of points is totally positive, then the basis is deemed totally positive on . If a function space has a totally positive basis, then the other totally positive basis can be formed by multiplying the optimal normalized totally positive basis (B-basis) by a totally positive matrix. Moreover, this basis is unique in space and has optimal shape preservation properties [40–42].
Assume that -dimensional is an -dimensional EC-space on closed bounded interval , which possesses constants, where . Select functions in so that forms a basis of . We set a mother function . This function being on , for arbitrary nonnegative integer , we can consider its osculating flat of order at , defined as the affine space passing through and the direction of which is spanned by , that is,
For arbitrary positive integers , with , and arbitrary pairwise distinct in , if ever the osculating flats , , possess a unique common point, then we define the blossom of as the function such thatThe arbitrary -tuple is equivalent to up to permutation, where
Reference [37] showed that the blossom that exists in space deduces three important properties, namely, pseudo-affinity, diagonal, and symmetry. In addition, we must emphasize a useful conclusion that the blossom that exists in space is equal to that in space , which is an EC- space on . Additional details can be seen in Theorem 3.1 in [38].
For arbitrary , the points , , are defined by the Chebyshev–Bézier points of about . When , , for all . However, when , the Chebyshev–Bézier points are obtained asMoreover, when , a de Casteljau algorithm starting from points can be developed by using the three important properties of the blossom. At the nth step of this algorithm, the values of can be obtained asGiven that is a set of bases in the function of , the affine flat of the mother function spans the whole space . Therefore, points are affinely independent. Furthermore, , , from a basis of [37].
2.2. Construction of Trigonometric Basis Function
For arbitrary real numbers , , we consider constructing a basis function in the trigonometric function space . We can easily obtain the corresponding mother function, which is defined as First, let us prove that the spaceis a three-dimensional EC-space on . Therefore, according to Theorem 2.1 of [38], possesses a blossom, indicating that is suitable for curve and surface designs.
Theorem 1. For arbitrary real numbers , is a three-dimensional EC-space on .
Proof. For arbitrary , we consider a linear combinationFor , from (9), we can easily obtain . In the same way, for , from (9), we can obtain . Thus, . Therefore, is a three-dimensional space.
Below, we prove that is an ECC-space in . For arbitrary , let andDirect computation indicates thatThus, for the Wronskian of and , For , we obtain the definition of the following weight functions:where are three arbitrary positive real numbers. All of the three weight functions are bounded, positive, and on closed bounded . Next, we discuss the following ECC-spaces: We can easily verify that these functions are linear combinations of Thus, is an ECC-space on . Furthermore, are any subintervals of ; hence, is also an ECC-space on .
Proof that is an EC-space on is provided. Thus, we must verify that the arbitrary nonzero element of has, at most, two zeros (counting multiplicities) on .
Consider the arbitrary nonzero functionwhere . has, at most, two zeros in because is an ECC-space in . Initially, is assumed to vanish at 0; then, we can obtain . Under these circumstances, if , then has two singular zeros at 0 and . If , then 0 is a double zero of . If , then has a singular zero at 0 but does not have a zero on . If , then has a singular zero at 0 but not at . In addition, for the function,direct computation yields where is a monotonic function on . From these values, together with , we can see that has exactly one zero in . Thus, we can immediately conclude that has one zero in . Similarly, if vanishes at , then function has, at most, two zeros on (counting multiplicities as far as possible up to 2). In summary, the space is an EC-space on .
Therefore, is an EC-space on . According to Theorem 3.1 in [43], a blossom exists in . This theorem also means that is suitable for curve and surface designs. According to Theorem 2.18 in [43], we can also infer that has a B-basis on .
Theorem 2. The four Chebyshev–Bézier points of the mother function defined in (7) are given by Moreover, the related new cubic trigonometric Bernstein-like basis of are given by
Proof. Through the definition of given in (7), we can easily obtain Thus, by simply computing, we have Thus, for any , from together with , we obtain the expressions of , .
Remark 3. From the expressions of the basis function given in (22), we can infer that (22) has important properties, such as linear independence, nonnegativity, and partition of unity. In addition, for all , we have the following end-point properties:Thus, system is precisely the B-basis of , which implies that (22) possesses totally positive and optimal shape preservation properties [33].
We define a cubic trigonometric Bernstein-like basis with denominator shape parameters given in (22) as , , or , and , to facilitate the following discussion and distinguish the traditional literature. Figure 1 shows images of the DTB-like basis under different shape parameters.
2.3. DTB-Like Curve with Denominator Shape Parameters
Definition 4. Given control points in or , are called a cubic DTB-like curve with two denominator shape parameters and .
Thus, the corresponding DTB-like curve given in (26) has the properties of affine invariance, convex hull, and variation diminishing, which are crucial properties in curve design, given that (22) possesses the properties of partition of unity, nonnegativity, and total positivity. Moreover, we have the following end-point property:
For arbitrary , the curve given in (26) has the end-point interpolation property, and and are the tangent lines of the curve at points and , respectively. From these properties, we can easily find that the curve given in (26) has similar geometric properties to the classical cubic Bézier curve.
The corner cutting algorithm is a steady and high-efficiency algorithm for generating the presented DTB-like curves. We rewrite (26) into the following matrix to develop the algorithm:By rewriting the curve expression into matrix form, we can rapidly obtain this algorithm. Figure 2 shows an example of this algorithm.
In addition, for , we can rewrite (26) into the following form:
For arbitrary fixed , monotonically decreases with respect to the shape parameter . This phenomenon also means that, as shape parameter increases, the generated curve moves in the same direction as the edge . By contrast, as shape parameter decreases, the opposite is true for the generated curve. On the edge , parameter has similar influences. When , as the shape parameters increase or decrease, the generated curve moves to the edge in the same or opposite direction, respectively. Thus, the two denominator shape parameters have a tension effect. Figure 3 shows the generated curves for different shape parameter values.
2.4. Representation of Elliptic and Parabolic Arcs
For , if the control points are , , , and , then the coordinates of the generated curve arewhere is a quarter of an elliptic arc whose center is located at . By constraining the parameter on the desired interval , we can obtain an arc of an ellipse whose starting and ending angles are and , respectively.
Furthermore, for , , if the control points , and with respective coordinates , and , then we obtain the following from (26):which presents a segment of the parabola , .
The discussion indicates that any arc of an ellipse or parabola can be exactly represented by using the proposed DTB-like curves. Figure 4 shows the elliptic and parabolic segments generated by using the cubic DTB-like curves (marked with solid black lines).
3. DT B-Spline-Like Basis Function
3.1. Construction of Trigonometric B-Spline-Like Basis Functions
Given a sequence of knots , we refer to as a knot vector. Let , and , , for arbitrary real numbers , , we construct trigonometric B-spline-like basis functions with the following forms: where , are the DTB-like basis functions given in (22).
To determine coefficient values of the basis functions (32), we add two restricting conditions: (1) including continuity at each knot and (2) forming a partition of unity on . Direct computation shows that
We will provide the following definition to facilitate the following discussion and distinguish the traditional literature.
Definition 5. For arbitrary real numbers , given a knot vector , with the coefficients , (32) expressions are defined as DT B-spline-like basis with two denominator shape parameters.
Remark 6. The DT B-spline-like basis functions are constructed in the spacewhere
In particular, for , and , direct computation yieldsfrom which we can immediately obtain the following explicit expressions of :
Given an equidistant knot vector, we call a uniform DT B-spline-like basis function, and the related knot vector is called a uniform knot vector. On the contrary, given a nonequidistant knot vector, we call and a nonuniform DT B-spline-like basis function and a nonuniform knot vector, respectively. Figure 5 illustrates the DT B-spline-like basis function under different shape parameter values. Before discussing the properties of the DT B-spline-like basis function, we prove the following lemma, which is crucial for the discussion of the curves and surfaces.
Lemma 7. For all possible , , the coefficients have the following properties:
Proof. The aforementioned expressions of the coefficients show that and . Straightforward computation yieldsSimilarly, . Thus, we have and . For and , we have Direct computation provides Furthermore, Finally, These expressions simplify the lemma.
3.2. Properties of the DT B-Spline-Like Basis Functions
Theorem 8. For , we have .
Proof. For , , for . By using Lemma 7, we have
Theorem 9. If , then for .
Proof. If , then for all possible . Thus, from the totally positive values of , we can immediately induce that for .
Theorem 10. For all possible , , the system is linearly independent from .
Proof. For , , letFor , direct computation yields where . Thus, we haveConsidering that , we can obtain the following determinant with respect to the aforementioned linear system of equations:Therefore, for .
Theorem 11. For , , , the system is a totally positive basis of the space span .
Proof. For , , we can obtainwhere and , . Given that the system is the B-basis of the space , according to Theorem 4.2 in [41], we can conclude that is a nonsingular stochastic and totally positive matrix.
For arbitrary , we have for all possible , . Moreover, we can conclude that is stochastic from Lemma 7. Straightforward computation provides This verification shows that is also a totally positive matrix and thus proves the theorem.
Theorem 12. For arbitrary , given a nonuniform knot vector, has continuity at each knot.
Proof. Without loss of generality, we consider the knot . For arbitrary , we have From the aforementioned calculations and Lemma 7, we can easily find that the theorem is established at knot . The case of the continuity at other knots can be discussed similarly.
Theorem 13. For arbitrary , given a nonuniform knot vector, has continuity at each knot.
Proof. We use mathematical induction to prove that the -order derivative of basis function (22) has the following form:When , the derivative of the basis functions (22) isThese forms are satisfied when .
We assume that the aforementioned forms are also satisfied when . Therefore, the -order derivative of the basis functions (22) isBy direct computing, we haveThese forms are satisfied when . In summary, the -order derivative of the basis functions (22) has the form of (55). Finally, we prove that the basis function is continuous at each knot.
Without loss of generality, we first consider the continuity at the knot . From here and Remark 6, direct computation gives Thus, we can immediately conclude that for a uniform knot vector and all . In summary, the theorem is established at knot . The continuity of the basis functions with respect to other knots can be discussed similarly.
3.3. DT B-Spline-Like Curves
Definition 14. Given control points in or and a knot vector , for arbitrary real numbers , ,is called a DT B-spline-like curve with two denominator shape parameters .
For , , we rewrite (60) as the following curve segment:From Theorems 9 and 10, possesses affine invariance and lies in the convex hull of the points for . Theorem 11 shows that possesses the variation diminishing property, indicating that is suitable for curve design. From Theorem 12, we can introduce the following theorem.
Theorem 15. Given a non-uniform knot vector, has continuity at each knot. Furthermore, considering the simple knot , we have
Theorem 16. Given a uniform knot vector, has continuity for all at each knot.
Proof. Without loss of generality, we consider the knot . From Theorem 15 and Remark 6, for all , we have We can immediately conclude that for a uniform knot vector and all . In summary, the theorem is established at knot . The continuity of with respect to other knots can be similarly discussed.
The DT B-spline-like curve proposed in this study possesses two denominator shape parameters . Therefore, when the control points are unchanged, we can easily adjust the proposed curve shape by changing the parameter values. From (61), the shape parameters affect the curve segment , . Therefore, shape parameters affect four curve segments , and affects four curve segments . Furthermore, the shape of the proposed curve can be predicted on the basis of the shape parameter value. As and increase, the coefficients of control points and decrease while the coefficients of and increase. Therefore, as and increase simultaneously, the curve segments tend to . If and increase, the curve segments will tend to control point or , respectively.
Figure 6 shows with different denominator shape parameters. The left figure shows , which is generated by setting all (black lines); the blue line is generated by changing one to 4; and the green line is generated by changing one to 4. The right figure shows that is generated by setting all and .
3.4. DT B-Spline-Like Surface
Definition 17. Given control points in and two knot vectors , , for arbitrary real numbers , ,is called a DT B-spline-like surface with four denominator shape parameters, namely, , where belong to basis function and belong to basis function .
Theorem 18. Given two nonuniform knot vectors and , the DT B-spline-like surface possesses continuity at each knot.
Theorem 19. For , , the DT B-spline-like surface possesses continuity for all at each knot.
Proof. Without loss of generality, we consider the continuity at the region . For , , , , , for , . From Definition 17, we haveWe prove that the continuity of the direction and the continuity of the direction can be similarly discussed. Considering the continuity at the knot , we haveFrom Theorem 16, for all , we haveIn summary, the theorem is established at knot . The continuity of given in (64) with respect to other knots can be similarly discussed.
The given spline surface possesses four local denominator shape parameters, namely, . By changing the value of the shape parameter, we can adjust the shape of flexibly. According to (66), shape parameters , , , affect the surface patch . Therefore, shape parameters affect surface patch , shape parameters affect surface patch , shape parameters affect the surface patch , and shape parameters affect the surface patch . These results show that parameters serve to local control tension in the surface. Increasing locally moves the surface , toward the control polygon. Figure 7 shows with different shape parameters. The figure on the left shows generated when all shape parameters are set . The figure on the right shows generated when all shape parameters are set .
4. DT BB-Like Basis with Three Denominator Shape Parameters
4.1. DT BB-Like Basis
Definition 20. For , , the 10 functions given below are considered new cubic DT BB-like basis functions with three denominator shape parameters , , and over the triangular domain .When , the 10 DT BB-like basis functions degenerate to the DT B-like basis functions (notice ) given in (22). Thus, the DT BB-like basis is the extension of over the triangular domain.
4.2. Properties of the DT BB-Like Basis
In this subsection, we will give important properties of DT BB-like basis functions given in (69).
Theorem 21. The basis functions given in (69) have the following properties.
(a) Nonnegativity: , , .
(b) Partition of unity: .
(c) Symmetry: for all , we have (d) Linear independence: are linearly independent.
Proof. We include the proofs of (a) and (d) only. The remaining cases can be directly computed.
(a) For any and , is apparent. Furthermore, for , by directly computing, we have(d) For , we consider a linear combinationFor , we can immediately induce that because the DT B-like basis given in (22) possesses the property of linear independence. In the same way, for , we have , . Finally, we have .
These findings imply the theorem.
Figure 8 shows four images of DT BB-like basis functions with the parameter values .
4.3. DT BB-Like Patch
Definition 22. Let , given control points and a domain triangle . We callthe DT BB-like patch over the triangular domain with three shape denominator parameters .
Next, we provide the properties of the DT BB-like patch given in (73).
(a) Affine invariance and convex hull property: the related patch (73) has affine invariance and convex hull property because the basis functions (69) possess the properties of partition of unity and nonnegativity.
(b) Geometric property at the corner points: straightforward computation gives (c) Corner point tangent plane: let . We have(d) Boundary property: For , is a DTB-like curve given in (26), which can be rewritten as
This discussion suggests that possesses two shape parameters, namely, and , when . In the same way, when and , and possess shape parameters and , respectively. The DTB-like curve can represent elliptic and parabolic arcs when the shape parameters assume a special value (Section 2). Thus, for , the three boundaries of the DT BB-like patch can be arcs of an ellipse, circle, and parabola, respectively. Figure 9 shows DT BB-like patches generated by setting . The patch whose boundaries are elliptic, circular, and parabolic arcs is shown on the left figure. Its control points are The related boundary parametric equations are ; and where . The right figure shows a patch that possesses three same boundaries as a quarter of the unit circle. The related control points are , , , , , , , , , . The three related parametric equations of the boundaries are ; ; and , , , , where .
(e) Shape adjustable property: by changing the value without changing the control net, we can easily adjust the DT BB-like patch flexibly. As the three denominator shape parameters increase, the patch approaches the control net. Therefore, the three denominator shape parameters are three tension parameters. The three boundaries of the DT BB-like patch are determined by the control points , independent of . Therefore, the proposed DT BB-like patch (73) can still change the shape by adjusting the control point without changing the three boundaries. In addition, according to the properties of the boundary curve given in (26) of the proposed DT BB-like patch, the three denominator shape parameters , and are independent of the boundary curves , and , respectively. This phenomenon also means that when changing the value of a certain denominator parameter, one of the three boundary curves will not be affected. Figure 10 shows the DT BB-like patch under different denominator shape parameters.
4.4. De Casteljau-Type Algorithm
We develop a de Casteljau algorithm that can calculate the DT BB-like patch given in (73). For arbitrary , let andThen we rewrite (73) as Furthermore, by settingwe have
For , checking that and (by using ) is easy. Thus (79) and (81) describe the de Casteljau-type algorithm for calculating the DT BB-like patch given in (73).
4.5. Composite Triangular DT BB-Like Patches
Let two DT BB-like patches be If the control points satisfythen (82) and (83) possess the common boundary curve . Therefore, the two patches form a surface with continuity.
Differentiating with respect to yields
For and , we differentiate about . Thus, we have
For the arbitrary value of , refer to [44]. If the vectors defined by (85) to (87) are coplanar, then the conditions for continuous smooth jointing can be achieved. Thus, it can be expressed aswhere and are constants. In summary, we can obtain the following rules:
Thus, we can summarize the following theorems.
Theorem 23. For , if conditions (84) and (89) hold, then the surface connecting (82) with (83) is continuous.
The aforementioned theorem shows that the two DT BB-like patches that connect the conditions are similar to those of the two triangular Bernstein–Bézier-like patches. Detailed content is available in [44]. The only difference is that we can obtain different continuous surfaces by changing the value of the denominator shape parameter. Figure 11 shows the continuous surface under different shape parameters, where .
5. Conclusion
In this study, the proposed DT B-like basis function forms a set of optimal normalized totally positive bases under the framework of the ECC-space, which leads to the DT BB-spline basis function and the DT-BB-like basis function. Curve and surface construction and related discussions are based on these kinds of basis functions. Compared with the traditional Bézier method and the B-spline technique, the proposed method not only retains all the remarkable properties of the traditional method, such as variation diminishing, but can also accurately represent special industrial curves, such as parabolic and elliptical arcs. In special cases, the B-spline curves and surfaces constructed in this study can automatically reach continuity, thereby satisfying the geometric design requirements of high-order continuity, which is not possible in the traditional literature. In addition, this study introduces a new method for the construction of triangular domain patches. This technique can flexibly adjust the patch with parameters and can accurately represent the boundary as a parabolic arc, an elliptical arc, or even an arc surface. Meanwhile, a de Casteljau algorithm is given to efficiently generate triangular domain patches and the connecting conditions of the patch. Although the method in this study solves the problem of the traditional method and has many advantages, the construction of the basis function is only the first step. In the design of a curve or surface that is closely in-line with the requirements of the geometric industry, then many problems still need to be addressed. These issues include the accurate quantitative analysis of the influence of the denominator parameter on the DT B-like curve and the DT BB-spline curve; shape analysis of the DT B-like curve and the DT BB-spline curve (convexity, cusp, inflection point, monotonicity, heavy node, etc.); and the higher-order continuous problem analysis of the DT BB-like patch on the triangular domain, except for the continuity. These issues will be the focus of future research.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by National Natural Science Foundation of China (61861040), the Gansu Education Department Science and Technology Achievement Transformation project (Project no. 2017D-09) and Gansu Province Science and Technology project funding (no. 17YF1FA119).