G^{2} continuity conditions for generalized Bézierlike surfaces with multiple shape parameters
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Abstract
In order to tackle the problem of shape design and shape adjustment of complex surfaces in engineering, continuity conditions between generalized Bézierlike surfaces with multiple shape parameters are studied in this paper. Firstly, the geometric model of the generalized Bézierlike surfaces is built by blending a number of Bézierlike curves with independent shape parameters. Secondly, based on the terminal properties and linear independence of Bernsteinlike basis functions, the conditions for G^{2} continuity between two adjacent generalized Bézierlike surfaces are derived, and then simplified by choosing appropriate shape parameters. Finally, some properties and applications of the smooth continuity between generalized Bézierlike surfaces are discussed. The modeling examples show that the proposed method is effective and easy to implement, which can greatly improve the ability to construct complex surfaces by using the generalized Bézierlike surfaces.
Keywords
Bernsteinlike basis functions generalized Bézierlike surfaces shape parameter G^{2} continuity conditionsMSC
65D07 65D10 65D17 65D18 68U05 68U071 Introduction
In the practical applications, since the appearance modelings of many products in industry are quite complex, they often cannot be described by a single surface in many cases. Thus, there is a need to design such products using adjacent surfaces. The smooth continuity among multiple surface patches with certain smooth constraints is usually used to achieve the appearance design of complex products. The ultimate aim of smooth continuity is to make adjacent surface patches satisfy certain smooth conditions so that the complex piecewise surface composed of these surface patches has global smoothness visually. Parametric surfaces, which are not only the standard form for the mathematical description of product appearance in CAD/CAM, but also a powerful tool for various shape designs and geometric representations, have received much attention since the 1960s. Thus the smooth continuity between parametric surfaces is an important method to construct complex surfaces and also significant research in the CAD/CAM system [1, 2].
There are two kinds of measuring standards established for the continuity of piecewise parametric surfaces [3]: (1) parametric continuity, which is usually called \(\mathrm{C}^{n}\) continuity; (2) geometric continuity, or \(\mathrm{G}^{n}\) continuity for short. However, the parametric continuity of a surface is relevant to its selected parameter and is usually valid under certain ones. In addition, if the common boundary of two adjacent surfaces is irregular, even though the two surfaces satisfy C^{1} continuity at the joint, it does not necessarily mean that the two surfaces possess a common tangent plane at any point on their common boundary. That is, the piecewise surface composed of the two adjacent surfaces may not be smooth at the joint. So the smooth continuity between surfaces cannot be exactly measured only by the parametric continuity [3]. In addition, the smoothness of surfaces is a kind of geometric characteristic. Therefore, in constructing smooth piecewise surfaces, people usually consider only geometric continuity, namely, \(\mathrm{G}^{n}\) continuity, which is irrelevant to the selected parameters. In practical application, adjacent surfaces usually only need to reach G^{1} continuity, which means that adjacent surfaces need to possess a common tangent plane or surface normal at any point on their common boundary; while in some situations with high demand for smoothness, adjacent surfaces are required to reach G^{2} continuity (namely, curvature continuity) [3]. At present, owing to their simple and intuitive definition and some outstanding properties, Bézier parametric surfaces have long been one of the important methods for representing surfaces in the CAD/CAM system. However, the Bézier model still has a weakness that the shape of a Bézier surface is uniquely determined by its control mesh points. In order to overcome this weakness, scholars proposed rational Bézier surfaces and NURBS surfaces, whose shapes can be modified or adjusted by changing their weight factors on the condition of given control mesh points. However, the introduction of rational fractions also brought in some other drawbacks such as complex calculation, inconvenience for integrals, higherorder expressions resulting from repeated differentiation, etc. [4]. In addition, though the smooth continuity technologies of Bézier, rational Bézier and NURBS surfaces, which can be used to construct various complex surfaces, have been widely researched in [5, 6, 7, 8, 9, 10], the drawbacks of these surfaces also exist in the piecewise surfaces composed of them. All of these might get the design of complex surfaces in trouble (such as the problem of shape adjustment).
In order to reserve the advantages of Bézier model and improve the shape adjustability of curves and surfaces, scholars have constructed many Bézier curves and surfaces with shape parameters [11, 12, 13, 14, 15, 16, 17, 18]. The common features of these curves and surfaces are as follows: (1) they inherit most of properties of Bézier curves and surfaces; (2) they all have shape parameters used to adjust the shape of these curves and surfaces handily; (3) the absence of rational fractions in their expressions makes them simpler than rational Bézier and NURBS curves and surfaces. Thus these curves and surfaces have extensive applications in describing complex curves and surfaces. However, the expressions of these Bézier curves and surfaces with shape parameters are polynomials; and consequently, they face the problem of smooth continuity in constructing complex curves and surfaces. Therefore, when researchers defined their curves and surfaces with shape parameters in [11, 12, 13, 14, 15], they also further studied the C^{1}, C^{2} or G^{1}, G^{2} continuity conditions of their proposed curves, but the continuity conditions of these surfaces have not been studied until now (note: the continuity conditions of the surfaces in [16, 17, 18] are also not studied). Compared with the research on smooth continuity between Bézier curves with shape parameters, the corresponding research on Bézier surfaces with shape parameters has not been extensively done and the relevant research results are relatively few. In this paper, we make some improvements to the Bézierlike surfaces in [17] and construct a kind of highorder generalized Bézierlike surfaces associated with multiple shape parameters. To improve the ability of describing complex surfaces by using the proposed surfaces, we lay emphasis on the study of G^{2} continuity conditions of these surfaces.
The remainder of the paper is organized as follows. The definition of generalized Bézierlike surfaces is given in Section 2. In Section 3, we propose the G^{2} continuity conditions for generalized Bézierlike surfaces. Some examples of G^{2} smooth continuity between generalized Bézierlike surfaces are given in Section 4. In Section 5, we discuss the shape adjustment of piecewise surfaces. At last, some conclusions are given in Section 6.
2 Generalized Bézierlike surfaces with shape parameters
2.1 Definition of Bernsteinlike basis functions
Definition 1
2.2 Construction of generalized Bézierlike surfaces
Similar to the form of classical tensorproduct Bézier surfaces (but slightly different), a kind of generalized Bézierlike surfaces associated with \(m+2\) shape parameters can be constructed by blending \(m+1\) Bézierlike curves with independent shape parameters.
Definition 2
Remark 1
The generalized Bézierlike surfaces inherited most of the properties of classical Bézier surface, such as angular point interpolation property, boundary property, degeneracy, symmetry, convex hull property, geometric, affine invariance, etc.
Remark 2
The generalized Bézierlike surfaces have the following advantages: on the condition of keeping the control mesh points of a surface unchanged, the shape of the surface can also be modified flexibly by changing its shape parameters, and the surface has \(3^{m+2}1\) ways to approximate its control mesh. Especially when all the shape parameters equal 0, the generalized Bézierlike surfaces degenerate into classical Bézier surfaces of degree \((m, n)\).
2.3 Influence rule of the shape parameters on generalized Bézierlike surfaces
In order to adjust the shape of the generalized Bézierlike surfaces effectively, the influence rule of the shape parameters on them is analyzed in details in this section. In other words, how will the shape of the surfaces change when one or multiple parameters change is particularly demonstrated to enable designers to modify the shape of the surfaces purposefully and efficiently.
Proposition 1
 (a)
the generalized Bézierlike surfaces will get nearer to (or farther away from) their control mesh when the shape parameter λ increases (or decreases).
 (b)
changing the value of the shape parameter λ, the position and shape of the boundary curves \(\boldsymbol{S}(0,v;\lambda,\gamma_{i})\) and \(\boldsymbol{S}(1,v;\lambda,\gamma_{i})\) as well as the position of the four corners of the generated surfaces will keep unchanged, while the position and shape of the boundary curves \(\boldsymbol{S}(u,0;\lambda,\gamma_{i})\) and \(\boldsymbol{S}(u,1;\lambda,\gamma_{i})\) will change.
Proposition 2
 (a)
with the increase (or decrease) of the shape parameters \(\gamma_{i}\) (\(i = 0,1, \ldots,m\)), the generalized Bézierlike surfaces will gradually get nearer to (or farther away from) their control mesh along the control polygon composed of the points \(\boldsymbol{P}_{i, j}\) (\(j = 0,1, \ldots,n\)). Therefore the shape parameters \(\gamma_{i}\) (\(i = 0,1, \ldots,m\)) mainly control the shape of the generalized Bézierlike surfaces near the control points \(\boldsymbol{P}_{i, 0},\boldsymbol{P}_{i, 1}, \ldots,\boldsymbol{P}_{i, n}\).
 (b)
with the single change of the shape parameter \(\gamma_{0}\) (or \(\gamma_{m}\)), the position and shape of the boundary curve \(\boldsymbol{S}(0,v;\lambda,\gamma_{i})\) (or \(\boldsymbol{S}(1,v;\lambda,\gamma_{i})\)) of generalized Bézierlike surfaces will change, while the position and shape of the other three boundary curves as well as the position of the four corners of the surfaces remain unchanged. With the change of the shape parameters \(\gamma_{i}\) (\(i = 1,2, \ldots,m  1\)), the position and shape of the four boundary curves as well as the position of the four corners of the generated surfaces remain unchanged.
Remark 3
 (a)
with the simultaneous increase (or decrease) of the shape parameters λ, \(\gamma_{i}\) (\(i = 0,1, \ldots,m\)), the generalized Bézierlike surfaces gradually get nearer to (or farther away from) their control mesh.
 (b)
fixing the values of the shape parameters λ, \(\gamma_{0}\), \(\gamma_{m}\), the shape of the generalized Bézierlike surfaces can be adjusted by changing the shape parameters \(\gamma_{i}\) (\(i = 1,2, \ldots,m  1\)) with the four boundary curves of the surfaces remaining unchanged.
3 G^{2} continuity conditions for generalized Bézierlike surfaces
3.1 Smooth continuity in the u direction
Theorem 1
Proof
If \(\boldsymbol{S}_{1}(u,v;\lambda_{1},\gamma_{i,1})\) and \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) need to reach G^{2} smooth continuity in the u direction at the joint, they are required to reach G^{1} smooth continuity at the joint first. In other words, the two surfaces need to possess a common tangent plane or surface normal at any point on their common boundary [1, 2, 3, 19, 20].
To sum up, if the two surfaces \(\boldsymbol{S}_{1}(u,v;\lambda_{1},\gamma_{i,1})\) and \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) satisfy (7), (11) and (18) simultaneously, they reach G^{2} smooth continuity in the u direction at the joint, and Theorem 1 gets proved. Obviously, if the two surfaces satisfy both (7) and (11), they reach G^{1} smooth continuity in the u direction at the joint. □
3.2 Smooth continuity in the direction of u and v
Theorem 2
Proof
To sum up, if the two surfaces \(\boldsymbol{S}_{1}(u,v;\lambda_{1},\gamma_{i,1})\) and \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) satisfy both (20) and (23), the two surfaces reach G^{2} smooth continuity in the direction of u and v at the joint, and Theorem 2 gets proved. Obviously, if the shape parameters and control mesh points of the two surfaces satisfy (20), the two surfaces reach G^{1} smooth continuity in the direction of u and v at the joint. □
3.3 Smooth continuity in the v direction
Similar to the G^{2} continuity conditions in the u direction between generalized Bézierlike surfaces, the following G^{2} continuity conditions in the v direction can be proved to be correct.
Theorem 3
Proof
The proof process of this theorem is similar to that of Theorem 1 and Theorem 2, so it is not covered here. □
Obviously, when all the shape parameters in \(\mbox{Theorem~1} \sim \mbox{Theorem~3}\) are equal to 0, these continuity conditions above degrade into the corresponding G^{2} continuity conditions for highorder classical Bézier surfaces; when \(\gamma_{0,1} = \gamma_{1,1} = \cdots = \gamma_{m_{1},1}\) and \(\gamma_{0,2} = \gamma_{1,2} = \cdots = \gamma_{m,2}\), the continuity conditions in \(\mbox{Theorem~1} \sim \mbox{Theorem~3}\) degrade into the corresponding G^{2} continuity conditions for Bézierlike surfaces in [17].
4 Steps and examples of G^{2} smooth continuity between generalized Bézierlike surfaces
4.1 Steps of G^{2} smooth continuity between generalized Bézierlike surfaces
Using the smooth continuity of generalized Bézierlike surfaces with shape adjustability, various complex surfaces can be designed handily and flexibly in engineering. In this section, we take the G^{2} smooth continuity in the u direction between two generalized Bézierlike surfaces as an example (the other two directions can be discussed similarly) to show the basic steps of G^{2} smooth continuity between generalized Bézierlike surfaces. On the basis of the conclusion in Theorem 1, the steps are as follows:
Step 1. According to designing requirement, give the order \(m_{1}\), \(n_{1}\) of the initial generalized Bézierlike surface \(\boldsymbol{S}_{1}(u,v;\lambda_{1},\gamma_{i,1})\) and its control mesh points \(\boldsymbol{P}_{i,j}^{1}\) (\(i = 0,1, \ldots,m_{1}\); \(j = 0,1, \ldots,n_{1}\)) as well as shape parameters \(\lambda_{1}\), \(\gamma_{i,1}\).
Step 2. Let \(m_{1} = m_{2}\), \(\lambda_{1} = \lambda_{2}\) and \(\boldsymbol{P}_{i,n_{1}}^{1} = \boldsymbol{P}_{i,0}^{2}\) (\(i = 0,1, \ldots,m_{1}\)), so that \(\boldsymbol{S}_{1}(u,v;\lambda_{1},\gamma_{i,1})\) and \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) possess a common boundary to reach G^{0} continuity.
Step 3. Give the values of the shape parameter \(\gamma_{i,2}\) and the constant \(f > 0\) as well as the other order \(n_{2}\) of the second generalized Bézierlike surface \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\). On the basis of Step 2, calculate the second row control points \(\boldsymbol{P}_{i,1}^{2}\) (\(i = 0,1, \ldots,m_{1}\)) of \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) according to (11).
Step 4. On the basis of Step 2 and Step 3, calculate the third row control points \(\boldsymbol{P}_{i,2}^{2}\) (\(i = 0,1, \ldots,m_{1}\)) of \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) according to (18).
Step 5. Given the remaining \(n_{2}  2\) control points \(\boldsymbol{P}_{i,j}^{2}\) (\(i = 0,1, \ldots,m_{2}\); \(j = 3,4, \ldots,n_{2}\)) of \(\boldsymbol{S}_{2}(u,v;\lambda_{2},\gamma_{i,2})\) freely, the G^{2} smooth continuity between two generalized Bézierlike surfaces in the u direction is achieved.
Repeating the steps above, G^{2} smooth continuity between multiple generalized Bézierlike surfaces will be achieved.
4.2 Examples of G^{2} smooth continuity between generalized Bézierlike surfaces
From the fourth and fifth equations of (24) and Figure 5, the scaling factor f between their normal vectors can be used to adjust the positions of the second and third row control points of the green surface \(\boldsymbol{S}_{2}\). The bigger (or smaller) the value of f is, the closer (or farther away) the control points \(\boldsymbol{P}_{i,1}^{2}\) (or \(\boldsymbol{P}_{i,2}^{2}\)) move to the control points \(\boldsymbol{P}_{i,0}^{2}\) (or \(\boldsymbol{P}_{i,1}^{2}\)), where \(\boldsymbol{P}_{i,0}^{2}\), \(\boldsymbol{P}_{i,1}^{2}\), \(\boldsymbol{P}_{i,2}^{2}\) (\(i = 0,1,2,3,4\)) are the first, second and third row control points of the green surface \(\boldsymbol{S}_{2}\).
From the smooth continuity result in Figure 5, the piecewise generalized Bézierlike surface composed of \(\boldsymbol{S}_{1}\) and \(\boldsymbol{S}_{2}\) is smooth and continuous at the joint, so the result of smooth continuity is quite good, and thus can better satisfy actual needs.
5 Shape adjustment of piecewise surfaces based on G^{2} smooth continuity
This section will focus on the shape adjustment of piecewise generalized Bézierlike surfaces with G^{2} smooth continuity. For simplicity, we take the smooth continuity between two generalized Bézierlike surfaces as an example to show the shape adjustment of piecewise surfaces. The situations for multiple surface patches can be discussed similarly, so they are not covered here. Compared with the smooth continuity between classical Bézier surfaces, the major advantage of the method in this paper is that apart from modifying control mesh points, we can also adjust the local or global shape of a piecewise surface by modifying its shape parameters with the overall smoothness of the surface remaining unchanged.
Proposition 3
 (a)
For a piecewise generalized Bézierlike surface with G ^{2} smooth continuityin the u direction, we can adjust the global shape of the surface by changing the shape parameters \(\lambda_{1}\) and \(\lambda_{2}\) simultaneously, but we cannot adjust its local shape by changing its shape parameters.
 (b)
For a piecewise generalized Bézierlike surface with G ^{2} smooth continuity in the direction of u and v, we can adjust its global shape by changing the shape parameters \(\lambda_{1}\), \(\gamma_{i,2}\) (\(i = 0,1, \ldots,m_{2}\)) simultaneously; meanwhile we can also adjust its local shape by changing the shape parameters \(\gamma_{i,2}\) (\(i = 3,4, \ldots,m_{2}\)).
 (c)
For a piecewise generalized Bézierlike surface with G ^{2} smooth continuity in the v direction, we can adjust its global shape by changing the shape parameters \(\gamma_{i,1}\) (\(i = 0,1, \ldots,m_{1}\)) and \(\gamma_{i,2}\) (\(i = 1,2, \ldots,m_{2}\)); meanwhile we can also adjust its local shape by changing the shape parameters \(\gamma_{i,1}\) (\(i = 0,1, \ldots,m_{1}  3\)) or \(\gamma_{i,2}\) (\(i = 3,4, \ldots,m_{2}\)).
Proof
(a) According to the equation \(\lambda_{1} = \lambda_{2}\) in (5), when we change the value of the parameter \(\lambda_{1}\) to adjust the shape of the surface \(\boldsymbol{S}_{1}\), the value of the parameter \(\lambda_{2}\) will also change necessarily to maintain the G^{2} smooth continuity, so does the shape of the surface \(\boldsymbol{S}_{2}\). Therefore we can adjust the global shape of the piecewise surface by changing the shape parameters \(\lambda_{1}\) and \(\lambda_{2}\) simultaneously.
(b) According to the equation \(\lambda_{1} = \gamma_{0,2} = \gamma_{1,2} = \gamma_{2,2}\) in (19), when we change the value of the parameter \(\lambda_{1}\) to adjust the shape of the surface \(\boldsymbol{S}_{1}\), the value of the shape parameters \(\gamma_{0,2}\), \(\gamma_{1,2}\), \(\gamma_{2,2}\) and the shape of the surface \(\boldsymbol{S}_{2}\) need to change necessarily to maintain the G^{2} smooth continuity. So we can change the shape parameters \(\lambda_{1}\) and \(\gamma_{i,2}\) (\(i = 0,1, \ldots,m_{2}\)) simultaneously to adjust the global shape of the piecewise surface. Furthermore, as the constraint equations for G^{2} smooth continuity in (19) do not contain the shape parameters \(\gamma_{i,2}\) (\(i = 3,4, \ldots,m_{2}\)), we can modify these parameters to adjust the shape of the surface \(\boldsymbol{S}_{2}\) so as to realize the local shape adjustment of the piecewise surface. In addition, by the proving method of conclusion (1), it can be proved that the shape parameters \(\lambda_{2}\), \(\gamma_{i,1}\) (\(i = 0,1, \ldots,m_{1}\)) cannot be used to adjust the local or global shape of the piecewise surface.
(c) Obviously, conclusion (3) can be proved to be correct by the proving method of conclusion (1) and (2), so its proof is not covered here. □
6 Conclusions

The proposed generalized Bézierlike surfaces of degree \((m, n)\) extend the conclusions of the Bézierlike surfaces given in [17].

For piecewise generalized Bézierlike surfaces with G^{2} smooth continuity, we can adjust their global and local shape by changing their shape parameters.

The G^{2} smooth continuity proposed in this paper is not only intuitive and easy to implement, but also offers more degrees of freedom for constructing complex surfaces in engineering design.
It is worth noting that the proposed methods in this paper are the first to consider G^{2} geometric continuity conditions for the generalized Bézierlike surfaces of degree \((m, n)\).
Notes
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No.51305344, No.11501443, and No.11626185). This work is also supported by the Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (No.2017JM5048), the Key Research and Development Program of Shaanxi Province of China (No.2017GY090).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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