Continuity conditions for QBézier curves of degree n
 Gang Hu^{1}Email author,
 Cuicui Bo^{1} and
 Xinqiang Qin^{1}
https://doi.org/10.1186/s1366001713903
© The Author(s) 2017
Received: 21 December 2016
Accepted: 1 May 2017
Published: 16 May 2017
Abstract
As a new method of representing curves, QBézier curves not only exhibit the beneficial properties of Bézier curves but also allow effective shape adjustment by changing multiple shape parameters. In order to resolve the problem of not being able to construct complex curves using a single curve, we study the geometric continuity conditions for QBézier curves of degree n. Following the analysis of basis functions and terminal properties of QBézier curves of degree n, the continuity conditions of \(\mathrm{G}^{1}\) and \(\mathrm{G}^{2}\) between two adjacent QBézier curves are proposed. In addition, we discuss the specific steps of smooth continuity for QBézier curves and analyze the influence rules of shape parameters for QBézier curves. The modeling examples show that the proposed method is effective and easy to achieve, making it useful for constructing complex curves for engineering design.
Keywords
MSC
1 Introduction
Parametric curves are not only an important research area in CAD/CAM, but also a powerful tool for shape design. Classical Bézier curves are constructed using Bernstein basis functions, which have a simple structure and are easy to use. The latter have already become one of the most important methods in the CAD/CAM field. However, the shapes of Bézier curves are only determined by the control points, which causes issues for engineering design. In order to overcome this shortcoming, rational Bézier curves can be used, and their shapes can be modified or adjusted by introducing weight factors without changing their control points. However, the introduction of rational fractions produces some other issues, such as complex calculations, cumbersome integrals, and repetitive differentiation [1, 2].
In order to maintain the advantages of the Bézier method and enhance the shape adjustability of the curves, scholars have constructed many nonrational Bézier curves with shape parameters [3–19]. A set of generalized Bernstein basis functions was proposed in [20], constructing a type of QBézier curve with multiple shape parameters. These generalized Bernstein basis functions inherited many of the beneficial properties of Bernstein basis functions, and the QBézier curves also inherited many beneficial properties of Bézier curves. Moreover, the QBézier curve also had flexible shape adjustability, with the shape of the curve being easily modified by changing shape parameters, thus creating complex curves with more degrees of freedom. Therefore, QBézier curves can be widely used in various CAD/CAM systems.
As the QBézier curve is a type of polynomial curve, it has inevitably inherited the instability that calculations of highorder polynomials suffer from. Consequently, the control of the polygon on the curve will be weakened when the degree of the QBézier curve is too high; by contrast, a lower degree cannot express a complex curve any better. Based on this, in order to describe a QBézier curve with more extensive applications in the CAD/CAM field, in this paper we derive the geometric continuity conditions between two adjacent QBézier curves by analyzing the basis functions and terminal properties of the QBézier curve. The resultant curves are flexible enough to be used in a wide variety of engineering design applications.
2 The family of QBézier curves
2.1 Generalized Bernstein basis functions
Definition 1
It can be easily proved that the generalized Bernstein basis functions \(b_{i,n}(t)\) of degree n have many properties similar to those of classical Bernstein basis functions of degree n, such as nonnegativity, partition of unity, symmetry, etc. [20]. Specifically, when the shape parameters are \(\lambda_{i} = 0\) (\(i = 1,2, \ldots,n\)), the generalized Bernstein basis functions of degree n degenerate into the classical Bernstein basis functions of degree n.
Theorem 1
The generalized Bernstein basis functions of degree n, as shown in (1), associated with the shape parameters \(\lambda_{i}\) (\(i = 1,2, \ldots,n\)), are linearly independent.
Proof
Since the Bernstein basis functions of degree \(n+1\) are linearly independent, we obtain \(\beta_{i} = 0\) (\(i = 0,1, \ldots,n + 1\)). Thus, it is obvious that \(\alpha_{i} = 0\) for \(i = 0,1, \ldots,n\), meaning that \(b_{i,n}(t)\) (\(i = 0,1, \ldots,n\)) are linearly independent. □
2.2 Definition and properties of QBézier curve
Definition 2
Theorem 2
Proof
For the terminal properties (7)(12) of the basis functions, as well as the definition of the QBézier curve, we can produce the terminal properties (6) of the QBézier curve, thus, proving Theorem 2. □
3 \(\mathrm{G}^{1}\) and \(\mathrm{G}^{2}\) smooth continuity conditions for QBézier curves
Given two adjacent QBézier curves \(\boldsymbol{r}_{1}(t) = \sum_{i = 0}^{n} \boldsymbol{P}_{i}b_{i,n}(t)\) with control points \(\boldsymbol{P}_{i}\) (\(i = 0,1, \ldots,n\)) and \(\boldsymbol{r}_{2}(t) = \sum_{i = 0}^{m} \boldsymbol{P}_{i}^{*}b_{i,m}(t)\) with control points \(\boldsymbol{P}_{i}^{*}\) (\(i = 0,1, \ldots,m\)), the continuity conditions \(\mathrm{G}^{1}\) and \(\mathrm{G}^{2}\) for QBézier curves are shown by the following equations.
3.1 Smooth continuity conditions of \(\mathrm{G}^{1}\) for QBézier curves
Theorem 3
Proof
By combining with (14), (15) can be expressed in the form of (13), thus proving Theorem 3. □
The geometric significance of \(\mathrm{G}^{1}\) continuity for two QBézier curves at the joint is that the control points \(\boldsymbol{P}_{n  1}\), \(\boldsymbol{P}_{n}\) (\(= \boldsymbol{P}_{0}^{*}\)) and \(\boldsymbol{P}_{1}^{*}\) should have collinear ordering when \(\boldsymbol{r}_{1}(t)\) and \(\boldsymbol{r}_{2}(t)\) combine.
Now, the continuity conditions of \(\mathrm{G}^{1}\) degrade into the corresponding \(\mathrm{C}^{1}\) continuity conditions.
3.2 Smooth continuity conditions of \(\mathrm{G}^{2}\) for QBézier curves
Theorem 4
Proof
In conclusion, if the two QBézier curves satisfy (13) and (23) simultaneously, then they reach \(\mathrm{G}^{2}\) smooth continuity at the joint, thus proving Theorem 4. □
This describes the smooth continuity conditions of \(\mathrm{C}^{2}\) for QBézier curves. Note that the \(\mathrm{C}^{2}\) smooth continuity conditions in [20] are incorrect because of an error in the secondorder terminal properties given in [20]. This was because equation (5.9) in [20] was incorrectly stated.
4 Steps and examples of smooth continuity for QBézier curves
Using the smooth continuity conditions between QBézier curves and combining with the flexible shape adjustability of these curves, we now take \(\mathrm{G}^{2}\) smooth continuity as an example to discuss the basic steps of smooth continuity between QBézier curves.
According to the proof of Theorem 3, the steps for smooth continuity for two QBézier curves are given by: ① for any degree n, with shape parameters \(\lambda_{i}\) (\(i = 1,2, \ldots,n\)) and control points \(\boldsymbol{P}_{i}\) (\(i = 0,1, \ldots,n\)) of the initial curve \(\boldsymbol{r}_{1}(t)\), then ② let \(\boldsymbol{P}_{n} = \boldsymbol{P}_{0}^{*}\) so that \(\boldsymbol{r}_{1}(t)\) and \(\boldsymbol{r}_{2}(t)\) have a common control point, which makes the curves reach \(\mathrm{G}^{0}\) continuity; ③ given the degree m and shape parameters \(\lambda_{i}^{ *}\) (\(i = 1,2, \ldots,m\)) of \(\boldsymbol{r}_{2}(t)\), as well as constant \(\alpha > 0\), according to the second equation in (16), calculate the second control point \(\boldsymbol{P}_{1}^{*}\) of \(\boldsymbol{r}_{2}(t)\). ④ On the basis of steps ② and ③, given an arbitrary constant γ, using the third equation in (16), calculate the third control point \(\boldsymbol{P}_{1}^{*}\) of \(\boldsymbol{r}_{2}(t)\). ⑤ Given the remaining \(m2\) control points \(\boldsymbol{P}_{i}^{*}\) (\(i = 3,4, \ldots,m\)) of \(\boldsymbol{r}_{2}(t)\), then we can achieve \(\mathrm{G}^{2}\) smooth continuity between two adjacent QBézier curves.
Obviously, repeating the above smooth continuity steps can achieve \(\mathrm{G}^{2}\) smooth continuity between multiple QBézier curves. A similar process can be used to obtain the steps for \(\mathrm{G}^{1}\) smooth continuity.
Example 1
Example 2
Example 3
Example 4
Example 5
5 Shape adjustment of the smooth continuity between QBézier curves
Compared to classical Bézier curves, QBézier curves have multiple shape parameters, allowing adjustment of the local or global shape. However, altering the control points at the same time as the shape parameters does not affect the smoothness of the curve. In this paper, we will now examine the issue of shape adjustment of \(\mathrm{G}^{1}\) and \(\mathrm{G}^{2}\) continuity using, as an example, the smooth continuity between two QBézier curves. A similar argument can be applied to multiple curves.
Proposition 1
In the case where the control points and \(\mathrm{G}^{1}\) continuity for the splicing curves are not changed, we can adjust the local and global shape of the splicing curves.
Proof
From Theorem 2, \(\mathrm{G}^{1}\) continuity only needs to have the same tangent direction at the common joint between adjacent QBézier curves, but modifying any shape parameters for part of curves simply impacts on the size of the tangent vector without changing the direction. Thus, Proposition 1 is proved. □
Specifically, referring to a QBézier curve with multiple shape parameters, the local shape of the splicing curves can all be modified so long as changing shape parameters. Such a property gives the QBézier curves their flexible shape adjustability.
Example 6
Similarly, we can prove the following proposition.
Proposition 2
Based on \(\mathrm{G}^{2}\) smooth continuity of the splicing curves, the following conclusions can be reached: ① If the control points and \(\mathrm{G}^{2}\) continuity for all the splicing curves are not changed, we can adjust the local shape of the splicing curves by altering shape parameters. ② If \(\mathrm{G}^{2}\) smooth continuity is unchanged, then global shape adjustment of the splicing curves can be achieved by altering shape parameters and control points.
Example 7
6 Conclusions

Our proposed \(\mathrm{G}^{1}\) and \(\mathrm{G}^{2}\) continuity conditions for QBézier curves of degree n extend the conclusions about the continuity condition given in [20].

For a piecewise generalized QBézier curve with \(\mathrm{G}^{1}\) or \(\mathrm{G}^{2}\) smooth continuity, we can adjust its global and local shape by changing the shape parameters.

The continuity conditions proposed in this paper are not only intuitive and easy to implement, but also offer more degrees of freedom for the construction of complex curves used in engineering design.
It is worth noting that the proposed methods in this paper are the first to consider the \(\mathrm{G}^{1}\) and \(\mathrm{G}^{2}\) geometric continuity conditions for QBézier curves.
Declarations
Acknowledgements
The authors are very grateful to the referees for their helpful suggestions and comments which have improved the paper. This work is supported by the National Natural Science Foundation of China (No. 51305344, No. 11501443, No. 11626185). This work is also supported by the Research Fund of Shaanxi, China (No. 2014K0522), the Research Fund of Department of Education of Shaanxi, China (No. 15JK1535).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Farin, G: Curves and Surfaces for CAGD: A Practical Guide, 5th edn. Academic Press, San Diego (2002) Google Scholar
 Mamar, E: Shape preserving alternatives to the rational Bézier model. Comput. Aided Geom. Des. 18(1), 3760 (2001) View ArticleGoogle Scholar
 Chen, Q, Wang, G: A class of Bézierlike curves. Comput. Aided Geom. Des. 20(1), 2939 (2003) View ArticleMATHGoogle Scholar
 Oruc, H, Phillips, G: QBernstein polynomials and Bézier curves. J. Comput. Appl. Math. 151(1), 112 (2003) MathSciNetView ArticleMATHGoogle Scholar
 Han, X: Cubic trigonometric polynomial curves with a shape parameter. Comput. Aided Geom. Des. 21(6), 535548 (2004) MathSciNetView ArticleMATHGoogle Scholar
 Han, X, Ma, Y, Huang, X: The cubic trigonometric Bézier curve with two shape parameters. Appl. Math. Lett. 22(3), 226231 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Farin, G: Class a Bézier curves. Comput. Aided Geom. Des. 23(7), 573581 (2006) MathSciNetView ArticleMATHGoogle Scholar
 Xiang, T, Liu, Z, Wang, W, Jiang, P: A novel extension of Bézier curves and surfaces of the same degree. J. Inf. Comput. Sci. 7(10), 20802089 (2010) Google Scholar
 Hu, G, Cao, H, Zhang, S, Wei, W: Developable Bézierlike surfaces with multiple shape parameters and its continuity conditions. Appl. Math. Model. 45(C), 728747 (2017) View ArticleGoogle Scholar
 Qin, X, Hu, G, Zhang, N, Shen, X, Yang, Y: A novel extension to the polynomial basis functions describing Bézier curves and surfaces of degree n with multiple shape parameters. Appl. Math. Comput. 223(C), 116 (2013) MathSciNetMATHGoogle Scholar
 Rachid, A, Yusuke, S, Taishin, N: GelfondBézier curves. Comput. Aided Geom. Des. 30(2), 199225 (2013) View ArticleMATHGoogle Scholar
 Yang, L, Zeng, X: Bézier curves and surfaces with shape parameter. Int. J. Comput. Math. 86(7), 12531263 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Hu, G, Qin, X, Ji, X, Wei, G, Zhang, S: The construction of λμBspline curves and its application to rotational surfaces. Appl. Math. Comput. 266, 194211 (2015) MathSciNetGoogle Scholar
 Liu, Z, Chen, X, Jiang, P: A class of generalized Bézier curves and surfaces with multiple shape parameters. J. Comput.Aided Des. Comput. Graph. 22(5), 838844 (2010) View ArticleGoogle Scholar
 Din, M, Wang, G: TBézier curves based on algebraic and trigonometric polynomials. Chinese J. Comput. 27(8), 10211026 (2004) MathSciNetGoogle Scholar
 Zhang, J, Frank, L, Zhang, H: Unifying Ccurves and Hcurves by extending the calculation to complex numbers. Comput. Aided Geom. Des. 22(9), 865883 (2005) MathSciNetView ArticleMATHGoogle Scholar
 Tan, J, Wang, Y, Li, Z: Subdivision algorithm, connection and applications of cubic HBézier curves. J. Comput.Aided Des. Comput. Graph. 21(5), 584588 (2009) Google Scholar
 Han, X, Huang, X, Ma, Y: Shape analysis of cubic trigonometric Bézier curves with a shape parameter. Appl. Math. Comput. 217(6), 25272533 (2010) MathSciNetMATHGoogle Scholar
 Qin, X, Hu, G, Yang, Y, Wei, G: Construction of PH splines based on HBézier curves. Appl. Math. Comput. 238(6), 460467 (2014) MathSciNetMATHGoogle Scholar
 Han, X, Ma, Y, Huang, X: A novel generalization of Bézier curve and surface. J. Comput. Appl. Math. 217(1), 180193 (2008) MathSciNetView ArticleMATHGoogle Scholar