Skip to main content
Log in

Continuity conditions for tensor product Q-Bézier surfaces of degree (\(m,\, n\))

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript


Based on a kind of Q-Bézier surfaces with shape parameters, the basic properties of the surfaces and the geometric significance of the shape parameters are analyzed. To resolve the problem of shape control and adjustment of composite surfaces, the continuity conditions for Q-Bézier surfaces of degree (m, n) are investigated. Taking advantage of the terminal properties of generalized Bernstein basis functions, we derive the conditions of \({G}^{1}\) and \({G}^{2}\) continuity between two adjacent Q-Bézier surfaces. In addition, the specific steps of smooth continuity between Q-Bézier surfaces and the shape adjustment function of shape parameters for composite surfaces are discussed. The modeling examples show that the proposed smooth continuity conditions are not only intuitive and easy to implement, but also greatly enhance the shape adjustability, which provide a useful method for constructing complex surfaces in engineering design.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others


  • Cao J, Wang GZ (2007) An extension. An extension of Bernstein-Bézier surface over the triangular domain. Progr Nat Sci 17(3):352–357

    Article  MATH  Google Scholar 

  • Cao J, Wang GZ (2008) A note on class a Bézier curves. Comput Aided Geom Des 25(7):523–528

    Article  MATH  Google Scholar 

  • Chen J (2013) Quasi-Bézier curves with shape parameters. J Appl Math Article ID:171392 1-9

  • Chen J, Wang GJ (2011) A new type of the generalized Bézier curves. Appl Math A J Chin Univ 26(1):47–56

    Article  MATH  Google Scholar 

  • Cheng SH, Zhang LT, Zhou YW et al (2010) Intersection point G\(^{1}\) continuity of three patches of adjacent cubic NURBS surfaces. Int Conf Comput Appl Syst Model 2(2010):72–76

    Google Scholar 

  • Chu LC, Zeng XM (2014) Constructing curves and triangular patches by Beta functions. J Comput Appl Math 260:191–200

    Article  MathSciNet  MATH  Google Scholar 

  • Degen WLF (1990) Explicit continuity conditions for adjacent Bézier surface patches. Comput Aided Geom Des 7(20):181–189

    Article  MATH  Google Scholar 

  • Farin G (2002) Curves and surfaces for CAGD: a practical guide, 5th edn. Academic Press, San Diego

    Google Scholar 

  • Farin G (2006) Class a Bézier curves. Comput. Aided Geom. Des. 23(7):573–581

    Article  MathSciNet  MATH  Google Scholar 

  • Han XA, Ma YC, Huang XL (2008) A novel generalization of Bézier curve and surface. J Comput Appl Math 217(1):180–193

    Article  MathSciNet  MATH  Google Scholar 

  • Hu G, Ji XM, Guo L et al (2014) The quartic generalized C-Bézier surface with multiple shape parameters and continuity condition. Mech Sci Technol Aerosp Eng 33(9):1359–1363

    Google Scholar 

  • Hu G, Cao HX, Zhang SX, Wei W (2017) Developable Bézier-like surfaces with multiple shape parameters and its continuity conditions. Appl Math Model 45(C):728–747

    Article  MathSciNet  Google Scholar 

  • Konno K, Tokuyama Y, Chiyokura H (2001) A G\(^{1}\) connection around complicated curve meshes using C\(^{1}\) NURBS boundary gregory Patches. Comput Aided Geom Des 33(4):293–306

    Article  Google Scholar 

  • Liu D (1990) GC\(^{1}\) continuity conditions between two adjacent rational Bézier surface patches. Comput Aided Geom Des 7(1):151–163

    Article  MATH  Google Scholar 

  • Liu Z, Chen X, Jiang P (2010) A class of generalized Bézier curves and surfaces with multiple shape parameters, \(\vartheta \). J Comput Aided Des Comput Graph 22(5):838–844

    Article  Google Scholar 

  • Mamar E (2001) Shape preserving alternatives to the rational Bézier model. Comput Aided Geom Des 18(1):37–60

    Article  Google Scholar 

  • Oruc H, Phillips GH (2003) q-Bernstein polynomials and Bézier curves. J Comput Appl Math 151(1):1–12

    Article  MathSciNet  MATH  Google Scholar 

  • Piegl L, Tiller W (1997) The NURBS book, 2nd edn. Springer, New York

    Book  MATH  Google Scholar 

  • Qin XQ, Hu G, Zhang NJ et al (2013) 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):1–16

    MathSciNet  MATH  Google Scholar 

  • Wang WT, Wang GZ (2005) Bézier curve with shape parameter. J Zhejiang Univ Sci A 6(6):497–501

    Article  Google Scholar 

  • Wang GJ, Wang GZ, Zheng JM (2001) Computer aided geometric design. Springer, Beijing

    Google Scholar 

  • Xiang TN, Liu Z, Wang WF (2010) A novel extension of Bézier curves and surfaces of the same degree. J Inf Comput Sci 7(10):2080–2089

    Google Scholar 

  • Yan LL, Liang JF (2011) An extension of the Bézier model. Appl Math Comput 218(6):2863–2879

    MathSciNet  MATH  Google Scholar 

  • Yang LQ, Zeng XM (2009) Bézier curves and surfaces with shape parameter. Int J Comput Math 86(7):1253–1263

    Article  MathSciNet  MATH  Google Scholar 

  • Zhu YP, Han XL (2013) A class of \(\alpha \beta \gamma \)-Bernstein-Bézier basis functions over triangular. Appl Math Comput 220:446–454

    MathSciNet  Google Scholar 

Download references


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 (Nos. 51305344, 11626185). This work is also supported by the Project Supported by Natural Science Basic Research Plan in Shaanxi Province of China (No. 2017JM5048).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gang Hu.

Additional information

Communicated by Cristina Turner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, G., Bo, C. & Qin, X. Continuity conditions for tensor product Q-Bézier surfaces of degree (\(m,\, n\)). Comp. Appl. Math. 37, 4237–4258 (2018).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification