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Quasi-quintic trigonometric Bézier curves with two shape parameters

  • Xuewen Tan
  • Yuanpeng ZhuEmail author
Article
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Abstract

In this work, we propose a family of six new quasi-quintic trigonometric blending functions with two shape parameters. Based on these blending functions, a class of quasi-quintic trigonometric Bézier curve is proposed, which has some properties analogous to the classical quintic Bézier curves. For the same control points, the resulting quasi-quintic trigonometric Bézier curves can be closer to the control polygon than the classical quintic Bézier curves. The shape of the quasi-quintic trigonometric Bézier curves can be flexibly adjusted by altering the values of the two shape parameters without changing their control points. Under the \({C^2}\) smooth connection conditions, the resulting composite quasi-quintic trigonometric Bézier curves can automatically reach \({C^2} \cap F{C^3}\) continuity.

Keywords

Trigonometric blending functions Trigonometric Bézier curves Shape parameters Smoothness 

Mathematics Subject Classification

65D07 65D17 

Notes

Acknowledgements

The research is supported by the National Natural Science Foundation of China (Grant No. 61802129), the Postdoctoral Science Foundation of China (Grant No. 2015M571931), the Fundamental Research Funds for the Central Universities (Grant No. 2017MS121) and the Natural Science Foundation Guangdong Province, China (Grant No. 2018A030310381).

References

  1. Bashir U, Abbas M, Awang M, Ali J (2013) A class of quasi-quintic trigonometric Bézier curve with two shape parameters. ScienceAsia 39S:11–15CrossRefGoogle Scholar
  2. Bashir U, Ali JM (2016) Rational cubic trigonometric Bézier curve with two shape parameters. Comput Appl Math 35:285–300MathSciNetCrossRefGoogle Scholar
  3. Chen QY, Wang GZ (2003) A class of Bézier-like curves. Comput Aided Geom Des 20(1):29–39CrossRefGoogle Scholar
  4. Costantini P, Manni C (2003) Geometric construction of spline curves with tension properties. Comput Aided Geom Des 20:579–599MathSciNetCrossRefGoogle Scholar
  5. Farin G (1993) Curves and surfaces for computer aided geometric design. Academic Press, San DiegozbMATHGoogle Scholar
  6. Han XL (2004) Cubic trigonometric polynomial curves with a shape parameter. Comput Aided Geom Des 21:535–548MathSciNetCrossRefGoogle Scholar
  7. Han XA, Ma YC, Huang XL (2009) The cubic trigonometric Bézier curve with two shape parameters. Appl Math Lett 22:226–231MathSciNetCrossRefGoogle Scholar
  8. Han XA, Ma YC, Huang XL (2010) Shape analysis of cubic trigonometric Bézier curves with a shape parameter. Appl Math Comput 217:2527–2533MathSciNetzbMATHGoogle Scholar
  9. Han XL, Zhu YP (2012) Curve construction based on five trigonometric blending functions. BIT 52:953–979MathSciNetCrossRefGoogle Scholar
  10. Han XL, Zhu YP (2014) Total positivity of the cubic trigonometric Bézier basis. J Appl Math 2014:1–5zbMATHGoogle Scholar
  11. Hoffmann M, Juhász I, Károlyi G (2014) A control point based curve with two exponential shape parameters. BIT 54:691–710MathSciNetCrossRefGoogle Scholar
  12. Hoschek J, Lasser D (1993) Fundamentals of computer aided geometric design. A.K. Peters, WellesleyzbMATHGoogle Scholar
  13. Mainar E, Penâ JM, Sánchez-Reyes J (2001) Shape preserving alternatives to the rational Bézier model. Comput Aided Geom Des 18(1):37–60Google Scholar
  14. Miscro MY, Ramli A, Ali JM (2017) Quintic trigonometric Bézier curve with two shape parameters. Sains Malays 46:825–831CrossRefGoogle Scholar
  15. Piegl L, Tiller W (1995) The NURBS book. Springer, New YorkCrossRefGoogle Scholar
  16. Wu XQ, Han XL, Luo S (2008) Quadratic trigonometric polynomial Bézier curves with a shape parameter. J Eng Graph 29:82–87Google Scholar
  17. Xie W, Li J (2018) \(C^3\) cubic trigonometric B-spline curves with a real parameter. J Natl Sci Found Sri 46(1):89–94Google Scholar
  18. Yan LL (2016) Cubic trigonometric nonuniform spline curves and surfaces. Math Probl Eng 2016:1–9MathSciNetzbMATHGoogle Scholar
  19. Zhu YP, Han XL (2015) New Trigonometric Basis Possessing Exponential Shape Parameters. J Comput Math 33:642–684MathSciNetCrossRefGoogle Scholar
  20. Zhu YP, Han XL, Jing Han (2012) Quartic trigonometric Bézier curves and shape preserving interpolation curves. J Comput Inform Syst 8:905–914Google Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.School of Business AdministrationSouth China University of TechnologyGuangzhouPeople’s Republic of China
  2. 2.Department of MathematicsSouth China University of TechnologyGuangzhouPeople’s Republic of China

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