Journal of Zhejiang University-SCIENCE A

, Volume 19, Issue 4, pp 304–314 | Cite as

Curve smoothing using a continuous function

  • Yu-fei Wu
  • Liang He
  • Zhi-dong Li


Current curve smoothing technologies provide a smoothed curve by joining together separate curves that have certain degrees of continuity at junctions. These technologies have found many applications in science and engineering. However, none of them can provide a smoothed curve using a single continuous function for arbitrary segmental curves. This paper reports a new approach that can be used to construct a single continuous function that joins an arbitrary number of different segmental curves, with the required degree of continuity at all junctions. The smoothness of transition at different junctions can be controlled by separate parameters to suit different needs. The combined continuous function can approach the original segmental functions asymptotically or match the original segmental functions “exactly” inside each segment by adjusting the smoothness parameter. This new approach may also find application outside the scope of curve smoothing/curving fitting in the future.


Curve smoothing Curve fitting Transition curve Path planning Tool path 






首次提出用一个连续函数替代原来由任意多非连 续区域函数构成的函数。该方法可视为一种新的 函数光滑算法。


1. 通过引入特殊的区域变量,并用该区域变量替 代原函数自变量的方法,将区域函数改造成在该 区域无限接近原函数而在区域外取值常数的函 数。2. 把所有的区域函数相乘得到一个连续函数 的方程。


1. 由任意多非连续区域函数构成的函数可以改造 成一个连续函数。2. 该连续函数在原非连续边界 的光滑程度可以由各个边界上独立的参数按需 调整。3. 该方法产生的连续函数没有摆动现象, 其形状与原始区域函数无限接近。该方程在数学 上是连续的,同时无限接近原始非连续函数,包 括原来在边界上函数值的非连续。


曲线光滑 区域变量 区域函数 连续函数 

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  1. Cai HH, Wang GJ, 2009. A new method in highway route design: joining circular arcs by a single C-Bézier curve with shape parameter. Journal of Zhejiang University-SCIENCE A, 10(4):562–569. Scholar
  2. CEB (Comite Euro-International du Beton), 1993. CEB-FIP Model Code 1990: Design Code FIB-Fédération Internationale du Beton.CrossRefGoogle Scholar
  3. Emami MM, Arezoo B, 2010. A look-ahead command generator with control over trajectory and chord error for NURBS curve with unknown arc length. Computer-Aided Design, 42(7):625–632. Scholar
  4. Erkorkmaz K, Altintas Y, 2001. High speed CNC system design. Part I: jerk limited trajectory generation and quintic spline interpolation. International Journal of Machine Tools and Manufacture, 41(9):1323–1345. Scholar
  5. Feng J, Li Y, Wang Y, et al., 2010. Design of a real-time adaptive NURBS interpolator with axis acceleration limit. The International Journal of Advanced Manufacturing Technology, 48(1–4): 227–241. Scholar
  6. Fraichard T, Scheuer A, 2004. From Reeds and Shepp’s to continuous-curvature paths. IEEE Transactions on Robotics, 20(6):1025–1035. Scholar
  7. Giuffre A, Pinto PE, 1970. Il comportamento del cemento armato per sollecitazioni cicliche di forte intensita. Giornale del Genio Civile, 5:391–408 (in Italian).Google Scholar
  8. Hognestad E, 1951. A Study of Combined Bending and Axial Load in Reinforced Concrete Members. Bulletin Series No. 399, Engineering Experiment Station, University of Illinois, Urbana, USA.Google Scholar
  9. Huh UY, Chang SR, 2014. A G² continuous path-smoothing algorithm using modified quadratic polynomial interpolation. International Journal of Advanced Robotic Systems, 11(2):1–11. Scholar
  10. Lin MT, Tsai MS, Yau HT, 2007. Development of a dynamicsbased NURBS interpolator with real-time look-ahead algorithm. International Journal of Machine Tools and Manufacture, 47(15):2246–2262. Scholar
  11. Liu X, Ahmad F, Yamazaki K, et al., 2005. Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics. International Journal of Machine Tools and Manufacture, 45(4–5): 433–444. Scholar
  12. Magid E, Keren D, Rivlin E, et al., 2006. Spline-based robot navigation. Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, p.2296-2301.CrossRefGoogle Scholar
  13. Nam SH, Yang MY, 2004. A study on a generalized parametric interpolator with real-time jerk-limited acceleration. Computer-Aided Design, 36(1):27–36. Scholar
  14. Pan J, Wu YF, 2014. Analytical modeling of bond behavior between FRP plate and concrete. Composites Part B: Engineering, 61:17–25. Scholar
  15. Park R, Paulay T, 1975. Reinforced Concrete Structures. John Wiley & Sons, New York.CrossRefGoogle Scholar
  16. Pateloup V, Duc E, Ray P, 2010. Bspline approximation of circle arc and straight line for pocket machining. Computer-Aided Design, 42(9):817–827. Scholar
  17. Richard RM, Abbott BJ, 1975. Versatile elastic-plastic stressstrain formula. Journal of the Engineering Mechanics Division, 101:511–515.Google Scholar
  18. Roy R, Bodduna K, Kumari N, et al., 2015. A fast and efficient mesh smoothing algorithm for 3D graphical models using cubic B-splines. In: Information Systems Design and Intelligent Applications. Springer, New Delhi, India, p.467-474. Scholar
  19. Walton DJ, Meek DS, 2009. G2 blends of linear segments with cubics and pythagorean-hodograph quintics. International Journal of Computer Mathematics, 86(9):1498–1511. Scholar
  20. Walton DJ, Meek DS, 2010. Cubic Bézier spiral segments for planar G2 curve design. Proceedings of the 7th International Conference on Computer Graphics, Virtual Reality, Visualisation and Interaction in Africa, ACM, p.21-26.Google Scholar
  21. Wang JB, Yau HT, 2009. Real-time NURBS interpolator: application to short linear segments. International Journal of Advanced Manufacturing Technology, 41(11-12): 1169–1185. Scholar
  22. Wu G, Sun ZY, Wu ZS, et al., 2012. Mechanical properties of steel-FRP composite bars (SFCBs) and performance of SFCB reinforced concrete structures. Advances in Structural Engineering, 15(4):625–636. Scholar
  23. Yang K, Sukkarieh S, 2010. An analytical continuouscurvature path-smoothing algorithm. IEEE Transactions on Robotics, 26(3):561–568. Scholar
  24. Yang X, Chen ZC, 2006. A practicable approach to G1 biarc approximations for making accurate, smooth and nongouged profile features in CNC contouring. Computer-Aided Design, 38(11):1205–1213. Scholar
  25. Yau HT, Wang JB, 2007. Fast Bezier interpolator with realtime look ahead function for high-accuracy machining. International Journal of Machine Tools and Manufacture, 47(10):1518–1529. Scholar
  26. Yeh SS, Hsu PL, 2002. Adaptive-feedrate interpolation for parametric curves with a confined chord error. Computer-Aided Design, 34(3):229–237. Scholar
  27. Zhang LB, You YP, He J, et al., 2011. The transition algorithm based on parametric spline curve for high-speed machining of continuous short line segments. International Journal of Advanced Manufacturing Technology, 52(1–4): 245–254. Scholar

Copyright information

© Zhejiang University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of EngineeringRMIT UniversityCarlton, VictoriaAustralia
  2. 2.Guangzhou Metro Design and Research Institute Co. Ltd.GuangzhouChina
  3. 3.Department of Architecture and Civil EngineeringCity University of Hong KongKowloon, Hong KongChina

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