Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Approximation of parabola and quadratic Bézier curve, by fewest circular-arcs within a tolerance-band

  • 163 Accesses

  • 1 Citations


This paper presents a new approximation method for parabola and quadratic Bézier curves by circular-arc within a tolerance-band, and it fully proves that the proposed method supports the fewest number of tangent circular-arcs. Thus, it proposed an approximation method on a specified type of general curve as parabola and QB-curve, to concentrate more on optimization techniques which is an interesting property in CAD/CAM for more efficiency. The approximation method defines tolerance-band by the exterior- and interior-offsets, and it defines some special circular-arcs within the tolerance-band which are used in the approximation by fewest circular-arcs. This article analyzes the proposed method with the possible arc approximations to prove and find out the conditions and features of a sequence of tangent arcs which consists of the minimized number of arcs. Finally, based on the proven minimized conditions and features, two algorithms represent the approximation methods for parabola and QB-curve which guarantee fewest number of arcs and they are not based on Bisection structure and biarc approximation, unlike most existing methods.

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


  1. 1.

    Walton DJ, Meek DS (1994) Approximating of quadratic Bézier curves by arc Splines. Comput Appl Math 54:107–120

  2. 2.

    Ahn YJ, Kim HO, Lee KY (1988) G 1 arc spline approximation of quadratic Bézier curves. Comput Aided Des 30(8):615–620

  3. 3.

    Yong JH, Hu SM, Sun JG (2000) Bisection algorithms for approximating quadratic Bézier curves by G 1 arc splines. Comput Aided Des 32(4):253–260

  4. 4.

    Marciniak K, Putz B (1984) Approximation of spirals by piecewise curves of fewest circular-arc segments. Comput Aided Des 16(2):87–90

  5. 5.

    Meek DS, Walton DJ (1992) Approximation of discrete data by G 1 arc splines. Comput Aided Des 24:301–306

  6. 6.

    Meek DS, Walton DJ (1993) Approximating quadratic NURBS curves by arc splines. Comput Aided Des 25:371–376

  7. 7.

    Meek DS, Walton DJ (1995) Approximating smooth planar curves by arc splines. Comput Appl Math 59:221–231

  8. 8.

    Walton DJ, Meek DS (1996) Approximation of a planar cubic Bézier spiral by circular-arcs. Comput Appl Math 75(1):47–56

  9. 9.

    Ong CJ, Wong YS, Loh HT, Hong XG (1996) An optimization approach for biarc curve fitting of B-spline curves. Comput Aided Des 28:951–959

  10. 10.

    Parkinson DB, Moreton DN (1991) Optimal biarc-curve fitting. Comput Aided Des 23:411–419

  11. 11.

    Bolton KM (1975) Biarc curves. Comput Aided Des 7:89–92

  12. 12.

    Mehlum E (1947) Nonlinear splines. In: Barnhill RE, Riesenfeld RF (Eds) Computer Aided Geometric Design, pp. 173–205

  13. 13.

    Farouki R, Neff C (1990) Algebraic properties of plane offset curves. Comput Aided Geom Des 7:101–127

  14. 14.

    Ris̆kus A (2006) Approximation of a cubic Bézier curve by circular-arcs and vice versa. Inf Technol Control 35(4):371– 378

  15. 15.

    Aichholzer O, Aurenhammer F, Hackl T, Jüttler B, Rabl M, S̆ír Z (2011) Computational and structural advantages of circular boundary representation. Int J Comput Geom Appl 21(1):47–69

  16. 16.

    Siahposhha SAH (2014) Two solutions for the Hausdorff-distance challenge in biarc approximation of quadratic Bézier curve. Int J Adv Manuf Technol, submitted manuscript

  17. 17.

    Siahposhha SAH (2014) A fast applicable algorithm for the optimized G 1 arc-spline approximation of quadratic Bézier curve. Int J Adv Manuf Technol, submitted manuscript

Download references

Author information

Correspondence to Seyed Amir Hossein Siahposhha.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Siahposhha, S.A.H. Approximation of parabola and quadratic Bézier curve, by fewest circular-arcs within a tolerance-band. Int J Adv Manuf Technol 76, 1653–1672 (2015). https://doi.org/10.1007/s00170-014-6316-3

Download citation


  • Minimized approximation
  • Parabola
  • Quadratic Bézier curve
  • Circular-arcs
  • Tolerance-band
  • Hausdorff-distance
  • Offset
  • Threshold