Skip to main content
SpringerLink
Log in

Using a biarc filter to compute curvature extremes of NURBS curves

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

A method to compute curvature minima and maxima of parametric curves (represented in NURBS format) is presented in this paper. Since the curvature changes vary rapidly along the path of (even smooth) curves, a biarc filter is employed to approximate the curvature function with a piecewise constant function. This allows the isolation of curvature extreme values that are found within-engineering tolerances via repeated biarc approximation followed by golden section search. Because the derivative of the curvature is numerically very unstable, only optimization without derivatives is feasible. However, given the excellent isolation property of biarc filters, curvature extremes are found within 10–20 steps even for high accuracy requirements ranging from 10−4 to 10−6.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

References

  1. Ahn YJ, Kim HO, Lee KY (1998) Arc spline approximation of quadratic Bezier curves. Comput Aided Des 30:615–620

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  3. Chuang SHF, Kao CZ (1999) One-sided arc approximation of B-spline curves for interference-free offsetting. Comput Aided Des 31:111–118

    Article  MATH  Google Scholar 

  4. Dannenberg L, Nowacki H (1985) Approximate conversion of surface representations with polynomial bases. Comput Aided Geometric Des 2:123–131

    Article  MATH  MathSciNet  Google Scholar 

  5. Degen WLF (1992) Best approximation of parametric curves by splines. In: Lyche T, Schumaker LL (eds) Mathematical methods in CAGD II. Academic Press, New York, NY, pp 171–184

    Google Scholar 

  6. Do Carmo M (1976) Differential geometry of curves and surfaces. Prentice Hall, NJ

    MATH  Google Scholar 

  7. Dokken T, Lyche T (1994) Spline conversion: existing solutions and open problems. In: Laurent P-J, Le Mehaute A, Schumaker LL (eds) Curves and surfaces in geometric design. A. K. Peters, Wellesley, pp 121–130

    Google Scholar 

  8. Drysdale RL, Rote G, Sturm A (2008) Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs. Comput Geometry 41:31–47

    Article  MATH  MathSciNet  Google Scholar 

  9. Eck M (1993) Degree reduction of Bezier curves. Comput Aided Geometric Des 10:237–251

    Article  MATH  MathSciNet  Google Scholar 

  10. Eck M, Hadenfeld J (1994) A stepwise algorithm for converting B-splines. In: Laurent P-J, Le Mehaute A, Schumaker LL (eds) Curves and surfaces in geometric design. A. K. Peters, Wellesley, pp 131–138

    Google Scholar 

  11. Farouki RT, Neff CA (1991) Analytic properties of plane offset curves. Comput Aided Geometric Des 7:83–99

    Article  MathSciNet  Google Scholar 

  12. Filip D, Magedson R, Markot R (1986) Surface algorithms using bounds on derivatives. Comput Aided Geometric Des 3:295–311

    Article  MATH  MathSciNet  Google Scholar 

  13. Floater M (1995) High-order approximation of conic sections by quadratic splines. Comput Aided Geometric Des 12:617–637

    Article  MATH  MathSciNet  Google Scholar 

  14. Hagen H, Hahmann S, Schreiber T (1995) Visualization and computation of curvature behavior of free-form curves and surfaces. Comput Aided Des 27:545–552

    Article  MATH  Google Scholar 

  15. Harik RF, Derigent WJ, Ris G (2008) Computer-aided process planning in aircraft manufacturing. Comput Aided Des Appl 5:953–962

    Google Scholar 

  16. Held M, Eibl J (2005) Biarc approximation of polygons within asymmetric tolerance bands. Comput Aided Des 37:357–371

    Article  Google Scholar 

  17. Holzle GE (1983) Knot placement for piecewise polynomial approximation of curves. Comput Aided Des 15:295–296

    Article  Google Scholar 

  18. Hoschek J (1987) Approximate conversion of spline curves. Comput Aided Geometric Des 4:59–66

    Article  MATH  MathSciNet  Google Scholar 

  19. Hoschek J (1992) Circular splines. Comput Aided Des 24:611–618

    Article  MATH  Google Scholar 

  20. Hoschek J, Schneider F-J (1994) Approximate conversion and data compression of integral and rational B-spline surfaces. In: Laurent P-J, Le Mehaute A, Schumaker LL (eds) Curves and surfaces in geometric design. A. K. Peters, Wellesley, pp 241–250

    Google Scholar 

  21. Hoschek J, Wissel N (1988) Optimal approximate conversion of spline curves and spline approximation of offset curves. Comput Aided Des 20:475–483

    Article  MATH  Google Scholar 

  22. Kallay M (1987) Approximating a composite cubic curve by one with fewer pieces. Comput Aided Des 19:536–543

    Google Scholar 

  23. Li Z, Meek DS (2005) Smoothing an arc spline. Comput Graph 29:576–587

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Meek DS, Walton DJ (2008) The family of biarcs that matches planar, two-point G 1 Hermite data. J Comput Appl Math 212:31–45

    Article  MATH  MathSciNet  Google Scholar 

  28. 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

    Article  Google Scholar 

  29. Park H (2004) Error-bounded biarc approximation of planar curves. Comput Aided Des 36:1241–1251

    Article  Google Scholar 

  30. Parkinson DB (1992) Optimized biarc curves with tension. Comput Aided Geometric Des 9:207–218

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  32. Patrikalakis NM (1989) Approximate conversion of rational splines. Comput Aided Geometric Des 6:155–165

    Article  MATH  MathSciNet  Google Scholar 

  33. Piegl LA, Tiller W (1997) The NURBS book. Springer-Verlag, New York, NY

    Google Scholar 

  34. Piegl LA, Tiller W (1997) Symbolic operators for NURBS. Comput Aided Des 29:361–368

    Article  Google Scholar 

  35. Piegl LA, Tiller W (2002) Data approximation using biarcs. Eng Comput 18:59–65

    Article  MATH  Google Scholar 

  36. Piegl LA, Tiller W (2002) Biarc approximation of NURBS curves. Comput Aided Des 34:807–814

    Article  Google Scholar 

  37. Pratt MJ, Goult RJ, Ye L (1993) On rational parametric curve approximation. Comput Aided Geometric Des 10:363–377

    Article  MATH  MathSciNet  Google Scholar 

  38. Qiu H, Cheng K, Li Y (1997) Optimal circular arc interpolation for NC tool path generation in curve contour manufacturing. Comput Aided Des 29:751–760

    Article  Google Scholar 

  39. Sabin MA (1977) The use of piecewise forms for the numerical representation of shape. Report 60/1977, Computer and Automation Institute, Hungarian Academy of Sciences

  40. Schoenherr J (1993) Smooth biarc curves. Comput Aided Des 25:365–370

    Article  MATH  Google Scholar 

  41. Sederberg T, Kakimoto M (1991) Approximating rational curves using polynomial curves. In: Farin G (ed) NURBS for curve and surface design. SIMA, Philadelphia, PA, pp 149–158

    Google Scholar 

  42. Sir Z, Feichtinger R, Juttler B (2006) Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics. Comput Aided Des 38:608–618

    Article  Google Scholar 

  43. Smarodzinava O, Rajab K, Piegl LA, Valavanis KP (2008) From CT to NURBS: bio-modeling with B-spline curves. The Visual Computer, submitted

  44. Wolter FE (1992) Approximation of high-degree and procedural curves. Eng Comput 8:61–80

    Article  Google Scholar 

  45. Xidias EK, Azariadis PN, Aspragathos NA (2008) Path planning of holonomic and non-holonomic robots using bump-surfaces. Comput Aided Des Appl 5:497–507

    Google Scholar 

  46. Yang X, Chen ZC (2008) A practical approach to G 1 biarc approximations for making accurate, smooth and non-gouged profile features in CNC contouring. Comput Aided Des 38:1205–1213

    Article  Google Scholar 

  47. Yang X, Wang G (2001) Planar point set fairing and fitting by arc splines. Comput Aided Des 33:35–43

    Article  Google Scholar 

  48. Yang H, Wang W, Sun J (2004) Control point adjustment for B-spline curve approximation. Comput Aided Des 36:639–652

    Article  Google Scholar 

  49. Yeung M, Walton DJ (1994) Curve fitting with arc splines for NC toolpath generation. Comput Aided Des 26:845–849

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

Download references

Acknowledgments

The work reported in this paper was supported by the US National Science Foundation under grant number DMI-0758231, awarded to the University of South Florida. All opinions, findings, conclusions and recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the National Science Foundation or the University of South Florida.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, University of South Florida, Tampa, FL, 33620, USA

    Les A. Piegl, Khairan Rajab & Volha Smarodzinava

  2. Department of Electrical and Computer Engineering, University of Denver, Denver, CO, 80208, USA

    Kimon P. Valavanis

Authors
  1. Les A. Piegl
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Khairan Rajab
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Volha Smarodzinava
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kimon P. Valavanis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Les A. Piegl.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Piegl, L.A., Rajab, K., Smarodzinava, V. et al. Using a biarc filter to compute curvature extremes of NURBS curves. Engineering with Computers 25, 379–387 (2009). https://doi.org/10.1007/s00366-009-0131-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-009-0131-8

Keywords

Search

Navigation