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Engineering with Computers

, Volume 23, Issue 3, pp 207–214 | Cite as

Approximation of involute curves for CAD-system processing

  • Fumitaka Higuchi
  • Shuuichi Gofuku
  • Takashi Maekawa
  • Harish Mukundan
  • Nicholas M. Patrikalakis
Original Article

Abstract

In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a Bézier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.

Keywords

Circle involute curves Involute gears Chebyshev approximation formula Bézier curves 

References

  1. 1.
    Abrams SL, Bardis L, Chryssostomidis C, Patrikalakis NM, Tuohy ST, Wolter F-E, Zhou J (1995) The geometric modeling and interrogation system Praxiteles. J Ship Prod 11(2):116–131Google Scholar
  2. 2.
    Barone S (2001) Gear geometric design by B-spline curve fitting and sweep surface modelling. Eng Comput 17(1):66-74zbMATHCrossRefGoogle Scholar
  3. 3.
    Buckingham E (1928) Spur gears. McGraw-Hill, New YorkGoogle Scholar
  4. 4.
    Dahlquist G, Björck Å (1974) Numerical methods. Prentice-Hall, Englewood CliffsGoogle Scholar
  5. 5.
    Farin G (1993) Curves and surfaces for computer aided geometric design: a practical guide, 3rd edn. Academic, BostonGoogle Scholar
  6. 6.
    Lawrence JD (1972) A catalogue of special plane curves. Dover, New YorkGoogle Scholar
  7. 7.
    Norton RL (2000) Machine design: an integrated approach, 2nd edn. Prentice Hall, Englewood CliffsGoogle Scholar
  8. 8.
    Patrikalakis NM (1989) Approximate conversion of rational splines. Comput Aided Geom Des 6(2):155–165zbMATHCrossRefGoogle Scholar
  9. 9.
    Patrikalakis NM, Maekawa T (2002) Shape interrogation for computer aided design and manufacturing. Springer, HeidelbergzbMATHGoogle Scholar
  10. 10.
    Piegl LA, Tiller W (1995) The NURBS book. Springer, New YorkzbMATHGoogle Scholar
  11. 11.
    Prautzsch H, Boehm W, Paluszny M (2002) Bézier and B-spline techniques. Springer, HeidelbergzbMATHGoogle Scholar
  12. 12.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1988) Numerical recipes in C. Cambridge University Press, CambridgeGoogle Scholar
  13. 13.
    Pro/ENGINEER (2005) Parametric Technology Corporation, MA http://www.ptc.com/
  14. 14.
    Wolter F-E, Tuohy ST (1992) Approximation of high degree and procedural curves. Eng Comput 8:61–80CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2007

Authors and Affiliations

  • Fumitaka Higuchi
    • 1
  • Shuuichi Gofuku
    • 1
  • Takashi Maekawa
    • 1
  • Harish Mukundan
    • 2
  • Nicholas M. Patrikalakis
    • 2
  1. 1.Department of Mechanical Engineering, Digital Engineering LaboratoryYokohama National UniversityYokohamaJapan
  2. 2.Department of Mechanical Engineering, Design LaboratoryMassachusetts Institute of TechnologyCambridgeUSA

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