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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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):116–131
Barone S (2001) Gear geometric design by B-spline curve fitting and sweep surface modelling. Eng Comput 17(1):66-74
Buckingham E (1928) Spur gears. McGraw-Hill, New York
Dahlquist G, Björck Å (1974) Numerical methods. Prentice-Hall, Englewood Cliffs
Farin G (1993) Curves and surfaces for computer aided geometric design: a practical guide, 3rd edn. Academic, Boston
Lawrence JD (1972) A catalogue of special plane curves. Dover, New York
Norton RL (2000) Machine design: an integrated approach, 2nd edn. Prentice Hall, Englewood Cliffs
Patrikalakis NM (1989) Approximate conversion of rational splines. Comput Aided Geom Des 6(2):155–165
Patrikalakis NM, Maekawa T (2002) Shape interrogation for computer aided design and manufacturing. Springer, Heidelberg
Piegl LA, Tiller W (1995) The NURBS book. Springer, New York
Prautzsch H, Boehm W, Paluszny M (2002) Bézier and B-spline techniques. Springer, Heidelberg
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1988) Numerical recipes in C. Cambridge University Press, Cambridge
Pro/ENGINEER (2005) Parametric Technology Corporation, MA http://www.ptc.com/
Wolter F-E, Tuohy ST (1992) Approximation of high degree and procedural curves. Eng Comput 8:61–80
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Higuchi, F., Gofuku, S., Maekawa, T. et al. Approximation of involute curves for CAD-system processing. Engineering with Computers 23, 207–214 (2007). https://doi.org/10.1007/s00366-007-0060-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-007-0060-3