Constraint-Based Design of Faired Parametric Curves

  • Dan Braha
  • Oded Maimon
Part of the Applied Optimization book series (APOP, volume 17)

Abstract

This chapter demonstrates techniques to allow relations on parametric curves in a variational design system. Constraints on the curves, which are normally represented as constrained nonlinear optimization problems, are reduced to systems of nonlinear equations (using the necessary conditions of the Non-Linear Programming). Additional degrees of freedom are constrained through fairing the curve and the resulting NLP is also reduced to its necessary conditions. Although the solution set of the necessary conditions contains the optimum, it contains many other solutions as well. The COAST design consistency algorithm introduced in Chapter 14 is extended to handle consistency when constraints take the form of relations between objects. Examples are given for elementary curves and for an apparel design system.

Keywords

Geometric Design Distance Constraint Nonlinear Optimization Problem Consistent Solution Composite Object 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Robert Light and David Gossard, “Modification of Geometric Models through Variational Geometry”, Computer-Aided Design, vol. 14, no. 4, pp 209–214, July 1982.CrossRefGoogle Scholar
  2. 2.
    V.C. Lin, D.C. Gossard, and R.A. Light, “Variational Geometry in Compter-Aided Design”, ACM Transactions on Computer Graphics, vol. 15, no. 3, pp 171–177, August 1981.CrossRefGoogle Scholar
  3. 3.
    S. Alasdair Buchanan and Alan de Pennington, “Constraint Definition System: A Computer Algebra Based Approach to Solving Geometric Problems”, Computer Aided Design, December, vol. 25, no. 12, pp. 741–750, 1993.MATHCrossRefGoogle Scholar
  4. 4.
    Horst Nowacki and Xinmin Lu, “Fairing Composite Polynomial Curves with Constraints”, Computer Aided Geometric Design, vol. 11, pp. 1–15, 1994.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    John A. Roulier, “Specifying the Arc Length of Bezier Curves”, Computer Aided Geometric Design, vol 10, pp 25–56, 1993.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Michel Bercovier and Arie Jacobi, “Minimization, Constraints, and Composite Bezier Curves”, Computer-Aided Geometric Design, vol. 11, pp 533–563, 1994.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Hans Hagen and Georges-Pierre Bonneau, “Variational Design of Smooth Rational Bezier Curves”, Computer-Aided Geometric Design, vol. 8, pp 393–399, 1991.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Barry Fowler and Richard Bartels, “Constraint-Based Curve Manipulation”, IEEE Computer Graphics and Applications, pp. 43–49, September 1993.Google Scholar
  9. 9.
    John A. Gregory and Muhammad Sarfraz, “Interactive Curve Design using C2 Rational Splines”, Computers and Graphics, vol. 18, no. 2, pp 153–159, 1994.CrossRefGoogle Scholar
  10. 10.
    Wieger Wesselink and Remco C. Veltcamp, “Interactive Design of Constrained Variational Curves”, Computer Aided Geometric Design, vol. 12, pp 533–546, 1995.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    William H. Press et al., Numerical Recipes: The Art of Scientific Computing, 2nd Edition, Cambridge University Press, 1992.Google Scholar
  12. 12.
    Arnold Neumaier, “Rigorous Sensitivity Analysis for Parameter-Dependent Systems of Equations”, Journal of Mathematical Analysis and Applications, vol. 144, pp 16–25, 1989.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Helen J. Armstrong, Pattern making for Fashion Design, Harper & Row, New York, 1987.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Dan Braha
    • 1
  • Oded Maimon
    • 2
  1. 1.Department of Industrial EngineeringBen Gurion UniversityBeer ShevaIsrael
  2. 2.Department of Industrial EngineeringTel-Aviv UniversityTel-AvivIsrael

Personalised recommendations