Fat Arcs for Implicitly Defined Curves

  • Szilvia Béla
  • Bert Jüttler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5862)


We present an algorithm generating a collection of fat arcs which bound the zero set of a given bivariate polynomial in Bernstein–Bézier representation. We demonstrate the performance of the algorithm (in particular the convergence rate) and we apply the results to the computation of intersection curves between implicitly defined algebraic surfaces and rational parametric surfaces.


Parameter Domain Intersection Curve Curve Segment Control Polygon Pythagorean Hodograph 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ballard, D.H.: Strip trees: a hierarchical representation for curves. Commun. ACM 24(5), 310–321 (1981)CrossRefGoogle Scholar
  2. 2.
    Sederberg, T.W., White, S.C., Zundel, A.K.: Fat arcs: a bounding region with cubic convergence. Comput. Aided Geom. Des. 6(3), 205–218 (1989)CrossRefzbMATHGoogle Scholar
  3. 3.
    Yang, X.: Efficient circular arc interpolation based on active tolerance control. Computer-Aided Design 34(13), 1037–1046 (2002)CrossRefGoogle Scholar
  4. 4.
    Sabin, M.A.: The use of circular arcs for the interpolation of curves through empirical data points. Technical Report VTO/MS/164, British Aircraft Corporation (1976)Google Scholar
  5. 5.
    Marciniak, K., Putz, B.: Approximation of spirals by piecewise curves of fewest circular arc segments. Computer-Aided Design 16(2), 87–90 (1984)CrossRefGoogle Scholar
  6. 6.
    Qiu, H., Cheng, K., Li, Y.: Optimal circular arc interpolation for NC tool path generation in curve contour manufacturing. Computer-Aided Design 29(11), 751–760 (1997)CrossRefGoogle Scholar
  7. 7.
    Walton, D.S., Meek, D.J.: Approximating quadratic NURBS curves by arc splines. Computer-Aided Design 25(6), 371–376 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Walton, D.S., Meek, D.J.: An arc spline approximation to a clothoid. Journal of Computational and Applied Mathematics 170(1), 59–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Walton, D.S., Meek, D.J.: Spiral arc spline approximation to a planar spiral. Journal of Computational and Applied Mathematics 170(1), 59–77 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yong, J.H., Hu, S.M., Sun, J.G.: Bisection algorithms for approximating quadratic bézier curves by G 1 arc splines. Computer-Aided Design 32(4), 253–260 (2000)CrossRefzbMATHGoogle Scholar
  11. 11.
    Šír, Z., Feichtinger, R., Jüttler, B.: Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics. Comp.–Aided Design 38, 608–618 (2006)CrossRefGoogle Scholar
  12. 12.
    Held, M., Eibl, J.: Biarc approximation of polygons within asymmetric tolerance bands. Computer-Aided Design 37(4), 357–371 (2004)CrossRefGoogle Scholar
  13. 13.
    Lin, Q., Rokne, J.G.: Approximation by fat arcs and fat biarcs. Computer-Aided Design 34(13), 969–979 (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Aigner, M., Szilagyi, I., Schicho, J., Jüttler, B.: Implicitization and distance bounds. In: Mourrain, B., Elkadi, M., Piene, R. (eds.) Algebraic Geometry and Geometric Modeling, pp. 71–86. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Sederberg, T.W.: Planar piecewise algebraic curves. Comp. Aided Geom. Design 1, 241–255 (1984)CrossRefzbMATHGoogle Scholar
  16. 16.
    Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. AK Peters, Wellesley (1993)zbMATHGoogle Scholar
  17. 17.
    Xu, G., Bajaj, C.L., Xue, W.: Regular algebraic curve segments (i) – definitions and characteristics. Comput. Aided Geom. Des. 17(6), 485–501 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Bartoň, M., Jüttler, B.: Computing roots of polynomials by quadratic clipping. Comp. Aided Geom. Design 24, 125–141 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lee, K.: Principles of CAD/CAM/CAE Systems. Addison-Wesley, Reading (1999)Google Scholar
  20. 20.
    Pratt, M.J., Geisow, A.D.: Surface/surface intersection problem. In: Gregory, J. (ed.) The Mathematics of Surfaces II, pp. 117–142. Claredon Press, Oxford (1986)Google Scholar
  21. 21.
    Patrikalakis, N.M., Maekawa, T.: Shape interrogation for computer aided design and manufacturing. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  22. 22.
    Mourrain, B., Pavone, J.P.: Subdivision methods for solving polynomial equations. Journal of Symbolic Computation (2008) (accepted manuscript)Google Scholar
  23. 23.
    Gonzalez-Vega, L., Necula, I.: Efficient topology determination of implicitly defined algebraic plane curves. Comput. Aided Geom. Design, 719–743 (2002)Google Scholar
  24. 24.
    Dokken, T.: Approximate implicitization. In: Lyche, T., Schumaker, L. (eds.) Mathematical methods for curves and surfaces, pp. 81–102. Vanderbilt University Press, Nashville (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Szilvia Béla
    • 1
  • Bert Jüttler
    • 2
  1. 1.Doctoral Program in Computational MathematicsAustria
  2. 2.Institute of Applied GeometryJohannes Kepler UniversityLinzAustria

Personalised recommendations