Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Fat Arcs for Implicitly Defined Curves

  • Conference paper
Mathematical Methods for Curves and Surfaces (MMCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5862))

Abstract

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.

This work was supported by the Austrian Science Found (FWF) through the Doctoral Program in Computational Mathematics, subproject 3.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ballard, D.H.: Strip trees: a hierarchical representation for curves. Commun. ACM 24(5), 310–321 (1981)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  3. Yang, X.: Efficient circular arc interpolation based on active tolerance control. Computer-Aided Design 34(13), 1037–1046 (2002)

    Article  Google Scholar 

  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. Marciniak, K., Putz, B.: Approximation of spirals by piecewise curves of fewest circular arc segments. Computer-Aided Design 16(2), 87–90 (1984)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  7. Walton, D.S., Meek, D.J.: Approximating quadratic NURBS curves by arc splines. Computer-Aided Design 25(6), 371–376 (1993)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  12. Held, M., Eibl, J.: Biarc approximation of polygons within asymmetric tolerance bands. Computer-Aided Design 37(4), 357–371 (2004)

    Article  Google Scholar 

  13. Lin, Q., Rokne, J.G.: Approximation by fat arcs and fat biarcs. Computer-Aided Design 34(13), 969–979 (2002)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  15. Sederberg, T.W.: Planar piecewise algebraic curves. Comp. Aided Geom. Design 1, 241–255 (1984)

    Article  MATH  Google Scholar 

  16. Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. AK Peters, Wellesley (1993)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bartoň, M., Jüttler, B.: Computing roots of polynomials by quadratic clipping. Comp. Aided Geom. Design 24, 125–141 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lee, K.: Principles of CAD/CAM/CAE Systems. Addison-Wesley, Reading (1999)

    Google Scholar 

  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. Patrikalakis, N.M., Maekawa, T.: Shape interrogation for computer aided design and manufacturing. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  22. Mourrain, B., Pavone, J.P.: Subdivision methods for solving polynomial equations. Journal of Symbolic Computation (2008) (accepted manuscript)

    Google Scholar 

  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. 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 

Download references

Author information

Authors and Affiliations

  1. Doctoral Program in Computational Mathematics, Austria

    Szilvia Béla

  2. Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria

    Bert Jüttler

Authors
  1. Szilvia Béla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bert Jüttler
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Oslo, Norway

    Morten Dæhlen  & Tom Lyche  & 

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  3. INSA de Rennes, Rennes, France

    Jean-Louis Merrien

  4. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Béla, S., Jüttler, B. (2010). Fat Arcs for Implicitly Defined Curves. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, JL., Mørken, K., Schumaker, L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11620-9_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-11620-9_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11619-3

  • Online ISBN: 978-3-642-11620-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics