Skip to main content
Springer Nature Link
Log in

Envelope computation in the plane by approximate implicitization

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Given a rational family of planar rational curves in a certain region of interest, we are interested in computing an implicit representation of the envelope. The points of the envelope correspond to the zero set of a function (which represents the envelope condition) in the parameter space combining the curve parameter and the motion parameter. We analyze the connection of this function to the implicit equation of the envelope. This connection enables us to use approximate implicitization for computing the (exact or approximate) implicit representation of the envelope. Based on these results, we formulate an algorithm for computing a piecewise algebraic approximation of low degree and illustrate its performance by several examples.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Abdel-Malek K., Yang J., Blackmore D., Joy K.: Swept volumes: fundation, perspectives, and applications. Int. J. Shape Modeling 12(1), 87–127 (2006)

    Article  MATH  Google Scholar 

  2. Alcazar J.: Good global behavior of offsets to plane algebraic curves. J. Symb. Comput. 43(9), 659–680 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alcazar J., Schicho J., Sendra J.: A delineability-based method for computing critical sets of algebraic surfaces. J. Symb. Comput. 42(6), 678–691 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alcazar J., Sendra J.: Local shape of offsets to algebraic curves. J. Symb. Comput. 42(3), 338–351 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arnold V., Gusein-Zade S., Varchenko A.: Singularities of Differentiable Maps. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  6. Bajaj, C.: The emergence of algebraic curves and surfaces in geometric design. In: Directions in Geometric Computing. pp. 1–28 (1993)

  7. Bottema O., Roth B.: Theoretical Kinematics. Dover Publications, New York (1990)

    MATH  Google Scholar 

  8. Dokken, T.: Approximate implicitization. In: Mathematical Methods for Curves and Surfaces, Oslo 2000, pp. 81–102 (2001)

  9. Dokken, T., Kellermann, H., Tegnander, C.: An approach to weak approximation implicitization. In: Mathematical Methods for Curves and Surfaces, Oslo 2000, pp. 103–112 (2001)

  10. Dokken, T., Thomassen, J.: Overview of approximate implicitization. In: Topics in Algebraic Geometry and Geometric Modeling: Workshop on Algebraic Geometry and Geometric Modeling, July 29–August 2, 2002, Vilnius University, Lithuania, vol. 334, pp. 169–184. American Mathematical Society (2003)

  11. Dokken, T., Thomassen, J.: Weak approximate implicitization. In: IEEE International Conference on Shape Modeling and Applications, pp. 204–214. SMI 2006 (2006)

  12. Elkadi M., Mourrain B.: Residue and implicitization problem for rational surfaces. Appl. Algebra Eng. Commun. Comput. 14(5), 361–379 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Farouki R.: Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  14. Flaquer J., Garate G., Pargada M.: Envelopes of moving quadric surfaces. Comput. Aided Geom. Des. 9(4), 299–312 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Galligo, A., van Hoeij, M.: Approximate bivariate factorization: a geometric viewpoint. In: Proceedings of International Workshop on Symbolic-Numeric Computation, pp. 1–10. ACM (2007)

  16. Golub G., Van Loan C.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  17. Hoffmann C.: Implicit curves and surfaces in cagd. IEEE Comput. Graph. Appl. 13(1), 79–88 (1993)

    Article  Google Scholar 

  18. Kim Y., Varadhan G., Lin M., Manocha D.: Fast swept volume approximation of complex polyhedral models. Comput. Aided Des. 36(11), 1013–1027 (2004)

    Article  Google Scholar 

  19. Kreyszig E.: Differential Geometry. Dover, New York (1991)

    Google Scholar 

  20. Patrikalakis N., Maekawa T.: Shape Interrogation for Computer Aided Design and Manufacturing. Springer, Berlin (2002)

    MATH  Google Scholar 

  21. Peternell M., Pottmann H., Steiner T., Zhao H.: Swept volumes. Comput. Aided Des. Appl. 2, 599–608 (2005)

    Google Scholar 

  22. Pottmann, H., Peternell, M.: Envelopes-computational theory and applications. In: Spring Conference on Computer Graphics, pp. 3–23. Comenius University, Bratislava (2000)

  23. Rabl M., Jüttler B., Gonzalez-Vega L.: Exact envelope computation for moving surfaces with quadratic support functions. In: Mourrain, B., Lenarcic, W. (eds) Advances in Robot Kinematics: Analysis and Design, pp. 283–290. Springer, Berlin (2008)

    Chapter  Google Scholar 

  24. San Segundo F., Sendra J.: Partial degree formulae for plane offset curves. J. Symb. Comput. 44(6), 635–654 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sederberg T.: Planar piecewise algebraic curves. Comput. Aided Geom. Des. 1(3), 241–255 (1984)

    Article  MATH  Google Scholar 

  26. Sederberg T., Zheng J., Klimaszewski K., Dokken T.: Approximate implicitization using monoid curves and surfaces. Graph. Models Image Process. 61(4), 177–198 (1999)

    Article  Google Scholar 

  27. Sendra J., Winkler F., Perez-Diaz S.: Rational Algebraic Curves. Springer, Berlin (2007)

    Google Scholar 

  28. Shalaby M., Thomassen J., Wurm E., Dokken T., Jüttler B.: Piecewise approximate implicitization: experiments using industrial data. In: Mourrain, B., Elkadi, M., Piene, R. (eds) Algebraic Geometry and Geometric Modeling, pp. 37–52. Springer, Berlin (2006)

    Chapter  Google Scholar 

  29. Shanks, D.: Solved and unsolved problems in number theory. AMS Chelsea Pub, KKK (1993)

  30. Van Hoeij, M.: An algorithm for computing the Weierstrass normal form. In: Proceedings of International Symposium Symbolic and Algebraic Computation, pp. 90–95. ACM (1995)

  31. Wurm E., Thomassen J., Jüttler B., Dokken T.: Comparative benchmarking of methods for approximate implicitization. In: Neamtu, M., Lucian, M. (eds) Geometric Modeling and Computing: Seattle 2003, pp. 537–548. Nashboro Press, Brentwood (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Geometry, Johannes Kepler University of Linz, Linz, Austria

    Tino Schulz & Bert Jüttler

Authors
  1. Tino Schulz
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Bert Jüttler
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Tino Schulz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schulz, T., Jüttler, B. Envelope computation in the plane by approximate implicitization. AAECC 22, 265–288 (2011). https://doi.org/10.1007/s00200-011-0149-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-011-0149-1

Keywords

Mathematics Subject Classification (2000)