SpringerLink
Log in

The Offset to an Algebraic Curve and an Application to Conics

  • Conference paper
Computational Science and Its Applications – ICCSA 2005 (ICCSA 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3480))

Included in the following conference series:

  • 936 Accesses

Abstract

Curve offsets are important objects in computer-aided design. We study the algebraic properties of the offset to an algebraic curve, thus obtaining a general formula for its degree. This is applied to computing the degree of the offset to conics. We also compute an implicit equation of the generalised offset to a conic by using sparse resultants and the knowledge of the degree of the implicit equation.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (France)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 106.99
Price includes VAT (France)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

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. Anton, F.: Voronoi diagrams of semi-algebraic sets. PhD thesis, The University of British Columbia, Vancouver, British Columbia, Canada (January 2004)

    Google Scholar 

  2. Arrondo, E., Sendra, J., Sendra, J.R.: Genus formula for generalized offset curves. J. Pure Appl. Algebra 136(3), 199–209 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Canny, J.F., Emiris, I.Z.: A subdivision-based algorithm for the sparse resultant. J. ACM 47(3), 417–451 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cox, D., Little, J., O’Shea, D.: Using algebraic geometry. Springer, New York (1998)

    MATH  Google Scholar 

  5. Emiris, I.: A general solver based on sparse resultants: Numerical issues and kinematic applications. Technical Report 3110, SAFIR (1997)

    Google Scholar 

  6. Emiris, I.Z., Canny, J.F.: Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symbolic Comput. 20(2), 117–149 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Greuel, G.-M., Pfister, G.: A Singular introduction to commutative algebra. Springer, Berlin (2002)

    MATH  Google Scholar 

  8. Hoffmann, C.M., Vermeer, P.J.: Eliminating extraneous solutions in curve and surface operations. Internat. J. Comput. Geom. Appl. 1(1), 47–66 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Shafarevich, I.R.: Basic algebraic geometry, 2nd edn., vol. 1. Springer, Berlin (1994); Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid.

    Google Scholar 

  10. Sturmfels, B.: Sparse elimination theory. In: Computational algebraic geometry and commutative algebra (Cortona 1991). Sympos. Math., vol. XXXIV, pp. 264–298. Cambr. Univ. Press, Cambridge (1993)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, T2N 1N4, Canada

    François Anton

  2. Department of Informatics and Telecommunications, National Kapodistrian University of Athens, Panepistimiopolis, 15784, Greece

    Ioannis Emiris

  3. INRIA Sophia-Antipolis, B.P. 93, 2004, route des Lucioles, 06902, Sophia-Antipolis, France

    Bernard Mourrain & Monique Teillaud

Authors
  1. François Anton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ioannis Emiris
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernard Mourrain
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Monique Teillaud
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics and Computer Science, University of Perugia, via Vanvitelli, 1, I-06123, Perugia, Italy

    Osvaldo Gervasi

  2. Department of Computer Science, University of Calgary, 2500 University Drive N.W, T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  3. William Norris Professor, Head of the Computer Science and Engineering Department, University of Minnesota, USA

    Vipin Kumar

  4. Department of Chemistry, University of Perugia, Via Elce di Sotto, 8, P.O. Box, I-06123, Perugia, Italy

    Antonio Laganà

  5. Institute of High Performance Computing, IHCP, 1 Science Park Road, 01-01 The Capricorn, Singapore Science Park II, 117528, Singapore

    Heow Pueh Lee

  6. School of Computing, Soongsil University, Seoul, Korea

    Youngsong Mun

  7. Clayton School of IT, Monash University, 3800, Clayton, Australia

    David Taniar

  8. OptimaNumerics Ltd, P.O. Box, Belfast, United Kingdom

    Chih Jeng Kenneth Tan

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Anton, F., Emiris, I., Mourrain, B., Teillaud, M. (2005). The Offset to an Algebraic Curve and an Application to Conics. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2005. ICCSA 2005. Lecture Notes in Computer Science, vol 3480. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11424758_71

Download citation

  • DOI: https://doi.org/10.1007/11424758_71

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25860-5

  • Online ISBN: 978-3-540-32043-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation