Skip to main content
SpringerLink
Log in

Rational Conchoid and Offset Constructions: Algorithms and Implementation

  • Conference paper
Artificial Intelligence and Symbolic Computation (AISC 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8884))

  • 492 Accesses

Abstract

This paper is framed within the problem of analyzing the rationality of the components of two classical geometric constructions, namely the offset and the conchoid to an algebraic plane curve and, in the affirmative case, the actual computation of parametrizations. We recall some of the basic definitions and main properties on offsets (see [13]), and conchoids (see [15]) as well as the algorithms for parametrizing their rational components (see [1] and [16], respectively). Moreover, we implement the basic ideas creating two packages in the computer algebra system Maple to analyze the rationality of conchoids and offset curves, as well as the corresponding help pages. In addition, we present a brief atlas where the offset and conchoids of several algebraic plane curves are obtained, their rationality analyzed, and parametrizations are provided using the created packages.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

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.

References

  1. Arrondo, E., Sendra, J., Sendra, J.R.: Parametric Generalized Offsets to Hypersurfaces. Journal of Symbolic Computation 23(2–3), 267–285 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arrondo, E., Sendra, J., Sendra, J.R.: Genus Formula for Generalized Offset Curves. Journal of Pure and Applied Algebra 136(3), 199–209 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Farouki, R.T., Neff, C.A.: Analytic Properties of Plane Offset Curves. Comput. Aided Geom. Des. 7, 83–99 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Farouki, R.T., Neff, C.A.: Algebraic Properties of Plane Offset Curves. Comput. Aided Geom. Des. 7, 100–127 (1990)

    Google Scholar 

  5. Hoffmann, C.M.: Geometric and Solid Modeling. Morgan Kaufmann Publis. (1993)

    Google Scholar 

  6. Kerrick, A.H.: The limaçon of Pascal as a basis for computed and graphic methods of determining astronomic positions. J. of the Instit. of Navigation 6, 5 (1959)

    Google Scholar 

  7. Menschik, F.: The hip joint as a conchoid shape. J. of Biomechanics 30(9) (September 1997) 971-3 9302622

    Google Scholar 

  8. Peternell, M., Gruber, D.: Conchoid surfaces of quadrics. Journal of Symbolic Computation (2013), doi:10.1016/j.jsc, 07.003

    Google Scholar 

  9. Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of rational ruled surfaces. Comp. Aided Geom. Design 28, 427–435 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Peternell, M., Gruber, D., Sendra, J.: Conchoid surfaces of spheres. Comp. Aided Geom. Design 30(1), 35–44 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Peternell, M., Gotthart, L., Sendra, J., Sendra, J.: The Relation Between Offset and Conchoid Constructions. arXiv:1302.1859v2 [math.AG] (June 10, 2013)

    Google Scholar 

  12. Peternell, M., Pottmann, H.: A Laguerre geometric approach to rational offsets. Computer Aided Geometric Design 15, 223–249 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sendra, J., Sendra, J.R.: Algebraic Analysis of Offsets to Hypersurfaces. Mathematische Zeitschrif 234, 697–719 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Sendra, J., Sendra, J.R.: Rationality Analysis and Direct Parametrization of Generalized Offsets to Quadrics. Applicable algebra in Engineering, Communication and Computing 11, 111–139 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sendra, J., Sendra, J.R.: An Algebraic Analysis of Conchoids to Algebraic Curves. Applicable Algebra in Engineering, Communication and Computing 19, 413–428 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Sendra, J., Sendra, J.R.: Rational parametrization of conchoids to algebraic curves, Applicable Algebra in Engineering. Communication and Computing 21(4), 413–428 (2010)

    MathSciNet  Google Scholar 

  17. Sendra, J.R., Sevilla, D.: Radical Parametrizations of Algebraic Curves by Adjoint Curves. Journal of Symbolic Computation 46, 1030–1038 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  18. Sendra, J.R., Sevilla, D.: First Steps Towards Radical Parametrization of Algebraic Surfaces. Computer Aided Geometric Design 30(4), 374–388 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sultan, A.: The Limaçon of Pascal: Mechanical Generating Fluid Processing. J. of Mechanical Engineering Science 219(8), 813–822 (2005) ISSN.0954-4062

    Google Scholar 

  20. Weigan, L., Yuang, E., Luk, K.M.: Conchoid of Nicomedes and Limaçon of Pascal as Electrode of Static Field and a Wavwguide of High Frecuency Wave. Progres. In: Electromagnetics Research Symposium, PIER, vol. 30, pp. 273–284 (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dpto. de Matemática Aplicada a las TIC, Universidad Politécnica de Madrid, Madrid, Spain

    Juana Sendra

  2. Research Center on Software Technologies and Multimedia Systems for Sustainability, Madrid, Spain

    Juana Sendra & David Gómez

  3. ETSI Sistemas de Telecomunicación, Universidad Politécnica de Madrid, Spain

    Valerio Morán

Authors
  1. Juana Sendra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Gómez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valerio Morán
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Information Technologies, University of Huelva, Crta. Palos de la Frontera, s/n, 21810, Palos de la Frontera, Spain

    Gonzalo A. Aranda-Corral

  2. Institute for Theoretical Informatics, Karlsruhe Institute of Technology, Am Fasanengarten 5, 76131, Karlsruhe, Germany

    Jacques Calmet

  3. Departamento de Ciencias de la Computación e Inteligencia Artificial, Universidad de Sevilla, Avda. Reina Mercedes, s/n, 41012, Sevilla, Spain

    Francisco J. Martín-Mateos

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Sendra, J., Gómez, D., Morán, V. (2014). Rational Conchoid and Offset Constructions: Algorithms and Implementation. In: Aranda-Corral, G.A., Calmet, J., Martín-Mateos, F.J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2014. Lecture Notes in Computer Science(), vol 8884. Springer, Cham. https://doi.org/10.1007/978-3-319-13770-4_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-13770-4_15

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-13769-8

  • Online ISBN: 978-3-319-13770-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation