Skip to main content
SpringerLink
Log in
Book cover

International Conference on Curves and Surfaces

Curves and Surfaces 2010: Curves and Surfaces pp 431–444Cite as

On a Special Class of Polynomial Surfaces with Pythagorean Normal Vector Fields

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6920))

Abstract

Rational shapes with rational offsets, especially Pythagorean hodograph (PH) curves and Pythagorean normal vector (PN) surfaces, have been thoroughly studied for many years. However compared to PH curves, Pythagorean normal vector surfaces were introduced using dual approach only in their rational version and a complete characterization of polynomial surfaces with rational offsets, i.e., a polynomial solution of the well-known surface Pythagorean condition, still remains an open and challenging problem. In this contribution, we study a remarkable family of cubic polynomial PN surfaces with birational Gauss mapping, which represent a surface counterpart to the planar Tschirnhausen cubic. A full description of these surfaces is presented and their properties are discussed.

This is a preview of subscription content, 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   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Pottmann, H.: Rational curves and surfaces with rational offsets. Computer Aided Geometric Design 12(2), 175–192 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Farouki, R., Sakkalis, T.: Pythagorean hodographs. IBM Journal of Research and Development 34(5), 736–752 (1990)

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  4. Lü, W., Pottmann, H.: Rational parameterization of quadrics and their offsets. Computing 57, 135–147 (1996)

    Article  MathSciNet  Google Scholar 

  5. Arrondo, E., Sendra, J., Sendra, J.R.: Parametric generalized offsets to hypersurfaces. Journal of Symbolic Computation 23, 267–285 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Arrondo, E., Sendra, J., Sendra, J.R.: Genus formula for generalized offset curves. Journal of Pure and Applied Algebra 136, 199–209 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Maekawa, T.: An overview of offset curves and surfaces. Computer-Aided Design 31, 165–173 (1999)

    Article  MATH  Google Scholar 

  8. Sendra, J.R., Sendra, J.: Algebraic analysis of offsets to hypersurfaces. Mathematische Zeitschrift 237, 697–719 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Vršek, J., Lávička, M.: On convolution of algebraic curves. Journal of symbolic computation 45(6), 657–676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Farouki, R.T., Pottmann, H.: Polynomial and rational Pythagorean-hodograph curves reconciled. In: Proceedings of the 6th IMA Conference on the Mathematics of Surfaces, pp. 355–378. Clarendon Press, New York (1996)

    Google Scholar 

  11. Jüttler, B.: Triangular Bézier surface patches with linear normal vector field. In: Cripps, R. (ed.) The Mathematics of Surfaces VIII. Information Geometers, pp. 431–446 (1998)

    Google Scholar 

  12. Lü, W.: Offset-rational parametric plane curves. Comput. Aided Geom. Des. 12(6), 601–616 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lávička, M., Bastl, B.: Rational hypersurfaces with rational convolutions. Computer Aided Geometric Design 24(7), 410–426 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Peternell, M., Odehnal, B.: Convolution surfaces of quadratic triangular Bézier surfaces. Computer Aided Geometric Design 25, 116–129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bastl, B., Jüttler, B., Kosinka, J., Lávička, M.: Computing exact rational offsets of quadratic triangular Bézier surface patches. Computer-Aided Design 40, 197–209 (2008)

    Article  MATH  Google Scholar 

  16. Ueda, K.: Pythagorean-hodograph curves on isothermal surfaces. In: Cripps, R. (ed.) The Mathematics of Surfaces. VIII. Proceedings of the 8th IMA Conference Held in Birmingham, GB, pp. 339–353. Information Geometers, Winchester (1998)

    Google Scholar 

  17. Lávička, M., Bastl, B.: PN surfaces and their convolutions with rational surfaces. Computer Aided Geometric Design 25, 763–774 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, G.I., Lee, S.: Pythagorean-hodograph preserving mappings. J. Comput. Appl. Math. 216(1), 217–226 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gravesen, J., Jüttler, B., Šír, Z.: On rationally supported surfaces. Computer Aided Geometric Design 25, 320–331 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Šír, Z., Gravesen, J., Jüttler, B.: Curves and surfaces represented by polynomial support functions. Theoretical Computer Science 392(1-3), 141–157 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lávička, M., Bastl, B., Šír, Z.: Reparameterization of curves and surfaces with respect to their convolution. In: Dæhlen, M., Floater, M., Lyche, T., Merrien, J.-L., Mørken, K., Schumaker, L.L. (eds.) MMCS 2008. LNCS, vol. 5862, pp. 285–298. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  22. Farin, G.: Curves and surfaces for CAGD: a practical guide. Morgan Kaufmann Publishers Inc., San Francisco (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Applied Sciences, Department of Mathematics, University of West Bohemia, Univerzitní 8, 301 00, Plzeň, Czech Republic

    Miroslav Lávička & Jan Vršek

Authors
  1. Miroslav Lávička
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jan Vršek
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. INRIA Sophia-Antipolis, 2004 Route des Lucioles, BP 93, 06902, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. Laboratoire LJK, University Joseph Fourier, BP 53, 38420, Grenoble Cedex 9, France

    Patrick Chenin

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, 75005, Paris, France

    Albert Cohen

  4. Laboratoire de Mathématiques, INSA Rouen, Avenue de l’Université, 76800, St. Etienne du Rouvray, France

    Christian Gout

  5. University of Oslo, DMA, Blindern, P.O.Box 1053, 0316, Oslo, Norway

    Tom Lyche

  6. Laboratoire LJK, Université Joseph Fourier, BP 53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

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

    Larry Schumaker

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Lávička, M., Vršek, J. (2012). On a Special Class of Polynomial Surfaces with Pythagorean Normal Vector Fields. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-27413-8_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27412-1

  • Online ISBN: 978-3-642-27413-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation