Skip to main content
SpringerLink
Log in

Shape preservation regions for six-dimensional spaces

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

We analyze the critical length for design purposes of six-dimensional spaces invariant under translations and reflections containing the functions 1, cos t and sin t. These spaces also contain the first degree polynomials as well as trigonometric and/or hyperbolic functions. We identify the spaces whose critical length for design purposes is greater than 2π and find its maximum 4π. By a change of variables, two biparametric families of spaces arise. We call shape preservation region to the set of admissible parameters in order that the space has shape preserving representations for curves. We describe the shape preserving regions for both families.

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

Access this article

Log in via an institution

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

References

  1. J.M. Carnicer, E. Mainar and J.M. Peña, Representing circles with five control points, Comput. Aided Geom. Design 20 (2003) 501–511.

    Article  MathSciNet  MATH  Google Scholar 

  2. J.M. Carnicer, E. Mainar and J.M. Peña, Critical length for design purposes and extended Chebyshev spaces, Constr. Approx. 20 (2004) 55–71.

    Article  MathSciNet  MATH  Google Scholar 

  3. J.M. Carnicer and J.M. Peña, Shape preserving representations and optimality of the Bernstein basis, Adv. Comput. Math. 1 (1993) 173–196.

    Article  MathSciNet  MATH  Google Scholar 

  4. J.M. Carnicer and J.M. Peña, Totally positive bases for shape preserving curve design and optimality of B-splines, Comput. Aided Geom. Design 11 (1994) 635–656.

    Article  Google Scholar 

  5. T.N.T. Goodman, Shape preserving representations, in: Mathematical Methods in CAGD, eds. T. Lyche and L.L. Schumaker (Academic Press, Boston, 1989) pp. 333–351.

    Google Scholar 

  6. E. Mainar, J.M. Peña and J. Sánchez-Reyes, Shape preserving alternatives to the rational Bézier model, Comput. Aided Geom. Design 18 (2001) 37–60.

    Article  MathSciNet  MATH  Google Scholar 

  7. J.M. Peña, Shape preserving representations for trigonometric polynomial curves, Comput. Aided Geom. Design 14 (1997) 5–11.

    Article  MathSciNet  MATH  Google Scholar 

  8. J.M. Peña, ed., Shape Preserving Representations in Computer-Aided Geometric Design (Nova Science Publishers, Commack, NJ, 1999).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, University of Zaragoza, Spain

    J. M. Carnicer & J. M. Peña

  2. Departamento de Matemáticas, Estadística y Computación, University of Cantabria, Spain

    E. Mainar

Authors
  1. J. M. Carnicer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Mainar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. M. Peña
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. M. Carnicer.

Additional information

Communicated by Y. Xu

To our friend Mariano Gasca on occasion of his 60th birthday

Research partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Carnicer, J.M., Mainar, E. & Peña, J.M. Shape preservation regions for six-dimensional spaces. Adv Comput Math 26, 121–136 (2007). https://doi.org/10.1007/s10444-005-7505-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-005-7505-2

Keywords

Mathematics subject classifications (2000)

Search

Navigation