Skip to main content
SpringerLink
Log in

An analysis of cubic approximation schemes for conic sections

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

Abstract

In this paper a piece of a conic section is approximated by a cubic or piecewise cubic polynomial. The main tool is to define the two inner control points of the cubic as an affine combination, defined by λ ∈ [0, 1], of two control points of the conic. If λ is taken to depend on the weightw of the latter, a function λ(w) results which is used to distinguish between different algorithms and to analyze their properties. One of the approximations is a piecewise cubic havingG 4 continuity at the break points.

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. C. de Boor,A Practical Guide to Splines (Springer-Verlag, New York, 1978).

    Google Scholar 

  2. C. de Boor, K. Höllig and M. Sabin, High accuracy geometric Hermite interpolation, Computer-Aided Geom. Design4 (1987) 269–278.

    Article  Google Scholar 

  3. W. Degen, Best approximations of parametric curves by splines, inMathematical methods in CAGD, eds. T. Lyche and L.L. Schumaker, vol. 2 (Academic Press, Boston, 1992) 171–184.

    Google Scholar 

  4. W. Degen, High accurate rational approximation of parametric curves, Computer-Aided Geom. Design10 (1993) 293–313.

    Article  Google Scholar 

  5. T. Dokken, M. Dæhlen, T. Lyche and K. Mørken, Good approximation of circles by curvature-continuous Bézier curves, Computer-Aided Geom. Design7 (1990) 33–41.

    Article  Google Scholar 

  6. G. Farin,Curves and surfaces for computer aided geometric design (Academic Press, San Diego, 1988).

    Google Scholar 

  7. M.S. Floater, High order approximation of conic sections by quadratic splines, Computer-Aided Geom. Design12 (1995) 617–637.

    Article  Google Scholar 

  8. M.S. Floater, AnO(h 2n) Hermite approximation for conic sections, to appear in Computer-Aided Geom. Design.

  9. T.N.T. Goodman, Properties of β-splines, J. Approx. Th.44 (1995) 132–153.

    Article  Google Scholar 

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

    Google Scholar 

  11. J. Hoschek and F. Schneider, Spline conversion for trimmed rational Bézier- and B-spline surfaces, Comput. Aided Design22 (1990) 580–590.

    Article  Google Scholar 

  12. E. Lee, The rational Bézier representations for conics, inGoemetric Modeling: Algorithms and New Trends, ed. G. Farin, SIAM, Philadelphia (1987) 3–19.

    Google Scholar 

  13. K. Mørken, private communication.

  14. K. Mørken and K. Scherer, A general framework for high accuracy parametric interpolation, preprint.

  15. R. Schaback, Interpolation with piecewise quadratic visuallyC 2 Bézier polynomials, Computer-Aided Geom. Design6 (1989) 219–233.

    Article  Google Scholar 

  16. R. Schaback, Planar curve interpolation by piecewise conics of arbitrary type, Computer-Aided Geom. Design9 (1993) 373–389.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SINTEF, P.O. Box 124, Blindern, 0314, Olso, Norway

    Michael S. Floater

Authors
  1. Michael S. Floater
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by T. Sederburg

Rights and permissions

Reprints and permissions

About this article

Cite this article

Floater, M.S. An analysis of cubic approximation schemes for conic sections. Adv Comput Math 5, 361–379 (1996). https://doi.org/10.1007/BF02124751

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02124751

Keywords

Search

Navigation