Constructing Invariant Fairness Measures for Surfaces

  • Jens Gravesen
  • Michael Ungstrup

Abstract

The paper proposes a rational method to derive fairness measures for surfaces. It works in cases where isophotes, reflection lines, planar intersection curves, or other curves are used to judge the fairness of the surface. The surface fairness measure is derived by demanding that all the given curves should be fair with respect to an appropriate curve fairness measure. The method is applied to the field of ship hull design where the curves are plane intersections. The method is extended to the case where one considers, not the fairness of one curve, but the fairness of a one parameter family of curves. Six basic third order invariants by which the fairing measures can be expressed are defined. Furthermore, the geometry of a plane intersection curve is studied, and the variation of the total, the normal, and the geodesic curvature and the geodesic torsion is determined.

fairing invariants fairness measure surface design plane intersection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    P. Brunet, A. Vinacua, M. Vivo, N. Pla and A. Rodriguez, Surface fairing for ship hull design application, Math. Engrg. Industry 7 (1998) 179-193.Google Scholar
  2. [2]
    G. Brunnett and J. Wendt, Elastic splines with tension control, in: Mathematical Methods for Curves and Surfaces, eds. M. Dæhlen, T. Lyche, and L.L. Schumaker (Vanderbilt Univ. Press, Nashville, 1998).Google Scholar
  3. [3]
    M.P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ, 1976).Google Scholar
  4. [4]
    M.P. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992).Google Scholar
  5. [5]
    L. Euler, De curvis elasticis, in: Additamentum I zum Methodus Inveniendi Lineas Curvas Maxima Minimivi Proprietate Gaudentes (Lausanne, 1744). Translated to German by H. Linsenart in Abhandlungen über das Gleichgewicht und die Schwingungen der ebenen elastischen Kurven (Ostwalds Klassiker der exakten Wissenschaften, No. 175, Leipzig, 1910).Google Scholar
  6. [6]
    A. Ginnis and S. Wahl, Benchmark results in the area of curve fairing, in: [15], pp. 29-54.Google Scholar
  7. [7]
    J.H. Grace and A. Young, The Algebra of Invariants (Cambridge Univ. Press, Cambridge, 1903).Google Scholar
  8. [8]
    W.C. Graustein, Differential Geometry (Dover, New York, 1966).Google Scholar
  9. [9]
    G. Greiner, Variational design and fairing of spline surfaces, Computer Graphics Forum 13 (1994) 143-154.Google Scholar
  10. [10]
    G. Greiner, Modelling of curves and surfaces based on optimization techniques, in: [15], pp. 11-27.Google Scholar
  11. [11]
    M. Lipschutz, Differential Geometry (McGraw-Hill, New York, 1969).Google Scholar
  12. [12]
    E. Mehlum, Nonlinear splines, in: Computer Aided Geometric Design, eds. R.E. Barnhill and R.F. Riesenfeld (Academic Press, New York, 1974) pp. 173-208.Google Scholar
  13. [13]
    E. Mehlum and C. Tarrou, Invariant smoothness measures for surfaces, Adv. Comput. Math. 8 (1998) 49-63.Google Scholar
  14. [14]
    H.P. Moreton and C.H. Séquin, Functional optimization for fair surface design, ACM Computer Graphics 26 (1992) 167-176.Google Scholar
  15. [15]
    H. Nowacki and P.D. Kaklis, eds., Creating Fair and Shape-Preserving Curves and Surfaces (B.G. Teubner, Stuttgart/Leipzig, 1998).Google Scholar
  16. [16]
    H. Nowacki, G. Westgaard and J. Heimann, Creation of fair surfaces based on higher order fairness measures with interpolation constraints, in: [15], pp. 141-161.Google Scholar
  17. [17]
    I.R. Porteous, Geometric Differentiation, for the Intelligence of Curves and Surfaces (Cambridge Univ. Press, Cambridge, 1994).Google Scholar
  18. [18]
    J.A. Roulier and T. Rando, Designing faired parametric surfaces, Computer-Aided Design 23 (1991) 492-497.Google Scholar
  19. [19]
    R.F. Sarraga, Recent methods for surface optimization, Computer Aided Geom. Design 15 (1998) 417-436.Google Scholar
  20. [20]
    G.G. Schweikert, An interpolating curve using a spline in tension, J. Math. Phys. 45 (1966) 312-317.Google Scholar
  21. [21]
    D.J. Struik, Lectures on Classical Differential Geometry (Addison-Wesley, London, 1961).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Jens Gravesen
    • 1
  • Michael Ungstrup
    • 2
  1. 1.Department of MathematicsTechnical University of DenmarkKgs. LyngbyDenmark
  2. 2.Odense Steel Shipyard, Ltd.Odense CDenmark

Personalised recommendations