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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
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).
M.P. do Carmo, Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ, 1976).
M.P. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992).
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).
A. Ginnis and S. Wahl, Benchmark results in the area of curve fairing, in: [15], pp. 29-54.
J.H. Grace and A. Young, The Algebra of Invariants (Cambridge Univ. Press, Cambridge, 1903).
W.C. Graustein, Differential Geometry (Dover, New York, 1966).
G. Greiner, Variational design and fairing of spline surfaces, Computer Graphics Forum 13 (1994) 143-154.
G. Greiner, Modelling of curves and surfaces based on optimization techniques, in: [15], pp. 11-27.
M. Lipschutz, Differential Geometry (McGraw-Hill, New York, 1969).
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.
E. Mehlum and C. Tarrou, Invariant smoothness measures for surfaces, Adv. Comput. Math. 8 (1998) 49-63.
H.P. Moreton and C.H. Séquin, Functional optimization for fair surface design, ACM Computer Graphics 26 (1992) 167-176.
H. Nowacki and P.D. Kaklis, eds., Creating Fair and Shape-Preserving Curves and Surfaces (B.G. Teubner, Stuttgart/Leipzig, 1998).
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.
I.R. Porteous, Geometric Differentiation, for the Intelligence of Curves and Surfaces (Cambridge Univ. Press, Cambridge, 1994).
J.A. Roulier and T. Rando, Designing faired parametric surfaces, Computer-Aided Design 23 (1991) 492-497.
R.F. Sarraga, Recent methods for surface optimization, Computer Aided Geom. Design 15 (1998) 417-436.
G.G. Schweikert, An interpolating curve using a spline in tension, J. Math. Phys. 45 (1966) 312-317.
D.J. Struik, Lectures on Classical Differential Geometry (Addison-Wesley, London, 1961).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gravesen, J., Ungstrup, M. Constructing Invariant Fairness Measures for Surfaces. Advances in Computational Mathematics 17, 67–88 (2002). https://doi.org/10.1023/A:1015229622042
Issue Date:
DOI: https://doi.org/10.1023/A:1015229622042