Abstract
Surface fitting is still an area of great research interest. In this paper two particular surface fitting problems are discussed. First, various strategies to approximate complex free-form shapes are described, with special emphasis on the so-called functionally decomposed surface representation. This is strongly related to surface fitting when the set of data points to be approximated is incomplete; i.e. there are areas which belong to other surfaces or which are just not accessible. In these cases we have to somehow "bridge" the unknown areas, and produce an overall pleasing surface, which smoothly connects the surface portions where data is available. The algorithm described is a combination of previous approaches to minimize hybrid objective functions, with several practical improvements. The second problem aims at fitting surfaces where not only positional data but surface normal data needs to be approximated as well. After analyzing the consistency of the data and possible objective functions to be minimized, the solution, which was found to be the most suitable is presented. Both the above problems emerged due to practical demands and have been intensively tested using real data obtained from the automobile industry. Some colour pictures illustrate the results.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
G. Celniker and D. Gossard, Deformable Curve and Surface Finite Elements for Free-Form Shape Design, Computer Graphics (Procs of SIGGRAPH ’91), 25 4 (1991), 257–266
M.G. Cox, The least-squares solution of overdetermined linear equations having band or augmented band structure, IMA Journal of Num. Anal., 1 (1981), 3–22.
M.G. Cox, Direct versus iterative methods of solution for multivariate spline-fitting problems, IMA Journal of Num. Anal., 2 (1982), 73–81.
M.G. Cox, Algorithms for spline curves and surfaces, in: L. Piegl (ed.), Fundamental Developments of Computer-Aided Geometric Modeling, Academic Press, (1993), 51–76.
P. Dierckx, An algorithm for surface-fitting with spline functions, IMA Journal of Num. Anal., 1 (1981), 267–283.
U. Dietz, Erzeugung glatter Flächen aus Meßpunkten, Teschnische Hochschule Darmstadt, Preprint-Nr. 1717, (1995).
M. Eck and H. Hoppe, Automatic Reconstruction of B-spline Surfaces of Arbitrary Topological Type, to appear in Procs. of SIGGRAPH’96, (1996).
G. Farin and N. Sapidis, Curvature and fairness of curves and surfaces, IEEE CGA, 9 (1989), 52–57.
D.R. Forsey and R.H. Bartels, Surface fitting with hierarchical splines, ACM Transactions on Graphics, 14 2 (1995), 134–161.
M. von Golitschek and L. L. Schumaker, Data fitting by penalized least squares, in: J.C. Mason and M. G. Cox (eds.), Algorithms for Approximation II, Chapman and Hall, London, (1990), 210–227.
G. Greiner, Surface construction based on variational principles, in: P. J. Laurent, A. Le Méhauté, L. L. Schumaker (eds.), Wavelets, Images, and Surface Fitting, A K Peters, (1994), 277–286.
J. G. Hayes and J. Halliday, The least-squares fitting of cubic spline surfaces to general data sets, J. Inst. Maths. Applics., 14 (1974), 89–103.
J. Hoschek and D. Lasser, Fundamentals of Computer Aided Geometric Design, A K Peters, (1991).
J. Hoschek, Intrinsic parametrization for approximation, CAGD, 5 (1988), 27–31.
Reverse Engineering, Ed.: J. Hoschek and W. Dankwort, Teubner, Stuttgart, 1996.
W. Ma and J. P. Kruth, Parametrization of randomly measured points for least squares fitting of B-spline curves and surfaces, CAD 27(9) (1995) 663–675.
H.P. Moretón and C. Sequin, Functional optimization for fair surface design, ACM Computer Graphics (Procs of SIGGRAPH’92), 26 (1992), 167–176.
G. M. Nielson, Multivariate smoothing and interpolating splines. SIAM J. Numer. Anal., 11(2) (1974), 435–446.
Designing Fair Curves and Surfaces, Shape Quality in Geometric Modeling and Computer-Aided Design, Ed.: N. S. Sapidis, SIAM, 1994.
R.F. Sarraga, Recent methods for shape optimization, General Motors, Research Publication 8477, February 1996.
D. Terzopoulos and H. Qin, Dynamic NURBS with Geometric Constraints for Interactive Sculpting, ACM Transactions on Graphics, 13 2 (1994), 103–136.
T. Várady, R.R. Martin and J. Cox, Reverse Engineering of Geometric Models — An Introduction, Geometric Modelling Studies, GML 1996/4, Computer and Automation Research Institute, Budapest, (to appear in Computer-Aided Design), 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Hermann, T., Kovács, Z., Várady, T. (1997). Special Applications in Surface Fitting. In: Strasser, W., Klein, R., Rau, R. (eds) Geometric Modeling: Theory and Practice. Focus on Computer Graphics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60607-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-60607-6_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-61883-6
Online ISBN: 978-3-642-60607-6
eBook Packages: Springer Book Archive