×

The approximation of non-degenerate offset surfaces. (English) Zbl 0621.65003

Let S be a surface such that the unit surface normal n is defined at each point of S. An offset surface \(S_ 0\) to S is defined as an envelope of the continuum of vectors \(d\updownarrow n\) where d is a real number. There are some difficulties that arise when generating offset surfaces. In his previous paper [ibid. 2, 257-279 (1985; Zbl 0583.65095)] the author considered problems appearing when n is not defined at each point of S. The paper under review deals with the problem of reducing the complexity of the functional representation of \(S_ 0\). The exact functional representation of \(S_ 0\) is typically more complex than that of S. A method for approximation of \(S_ 0\) by bicubic patches is developed. An analysis of the accuracy of the method along with examples is presented.
Reviewer: J.Krč-Jediný

MSC:

65D15 Algorithms for approximation of functions
65D07 Numerical computation using splines
65S05 Graphical methods in numerical analysis
41A15 Spline approximation
41A63 Multidimensional problems
53A05 Surfaces in Euclidean and related spaces
68U20 Simulation (MSC2010)

Citations:

Zbl 0583.65095
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Acton, F.S., Numerical methods that (usually) work, (1970), Harper and Row New York · Zbl 0204.47801
[2] Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer aided geometric design, 1, 1-60, (1984) · Zbl 0604.65005
[3] Coons, S.A., Surfaces for computer aided design of space forms, ()
[4] de Boor, C., A practical guide to splines, (1978), Springer-Verlag New York · Zbl 0406.41003
[5] Dill, J.C.; Rogers, D.F., Color graphics and ship hull surface curvature, (), 197-205
[6] do Carmo, M.P., Differential geometry of curves and surfaces, (1976), Prentice-Hall Englewood Cliffs, NJ · Zbl 0326.53001
[7] Eisenhart, L.P., An introduction to differential geometry, (1947), Princeton University Press Princeton, NJ
[8] Farouki, R.T., Exact offset procedures for simple solids, Computer aided geometric design, 2, 257-279, (1985) · Zbl 0583.65095
[9] Farouki, R.T.; Hinds, J.K., A hierarchy of geometric forms, I.E.E.E. computer graphics and applications, 5, 5, 51-78, (1985)
[10] Faux, I.D.; Pratt, M.J., Computational geometry for design and manufacture, (1979), Ellis Horwood Ltd Chichester · Zbl 0395.51001
[11] Gauss, K.F., Disquisitiones generales circa superficies curves, Collected works, Vol. 4, 217-258, (1880), Göttingen
[12] Gordon, W.J., Free-form surface interpolation through curve networks, () · Zbl 0192.42201
[13] Kajiya, J.T., Ray tracing parametric patches, Computer graphics, 16, 3, 245-254, (1982)
[14] Klass, R., An offset spline approximation for plane cubic splines, Computer-aided design, 15, 5, 297-299, (1983)
[15] Kreyszig, I., Differential geometry, (1959), University of Toronto Press Toronto · Zbl 0088.13901
[16] Martin, R.R., Principal patches for computational geometry, ()
[17] Martin, R.R., Principal patches — A new class of surface patch based on differential geometry, ()
[18] Martin, R.R., Principal patches — a new method of creating sculptured surfaces, ()
[19] Phillips, G.M.; Taylor, P.J., Theory and applications of numerical analysis, (1973), Academic Press New York · Zbl 0312.65002
[20] Ralston, A.; Rabinowitz, P., A first course in numerical analysis, (1978), McGraw-Hill New York · Zbl 0408.65001
[21] Salmon, G.; Salmon, G., Modern higher algebra, (1885), Chelsea Publ. Co New York, reprinted · JFM 01.0025.02
[22] Sederberg, T.W., Implicit and parametric curves and surfaces for computer aided geometric design, () · Zbl 0688.65012
[23] Sederberg, T.W.; Anderson, D.C.; Goldman, R.N., Implicit representation of parametric curves and surfaces, Computer vision, graphics and image processing, 28, 72-84, (1984) · Zbl 0601.65008
[24] Semple, J.G.; Kneebone, G.T., Algebraic projective geometry, (1952), Oxford University Press Oxford · Zbl 0046.38103
[25] Sommerville, D.M.Y., Analytical geometry of three dimensions, (1951), Cambridge University Press Cambridge · Zbl 0008.40203
[26] Tiller, W.; Hanson, E.G., Offsets of two-dimensional profiles, I.E.E.E. computer graphics and applications, 4, 9, 36-46, (1984)
[27] Wang, W.P., Integration of solid geometric modeling for computerized process planning, ()
[28] Willmore, T.J., An introduction to differential geometry, (1959), Oxford University Press Oxford · Zbl 0086.14401
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.