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Exact and Approximate Conversion of Spline Curves and Spline Surfaces

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Computation of Curves and Surfaces

Part of the book series: NATO ASI Series ((ASIC,volume 307))

Abstract

A great majority of Computer Aided Design systems for free-form curve and surface modelling uses parametric polynomial representation. However, the representation schemes used within these systems nevertheless differ much with regard to the types of polynomial bases and the maximum polynomial degrees provided. Bernstein-Bézier, Schoenberg-B-Spline, Hermite-Coons type basis functions are frequently used in different systems. Polynomial degrees vary between 3 and about 20 as available upper bounds.

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References

  1. Bardis, L.; Patrikalakis, N.M.: Approximate Conversion of Rational B-Spline Patches. Computer Aided Geometric Design 1989.

    Google Scholar 

  2. Barry, P.J.; Goldman, R.N.: De Casteljau-type subdivision is peculiar to Bézier curves. Computer-aided design 20 (1988) 114–116.

    Article  MATH  Google Scholar 

  3. Bézier, P.: Numerical Control, Mathematics and Applications. Wiley 1972.

    MATH  Google Scholar 

  4. Böhm, W.: Darstellung und Korrektur symmetrischer Kurven und Flächen auf EDV-Anlagen. Computing 17 (1976) 79–85.

    Article  MathSciNet  MATH  Google Scholar 

  5. Böhm, W.: Über die Konstruktion von B-Spline Kurven. Computing 18 (1977) 161–166.

    Article  MathSciNet  MATH  Google Scholar 

  6. Böhm, W.: Cubic B-Spline-Curves and Surfaces in ComputerAided Geometric Design. Computing 19 (1977) 29–34.

    Article  MathSciNet  MATH  Google Scholar 

  7. Böhm, W.: Generating the Bézier Points of B-Spline Curves and Surfaces. Computer-aided design 13 (1981) 365–366.

    Article  Google Scholar 

  8. Böhm, W.; Farin, G.; Kahmann, J.: A survey of curve and surface methods in CAGD. Computer Aided Geometric Design 1 (1984) 1–60.

    Article  MATH  Google Scholar 

  9. Böhm, W.: Efficient evaluation of splines. Computing 33 (1984) 171–177.

    Article  MathSciNet  MATH  Google Scholar 

  10. Boor de, C.: On calculating with B-Splines. J. Approx. Theory 6 (1972) 50–62.

    Article  MATH  Google Scholar 

  11. Boor de, C.: A Practical Guide to Splines. Springer 1978.

    MATH  Google Scholar 

  12. Casteljau de, P.: Outillage méthodes calcul. Paris: André Citroen Automobiles SA 1959.

    Google Scholar 

  13. Cohen, E.; Riesenfeld, R.F.: General matrix representations for Bézier- and B-spline curves. Computers in Industry 3 (1982) 9–15.

    Article  Google Scholar 

  14. Cohen, E.; Schumaker, L. L.: Rates of convergence of control polygons. Computer Aided Geometric Design 2 (1985) 229–235.

    Article  MathSciNet  MATH  Google Scholar 

  15. Cox, M. G.: The numerical evaluation of B-splines. Nat Phys. Lab. England: Teddington 1971.

    Google Scholar 

  16. Dahmen, W.: Subdivision algorithms converge quadratically. Journal of Computational and Applied Mathematics 16 (1986) 145–158.

    Article  MathSciNet  MATH  Google Scholar 

  17. Dannenberg, L.; Nowacki, H.: Approximate conversion of surface representations with polynomial bases. Computer Aided Geometric Design 2 (1985) 123–132.

    Article  MathSciNet  MATH  Google Scholar 

  18. Davis, P.: Interpolation and Approximation. Dover 1975.

    MATH  Google Scholar 

  19. Farouki, R. T.; Rajan, V. T.: On the numerical condition of polynomials in Bernstein form. Computer Aided Geometric Design 4 (1987). 191–216.

    Article  MathSciNet  MATH  Google Scholar 

  20. Farouki, R.T.; Rajan, V.T.: Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design 5 (1988) 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  21. Forrest, A. R.: Interactive interpolation and approximation by Bézier polynomials. Computer Journal 15 (1972) 71–79.

    MathSciNet  MATH  Google Scholar 

  22. Geise, G.: Über berührende Kegelschnitte einer ebenen Kurve. Zeitsch. Angew. Math. Mech. 42 (1962) 297–304.

    Article  MATH  Google Scholar 

  23. Gonska, H.; Meier, J.: A bibliography on approximation of functions by Bernstein type operators, in Schumaker, L.L.; Chui, C.K. (ed.): Approximation theory IV. Academic Press (1983) 739–785.

    Google Scholar 

  24. Gordon, W. J.; Riesenfeld, R. F.: Bernstein-Bézier Methods for Computer Aided Design of Free-Form Curves and Surfaces. Jour. of Assoc. for Computing Machinery 21 (1974) 293–310.

    MathSciNet  MATH  Google Scholar 

  25. Hoschek, J.: Approximate conversion of spline curves. Computer Aided Geometric Design 4 (1987) 59–66.

    Article  MathSciNet  MATH  Google Scholar 

  26. Hoschek, J.: Intrinsic parametrization for approximation. Computer Aided Geometric Design 5 (1988) 27–31.

    Article  MathSciNet  MATH  Google Scholar 

  27. Hoschek, J.: Spline approximation of offset curves. Computer Aided Geometric Design 5 (1988) 33–40.

    Article  MathSciNet  MATH  Google Scholar 

  28. Hoschek, J.; Schneider, F.J.; Wassum, P.: Optimal Approximate Conversion of Spline Surfaces. Computer Aided Geometric Design 1989.

    MATH  Google Scholar 

  29. Hölzle, G.E.: Knot placement for piecewise polynomial approximation of curves. Computer-aided design 15 (1983) 295–296.

    Article  Google Scholar 

  30. Kallay, M.: Approximating a composite cubic curve by one with fewer pieces. Computer-aided design 19 (1987) 539–543.

    Article  MATH  Google Scholar 

  31. Lachance, M.A.: Chebyshew economization for parametric surfaces. Computer Aided Geometric Design 5 (1988) 195–208.

    Article  MathSciNet  MATH  Google Scholar 

  32. Lane, J. M.; Riesenfeld, R. F.: A theoretical development for the computer generation of piecewise polynomial surfaces. IEEE Transaction on Pattern Analysis and Machine Intelligence PAMI 2 (1980) 35–45.

    Article  MATH  Google Scholar 

  33. Lane, J.M.; Riesenfeld, R.F.: A geometric proof for the variation diminishing property of B-spline approximation. Journal of Approximation Theory 37 (1983) 1–4.

    Article  MathSciNet  MATH  Google Scholar 

  34. Lee, E.T.Y.: A simplified B-spline computation routine. Computing 29 (1982) 365–373.

    Article  MathSciNet  MATH  Google Scholar 

  35. Lee, E.T.Y.: Comments on some B-Spline-Algorithms. Computing 36 (1986) 229–238.

    Article  MathSciNet  MATH  Google Scholar 

  36. Loh, R.: Convex B-spline surfaces. Computer-aided design 13 (1981) 145–149.

    Article  Google Scholar 

  37. Lorentz, G.: Bernstein Polynomials. Toronto Press 1953.

    MATH  Google Scholar 

  38. Patrikalikis, N.M.: Approximate conversion of rational splines. Computer Aided Geometric Design 6 (1989) 155–166

    Article  MathSciNet  Google Scholar 

  39. Ramshaw, L.: Blossoming: A Connect-the-Dots Approach to Splines. Research Report 19, Digital Systems Research Center, Palo Alto 1987.

    Google Scholar 

  40. Riesenfeld, R.F.: Applications of B-spline approximation to geometrie problems of computer-aided design. Thesis Syrakus 1973.

    Google Scholar 

  41. Scheffers, G.: Anwendungen der Differential- und Integralrechnung auf Geometrie. Bd. I: Einführung in die Theorie der Kurven in der Ebene und im Raume, von Veit 1901.

    Google Scholar 

  42. Schoenberg, I.J.; Greville, T.N.E.: On spline functions, in Shisha, O. (ed.): Inequalities. Academic Press (1967) 255–291.

    Google Scholar 

  43. Seidel, H.P.: Knot insertion from a blossoming point of view. Computer Aided Geometric Design 5 (1988) 81–86.

    Article  MathSciNet  MATH  Google Scholar 

  44. Stark, E.L.: Bernstein-Polynome, 1912-1955. in Butzer, P.L.; Nagy, B.Sz.; Gorlich, E. (ed.):Functional Analysis and Approximation. Birkhäuser (1981) 443–461.

    Google Scholar 

  45. Stärk, E.: Mehrfach differenzierbare Bézier-Kurven und Bézier-Flächen. Diss. Braunschweig 1976.

    Google Scholar 

  46. Wassum, P.: VC — und VC -Übergangsbedingungen zwischen angrenzenden Rechtecks- und Dreiecks-Bézier-Flächen. Preprint 1233, Fachbereich Mathematik, Technische Hochschule Darmstadt 1989.

    Google Scholar 

  47. Watkins, M.A.; Worsey, A.J.: Degree reduction of Bézier curves. Computer-aided design 20 (1988) 398–405.

    Article  MATH  Google Scholar 

  48. Yamaguchi, F.: Curves and Surfaces in Computer Aided Geometric Design. Springer 1988.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Technical University Darmstadt, Schlossgartenstrasse 7, D-6100, Darmstadt, Federal Republic of Germany

    Josef Hoschek

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  1. Josef Hoschek
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Editor information

Editors and Affiliations

  1. Institut für Mathematik I, Freie Universität, Berlin, Germany

    Wolfgang Dahmen

  2. Department of Applied Mathematics, University of Zaragoza, Zaragoza, Spain

    Mariano Gasca

  3. IBM T.J. Watson Research Center, Yorktown Heights, NY, USA

    Charles A. Micchelli

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© 1990 Kluwer Academic Publishers

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Hoschek, J. (1990). Exact and Approximate Conversion of Spline Curves and Spline Surfaces. In: Dahmen, W., Gasca, M., Micchelli, C.A. (eds) Computation of Curves and Surfaces. NATO ASI Series, vol 307. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2017-0_3

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  • DOI: https://doi.org/10.1007/978-94-009-2017-0_3

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7404-9

  • Online ISBN: 978-94-009-2017-0

  • eBook Packages: Springer Book Archive

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