“Local control of bias and tension in beta-splines” by Barsky and Beatty

  • ©Brian A. Barsky and John C. Beatty

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    Local control of bias and tension in beta-splines

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Abstract:


    The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated. Using a restricted form of quintic Hermite interpolation, it is possible to allow distinct bias and tension parameters at each joint without destroying geometric continuity. This provides a new means of obtaining local control of bias and tension in piecewise polynomial curves and surfaces.

References:


    1. BARSKY, B.A. The Beta-spline: A local representation based on shape parameters and fundamental geometric measures. Ph.D. dissertation, Dept. of Computer Science, Univ. of Utah, Salt Lake City, Utah, Dec., 1981.
    2. BARSKY, B.A. Exponential and polynomial methods for applying tension to an interpolating spline curve. Computer Vision Graphic Image Processing 1983, to appear.
    3. BARSKY, B.A. A study of the parametric uniform B-spline curve and surface representations. Tech. Rep. CSD 83/118, Computer Science Div., Univ. of Calif., Berkeley, Calif., May 1983.
    4. BARSKY, B.A. The Beta-spline: A curve and surface representation for computer graphics and computer aided geometric design. To be published.
    5. BARSKY, B.A. Algorithms for the evaluation and perturbation of Beta-splines. To be published.
    6. BARSKY, B.A., BARTELS, R.H., AND BEATTY, J.C. An introduction to the use of splines in computer graphics. CS-83-9, Dept. of Computer Science, Univ. of Waterloo, Waterloo, Ontario, Canada, 1983.
    7. BARSKY, B.A., AND BEATTY, J.C. Varying the betas in Beta-splines. CS-82-49, Dept. of Computer Science, Univ. of Waterloo, Waterloo, Ontario, Canada, 1982. Also Tech. Rep. CSD 82/112, Computer Science Division, Univ. of Calif., Berkeley, Calif., Dec. 1982.
    8. BÉZIER, P.E. Emploi des Machines à Commande Numérique. Masson et Cie., Paris, 1970. English ed., Numerical Control—Mathematics and Applications, A. R. Forrest and A. F. Pankhurst, Trans.,Wiley, New York, 1972.
    9. BÉZIER, P.E. Essai de définition numérique des courbes et des surfaces expérimentales. Ph.D. dissertation, Univ. Pierre et Marie Curie, Paris, Feb. 1977.
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    15. FATEMAN, R.J. Addendum to the MACSYMA Reference Manual for the VAX. Univ. of Calif., Berkeley, Calif., 1982.
    16. FAUX, I.D., AND PRATT, M.J. Computational Geometry for Design and Manufacture. Wiley, New York, 1979.
    17. GORDON, W.J., AND RIESENFELD, R.F. B-spline curves and surfaces. In Computer Aided Geometric Design, R. E. Barnhill and R. F. Riesenfeld, Eds. Academic Press, New York, 1974, pp. 95-126.
    18. LANE, J.M. Shape operators for computer aided geometric design. Ph.D. dissertation, Univ. of Utah, Salt Lake City, Utah, June 1977.
    19. NIELSON, G.M. Some piecewise polynomial alternatives to splines under tension. In Computer Aided Geometric Design, R. E. Barnhill and R. F. Riesenfeld, Eds. Academic Press, New York, 1974, pp. 209-235.
    20. NIELSON, G.M. Computation of Nu-splines. Dept. of Mathematics, Arizona State Univ., Tempe, Ariz., June 1974.
    21. PILCHER, D.T. Smooth approximation of parametric curves and surfaces. Ph.D. dissertation, Univ. of Utah, Salt Lake City, Utah, Aug. 1973.
    22. RIESENFELD, R.F. Applications of B-spline approximation to geometric problems of computer-aided design. Ph.D. dissertation, Dept. of Systems and Information Science, Syracuse Univ., New York, May 1973.
    23. SCHWEIKERT, D.G. An interpolation curve using a spline in tension. J. Math. Phys. 45 (1966), 312-317.


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