The Community for Technology Leaders
RSS Icon
Issue No.06 - November/December (1993 vol.13)
pp: 24-32
<p>The rational Beta-spline representation, which offers the features of the rational form as well as those of the Beta-spline, is discussed. The rational form provides a unified representation for conventional free-form curves and surfaces along with conic sections and quadratic surfaces, is invariant under projective transformation, and possesses weights, which can be used to control shape in a manner similar to shape parameters. Shape parameters are an inherent property of the Beta-spline and provide intuitive and natural control over shape. The Beta-spline is based on geometric continuity, which provides an appropriate measure of smoothness in computer-aided geometric design. The Beta-spline has local control with respect to vertex movement, is affine invariant, and satisfies the convex hull property. The rational Beta-spline enjoys the benefit of all these attributes. The result is a general, flexible representation, which is amenable to implementation in modern geometric modeling systems.</p>
Brian A. Barsky, "Rational Beta-Splines for Representing Curves and Surfaces", IEEE Computer Graphics and Applications, vol.13, no. 6, pp. 24-32, November/December 1993, doi:10.1109/38.252550
1. K.J. Versprille,Computer-Aided Design Applications of the Rational B-Spline Approximation Form, doctoral dissertation, Syracuse Univ., Syracuse, N.Y., 1975.
2. L. Piegl and W. Tiller, "Curve and Surface Constructions Using Rational B-Splines,"Computer-Aided Design, Vol. 19, No. 9, Nov. 1987, pp. 485-498.
3. W. Tiller, "Rational B-splines for Curve and Surface Representation,"IEEE CG&A, Vol. 3, No. 6, Sept. 1983, pp. 61-69.
4. B.A. Barsky, "The beta-spline: A local representation based on shape parameters and fundamental geometric measures," Ph.D. dissertation, Univ. of Utah, Salt Lake City, Dec. 1981.
5. B.A. Barsky,Computer Graphics and Geometric Modeling Using Beta-Splines, Springer-Verlag, Heidelberg, W. Germany, 1988.
6. B.A. Barsky and T.D. DeRose, "Geometric Continuity of Parametric Curves: Three Equivalent Characterizations,"IEEE CG&A, Vol. 9, No. 6, Nov. 1989, pp. 60-68.
7. B.A. Barsky and T.D. DeRose, "Geometric Continuity of Parametric Curves: Constructions of Geometrically Continuous Splines,"IEEE CG&A, Vol. 10, No. 1, Jan. 1990, pp. 60-68.
8. G. Seroussi and B.A. Barsky, "An Explicit Derivation of Discretely Shaped Beta-spline Basis Functions of Arbitrary Order," inMathematical Methods in Computer-Aided Geometric Design II(Proc. 1991 Conf. on Curves, Surfaces, CAGD, and Image Processing), T. Lyche and L.L. Schumaker, eds., Academic Press, Boston, 1992, pp. 567-584.
9. G. Seroussi and B.A. Barsky,A Symbolic Derivation of Beta-splines of Arbitrary Order, Tech. Report No. UCB/CSD 91/633, Computer Science Div., Electrical Engineering and Computer Sciences Dept. University of California, Berkeley, Calif., June 1991. Also Hewlett-Packard Laboratories Tech. Report No. HPL-91-87, Palo Alto, Calif.
10. R.N. Goldman and B.A. Barsky, "Beta-Continuity and Its Application to Rational Beta-Splines,"Proc. Computer Graphics 89 Conf., Smolenice, Czechoslovakia, 1989, pp. 5-11. Also available as Tech. Report UCB/CSD 88/442, Computer Science Div., Dept. of Electrical Eng. and Computer Sciences, Univ. of California, Berkeley, Calif., 1988.
11. R.N. Goldman and B.A. Barsky, "On Beta-Continuous Functions and Their Application to the Construction of Geometrically Continuous Curves and Surfaces," inMathematical Methods in Computer Aided Geometric Design, T. Lyche and L.L. Schumaker, eds., Academic Press, Boston, 1989, pp. 299- 311.
12. R.H. Bartels, J.C. Beatty, and B.A. Barsky,An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publishers, Los Altos, Calif., 1987.
13. T.N.T. Goodman and K. Unsworth, "Generation of Beta-splines Curves Using a Recurrence Relation," inFundamental Algorithms for Computer Graphics, R.A. Earnshaw, ed., NATO Advanced Study Institute Series, Series F, Vol. 17, Springer-Verlag, New York, 1985, pp. 325-357.
14. G. Farin, "Visually C2 Cubic Splines,"Computer-Aided Design, Vol. 14, No. 3, May 1982, pp. 137-139.
15. W. Boehm, "Curvature Continuous Curves and Surfaces,"Computer Aided Geometric Design, Vol. 2, No. 4, Dec. 1985, pp. 313-323.
16. E. Cohen, "A New Local Basis for Designing with Tensioned Splines,"ACM Trans. Graphics, Vol. 6, No. 2, Apr. 1987, pp. 81-122.
17. B.A. Barsky, "Introducing the Rational Beta-spline,"Proc. 3d Int'l Conf. on Engineering Graphics and Descriptive Geometry, Vol. 1, American Society for Engineering Education, Vienna, July 1988, pp. 16-27.
18. B. Joe,Rational Beta-spline Curves and Surfaces and Discrete Beta-splines, Tech. Report No. TR87-04, Dept. of Computing Science, University of Alberta, Edmonton, Canada, April 1987.
19. B. Joe, "Multiple Knot and Rational Cubic Beta-splines,"ACM Trans. on Graphics, Vol. 8, No. 2, Apr. 1989, pp. 100-120.
20. W. Boehm, "Smooth Curves and Surfaces," inGeometric Modeling: Algorithms and New Trends, G. Farin, ed., SIAM, Philadelphia, 1987, pp. 175-184.
21. W. Boehm, "Rational Geometric Splines,"Computer Aided Geometric Design, Vol. 4, No. 1-2, July 1987, pp. 67-77.
22. B. Joe, "Quartic Beta-splines,"ACM Trans. on Graphics, Vol. 9, No. 3, July 1990, pp. 301-337.
23. T. Goodman, "Constructing Piecewise Rational Curves with Frenet Frame Continuity,"Computer-Aided Geometric Design, Vol. 7, Nos. 1-4, June 1990, pp. 15-32.
569 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool