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Parametric Curves, Part Two
January/February 1990 (vol. 10 no. 1)
pp. 60-68

Some observations are made concerning the source and nature of shape parameters. It is then described how Bezier curve segments can be stitched together with G/sup 1/ or G/sup 2/ continuity, using geometric constructions. These constructions lead to the development of geometric constructions for quadratic G/sup 1/ and cubic G/sup 2/ Beta-splines. A geometrically continuous subclass of Catmull-Rom splines based on geometric continuity and possessing shape parameters is discussed.

1. B.A. Barsky and T.D. DeRose, "Geometric Continuity of Parametric Curves: Three Equivalent Characterizations,"CG&A, Vol. 9, No. 6, Nov. 1989, pp. 60-68.
2. B.A. Barsky, "A Description and Evaluation of Various 3-D Models,"CG&A, Vol. 4, No. 1, Jan. 1984, pp. 38-52.
3. 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.
4. P.E. Bezier,Emploi des machines a commande numerique, Masson et Cie., Paris, 1970; trans. A.R. Forrest and A.F. Pankhurst,Numerical Control--Mathematics and Applications, John Wiley, London, 1972.
5. P.E. Bezier, "Mathematical and Practical Possibilities of UNISURF," inComputer Aided Geometric Design, R.E. Barnhill and R.F. Riesenfeld, eds., Academic Press, New York, 1974, pp. 127-152.
6. P.E. Bezier,Essai de definition numerique des courbes et des surfaces experimentales [An Attempt to Define Experimental Curves and Surfaces Numerically], doctoral dissertation, Pierre and Marie Curie University, Paris, 1977.
7. W. Boehm, G. Farin, and J. Kahmann, "A survey of curve and surface methods in CAGD,"Comput. Aided Geometric Des., vol. 1, no. 1, pp. 1-60, July 1984.
8. G. Farin,Curves and Surfaces for Computer Aided Geometric Design, Academic Press, Boston, 1988.
9. A. Fournier and B.A. Barsky, "Geometric Continuity with Interpolating Bezier Curves" (extended summary),Proc. Graphics Interface, Canadian Information Processing Soc., Toronto, 1985, pp. 337-341; revised forComputer-Generated Images: The State of the Art, N. Magnenat-Thalmann and D. Thalmann, eds., Springer-Verlag, Tokyo, 1985, pp. 153-158.
10. G. Farin, "Visually C2 Cubic Splines,"Computer-Aided Design, Vol. 14, No. 3, May 1982, pp. 137-139.
11. W. Boehm, "Curvature Continuous Curves and Surfaces,"Computer Aided Geometric Design, Vol. 2, No. 4, Dec. 1985, pp. 313-323.
12. T.D. DeRose, "Composing Bezier Simplexes,"ACM Trans. Graphics, Vol. 7, No. 3, Jan. 1988, pp. 198-221.
13. 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.
14. B.A. Barsky,Computer Graphics and Geometric Modeling Using Beta-Splines, Springer-Verlag, Heidelberg, W. Germany, 1988.
15. B.A. Barsky and J.C. Beatty, "Local Control of Bias and Tension in Beta-Splines,"ACM Trans. Graphics, Vol. 2, No. 2, Apr. 1983, pp. 109-134; also published inComputer Graphics(Proc. SIGGRAPH), Vol. 17, No. 3, July 1983, pp. 193-218.
16. B.A. Barsky and T.D. DeRose, "The Beta 2-Spline: A Special Case of the Beta-Spline Curve and Surface Representation,"CG&A, Vol. 5, No. 9, Sept. 1985, pp. 46-58; correction published in Letter to the Editor,CG&A, Vol. 7, No. 3, Mar. 1987, p. 15.
17. B.A. Barsky, T.D. DeRose, and M.D. Dippe, "An Adaptive Subdivision Method with Crack Prevention for Rendering Beta-Spline Objects," Tech. Report UCB/CSD 87/348, Computer Science Div., Dept. of Electrical Eng. and Computer Sciences, Univ. of California, Berkeley, Calif., 1987.
18. R.H. Bartels and J.C. Beatty, "Beta-Splines with a Difference," Tech. Report CS-83-40, Dept. of Computer Science, Univ. of Waterloo, Waterloo, Canada, 1984.
19. E. Cohen, "A New Local Basis for Designing with Tensioned Splines,"ACM Trans. Graphics, Vol. 6, No. 2, Apr. 1987, pp. 81-122.
20. T.N.T. Goodman, "Properties of Beta-Splines,"J. Approximation Theory, Vol. 44, No. 2, June 1985, pp. 132-153.
21. T.N.T. Goodman and K. Unsworth, "Generation of Beta-Spline Curves Using a Recurrence Relation," inFundamental Algorithms for Computer Graphics, R.A. Earnshaw, ed., NATO Advanced Study Inst. Series, Ser. F, Vol. 17, pp. 325-357.
22. T.N.T. Goodman and K. Unsworth, "Manipulating Shape and Producing Geometric Continuity inβ-Spline Curves,"CG&A, Vol. 6, No. 2, Feb. 1986, pp. 50-56.
23. B. Joe, "Rational Beta-Spline Curves and Surfaces and Discrete Beta-Splines," Tech. Report TR87-04, Dept. of Computing Science, Univ. of Alberta, Edmonton, Canada, 1987.
24. B. Joe, "Discrete Beta-Splines,"Computer Graphics(Proc. SIGGRAPH), Vol. 21, No. 4, July 1987, pp. 137-144.
25. B.A. Barsky and T.D. DeRose, "Geometric Continuity of Parametric Curves," Tech. Report UCB/CSD 84/205, Computer Science Div., Dept. of Electrical Eng. and Computer Sciences, Univ. of California, Berkeley, Calif., 1984.
26. J.M. Lane and R.F. Riesenfeld, "A Theoretical Development for the Computer Generation of Piecewise Polynomial Surfaces,"IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 2, No. 1, Jan. 1980, pp. 35-46.
27. N. Dyn and C.A. Micchelli, "Piecewise Polynomial Spaces and Geometric Continuity of Curves," Research Report 11390, IBM T.J. Watson Research Ctr., Yorktown Heights, N.Y., 1985.
28. E.E. Catmull and R.J. Rom, "A Class of Local Interpolating Splines," inComputer Aided Geometric Design, R.E. Barnhill and R.F. Riesenfeld, eds., Academic Press, New York, 1974, pp. 317-326.
29. T.D. DeRose and B.A. Barsky, "Geometric Continuity and Shape Parameters for Catmull-Rom Splines" (extended abstract),Proc. Graphics Interface, Canadian Information Processing Soc., Toronto, 1984, pp. 57-64.
30. T.D. DeRose and B.A. Barsky, "Geometric Continuity, Shape Parameters, and Geometric Constructions for Catmull-Rom Splines,"ACM Trans. Graphics, Vol. 7, No. 1, Jan. 1988, pp. 1-41.

Citation:
Brian A. Barsky, Tony D. DeRose, "Parametric Curves, Part Two," IEEE Computer Graphics and Applications, vol. 10, no. 1, pp. 60-68, Jan.-Feb. 1990, doi:10.1109/38.45811
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