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Issue No.02 - February (1986 vol.6)
pp: 57-64
Ronald Goldman , Control Data Corporation
ABSTRACT
Splines were originally studied in approximation theory where the focus is on approximating explicit functions of the form y = f(x) or z = f(x,y). These splines were later adopted by mathematicians and computer sicentists for use in computer-aided geometric design (CAGD) where the emphasis was shifted to parametric curves and surfaces. Initially the continuity conditions for splines developed in approximation theory were retained in CAGD, but it was soon realized that the old constraints were unnecessarily restrictive in this new context and that they could be relaxed without losing the essential property of smoothness. Beta-splines were developed to take advantage of this new freedom by introducing shape parameters into the constraint equations. These parameters could then be manipulated by a designer to change the shape of a curve of surface in an intuitively meaningful and useful way. Another seemingly unrelated context in which shape parameters appear is in blending functions constructed from discrete urn models. The purpose of this article is to begin to unify these two independent approaches to shape parameters, and in the process apply the techniques of urn models to gain some insight into the properties of Beta-splines.
INDEX TERMS
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CITATION
Ronald Goldman, "Urn Models and Beta-Splines", IEEE Computer Graphics and Applications, vol.6, no. 2, pp. 57-64, February 1986, doi:10.1109/MCG.1986.276693
REFERENCES
1. R.N.Goldman, "An Urnful of Blending Functions," IEEE CG&A Vol. 3, No. 7, pp. 49-54 July 1983
2. R.N.Goldman, "Polya's Urn Model and Computer-Aided Geometric Design," SIAM J. Algebraic and Discrete Methods Vol. 6, No. 1, pp. 1-28 Jan. 1985
3. R.N.Goldman, "Urn Models, Approximations, and Splines," submitted for publication
4. B.A.Barsky, "The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures," 1981
5. B.A.Barsky, Computer Graphics and Geometric Modeling Using Beta-splines , Springer-Verlag 1986
6. B.A.Barsky and J.C.Beatty, "Local Control of Bias and Tension in Beta-splines," ACM Trans. Graphics Vol. 2, No. 2, pp. 109-134 Apr. 1983
7. B.A.Barsky and T.D.DeRose, "Geometric Continuity of Parametric Curves," , Computer Science Division, University of California Oct. 1984
8. I.E.Faux and M.J.Pratt, Computational Geometry for Design and Manufacture , Ellis Horwood 1979
9. T.D.DeRose, "Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer-Aided Geometric Design," 1985
10. R.N.Goldman , "Urn Models and B-splines," in preparation
11. L.L.Schumaker, Spline Functions: Basic Theory , John Wiley and Sons 1981
12. R.N.Goldman and D.C.Heath, "Linear Subdivision is Strictly a Polynomial Phenomenon," Computer-Aided Geometric Design Vol. 1, No. 3, pp. 269-278 Dec. 1984
13. R.H.Bartels and J.C.Beatty, "Beta-splines With a Difference," , Department of Computer Science, University of Waterloo May 1984
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