Markov chains and computer aided geometric design: Part II

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Markov chains and computer aided geometric design: Part II—examples and subdivision matrices

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ACM Transactions on Graphics (TOG) archive
Volume 4 , Issue 1 (January 1985) table of contents

Pages: 12 - 40

Year of Publication: 1985

ISSN:0730-0301

Author

Ronald N. Goldman

Control Data Corporation, Arden Hills, MN

Publisher

ACM New York, NY, USA

Bibliometrics

Downloads (6 Weeks): 3, Downloads (12 Months): 18, Citation Count: 2

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abstract references cited by index terms review collaborative colleagues

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ABSTRACT

In Part I, Markov chains were shown to be associated with solutions to several standard problems in computer-aided geometric design. Constraints on these Markov chains were also derived. Examples are given here of Markov chains that either satisfy some of these constraints or solve one of these problems. Subdivision matrices are also studied in special detail.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

	1	CHUNG, K.L. Elementary Probability Theory with Stochastic Processes. Springer-Verlag, New York, 1975.
	2	COHEN, E., LYCHE, T., AND RIESENFELD, R. Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics. Comput. Graphics Image Proc. 14 (1980), 87-111.
	3	DEBOOR, C. On calculating with B~splines. J. Approx. Theory 6 (1972), 50-62.
	4	GOLDMAN, R.N. Using degenerate Bezier triangles and tetrahedra to subdivide Bezier curves. Comput. Aided Des. 14, 6 (1982), 307-311.
	5	GOLDMAN, R.N. Polya's urn model and computer aided geometric design. SIAM J. Alg. Discr. Meth. 6, 1 (1985), 1-28.
	6	Ronald N. Goldman, Markov chains and computer-aided geometric design: part I - problems and constraints, ACM Transactions on Graphics (TOG), v.3 n.3, p.204-222, July 1984 [doi>10.1145/3870.3978]
	7	KARLIN, S. Total positivity, absorption probabilities and applications. Trans. Amer. Math. Soc. 3, 1 (1964), 33-107.
	8	KELISKY, R.P., AND RIVLIN, T.J. Iterates of Bernstein polynomials. Pacific J. Math. 21, 3 {1967), 511-520.
	9	NIELSON, G.M., RIESENFELD, R.F., AND WEISS, N.A. Iterates of Markov operators. J. Approx. Theory 17, 4 (1976), 321-331.

CITED BY 2

	Richard R. Patterson, Projective transformations of the parameter of a Bernstein-Bézier curve, ACM Transactions on Graphics (TOG), v.4 n.4, p.276-290, Oct. 1985
	Ronald N. Goldman, Markov chains and computer-aided geometric design: part I - problems and constraints, ACM Transactions on Graphics (TOG), v.3 n.3, p.204-222, July 1984

INDEX TERMS

Classification:
F. Theory of Computation
F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
F.2.2 Nonnumerical Algorithms and Problems
Subjects: Geometrical problems and computations

I. Computing Methodologies
I.3 COMPUTER GRAPHICS
I.3.5 Computational Geometry and Object Modeling
Subjects: Curve, surface, solid, and object representations

J. Computer Applications
J.6 COMPUTER-AIDED ENGINEERING
Subjects: Computer-aided design (CAD)

General Terms:
Design, Theory

REVIEW

"Paolo E. Sabella : Reviewer"

This paper is a continuation of Part I [1], in which matrices Mjk that transform curve definitions were identified as Markov chains in certain cases. That is, they satisfy :1SkM more...

Collaborative Colleagues:

Ronald N. Goldman: colleagues

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