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Publication Type	Technical report
School or College	College of Engineering
Department	Computing, School of
Creator	Barsky, Brian A., 1954-
Title	Beta-spline: a local representation based on shape parameters and fundamental geometric measures
Date	1981-12
Description	Abstract: A mathematical technique for curve and surface representation has been developed which reflects an underlying geometric, as opposed to algebraic nature. Elementary differential geometry concepts are reviewed and the unit tangent and curvature vectors are used as fundamental geometric measures replacing parametric derivative vectors in the continuity constraints. In addition to being more appropriate, the use of these geometric measures adds degrees of freedom which can be captured to provide further control of shape. This control is via shape parameters, beta1 and beta2, which have the property that beta1=1 indicates continuity of the parametric first derivative vector and beta1=1 with beta2=0 indicates continuity of the parametric first and second derivative vectors. The geometrical interpretation of these shape parameters and their relation to the mathematical modeling of tension is also presented. In addition to inherent tension capability, the Beta-spline representation has local control; that is, the capability of modifying one portion of the shape without altering the remainder of the shape. The combination of the geometric nature of this technique with shape control via the shape parameters as well as the presence of local control capability forms a powerful representation which will be useful in computer graphics and computer aided geometric design and modeling.
Type	Text
Publisher	University of Utah
Subject	beta-spline; geometric curves; geometric surfaces; shape parameters
Subject LCSH	Curves--Computer simulation; Surfaces--Computer simulation; Computer-aided design; Computer graphics--Research; Three-dimensional imaging--Research
Language	eng
Series	University of Utah Computer Science Technical Report
Relation is Part of	ARPANET
Format Medium	application/pdf
Format Extent	238,152,680 bytes
File Name	Barsky-The_Beta_Spline.pdf
Conversion Specifications	Original scanned with Kirtas 2400 and saved as 400 ppi uncompressed TIFF. PDF generated by LIMB™ for CONTENTdm display
ARK	ark:/87278/s64f49w4
Setname	ir_computersa
ID	105814
Reference URL	https://collections.lib.utah.edu/ark:/87278/s64f49w4

Page Metadata

Title	Page 197
Setname	ir_computersa
ID	105793
OCR Text	Show 181 173. Fuchs, Henry; Kedem, Zvi M.; and Uselton, Sam P. "Optimal Surface Reconstruction from Planar Contours", Communications of the ACM, Vol. 20, No. 10, October 1977, pp. 693-702. 174. Gauss, Karl F. "General Investigations of Curved Surfaces", 1827. Translated by Hiltebeitel, Adam and Morehead, James in General Investigations of Curved Surfaces, Raven Press, Hewlett, N.Y., 1965. 175. Gauss, Karl F. "New General Investigations of Curved Surfaces", 1825. Translated by Hiltebeitel, Adam and Morehead, James in General Investigations of Curved Surfaces, Raven Press, Hewlett, N.Y., 1965. 176. Ghezzi, C. and Tisato, F. "Interactive Computer-Aided Design for Sculptured Surfaces". In Proceedings of the 1973 International Conference on Programming Languages for Numer1CallYlControlled Machine Tools, Budapest, edited by Hatvany, Joseph, North-Holland Publishing Company, Amsterdam, April 1973. 177. Gill, J. I. Computer-Aided Design of Shell Structures Using the Finite Element Method, Ph.D. Thesis, Cambridge University, Cambridge, England, 1972. 178. Godwin, A. N. "Family of Cubic Splines with One Degree of Freedom", Computer-Aided Design, Vol. 11, No.1, January 1979, pp. 13- 18. 179. Gordon, William J. "Spline-Blended Surface Interpolation through Curve Networks", Journal of Mathematics and Mechanics, Vol. 18, No. 10, 1969, pp. 931-952: Also Resear􁪽􋵐Publication GMR- 921, General Motors Research Laboratories, September 1969. 180. Gordon, William J. "Blending-Function Methods of Bivariate and Multivariate Interpolation and Approximation", SIAM Journal on Numerical Analysis, Vol. 8, 1971, pp. 158-177. 181. Gordon, William J. "Construction of Curvilinear Co-ordinate Systems and Applications to Mesh Generation", International Journal for Numerical Methods in Engineering, Vol. 7, 1973, pp. 461-477. 182. Gordon, William J. and Riesenfeld, Richard F. "B-spline Curves and Surfaces". In Computer Aided Geometric Design, edited by Barnhill, Robert E. and Riesenfeld, Richard F., Academic Press, New York, 1974, pp. 95-126. 183. Gordon, William J. and Riesenfeld, Richard F. "Bernstein-Bezier Methods for the Computer-Aided Design of Free Form Curves and Surfaces", Journal of the ACM, Vol. 21, No.2, April 1974, pp. 293- 310. Also Research Publication GMR-1176, General Motors Research Laboratories, March 1972.
Reference URL	https://collections.lib.utah.edu/ark:/87278/s64f49w4/105793