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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

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Title	Page 179
Setname	ir_computersa
ID	105775
OCR Text	Show REFERENCES 1. Barsky, Brian A. "Exponential and Polynomial Methods for Applying Tension to an Interpolating Spline Curve". Submitted for publication. 2. Barsky, Brian A. "A Study of the Parametric Uniform B-spline Curve and Surface Representations". Submitted for publication. 3. Barsky, Brian A. "End Conditions and Boundary Conditions for Uniform B-spline Curve and Surface Representations", Computers in Industry. Special Steven A. Coons Memorial Issue, to appear. 4. Barsky, Brian A. and Thomas, Spencer W. "TRANSPLINE Curve Representation System", Tech. Report No. UUCs-80-104, Department of Computer Science, University of Utah, April 1980. 5. Barsky, Brian A. and Thomas, Spencer W. "TRANSPLINE -- A System for Representing Curves Using Transformations among Four Spline Formulations", The Computer Journal, Vol. 24, No.3, August 1981, pp. 271-277. 6. Cline, Alan K. "Scalar- and Planar-Valued Curve Fitting Using Splines Under Tension", Communications of the ACM, Vol. 17, No.4, April 1974, pp. 218-220. 7. Coons, Steven A. "Surface Patches and B-spline Curves". In Computer Aided Geometric Design, edited by Barnhill, Robert E. and Riesenfeld, Richard F., Academic Press, 1974, pp. 1-16. 8. Faux, Ivor D. and Pratt, Michael J. Computational Geometry for Design and Manufacture, Ellis Horwood Ltd., 1979. 9. 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. 10. 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.
Reference URL	https://collections.lib.utah.edu/ark:/87278/s64f49w4/105775