Page 134

Contents | 134 of 217

Download PDF | | Reference URL | Gallery View | Parent Record

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 134
Setname	ir_computersa
ID	105730
OCR Text	Show 118 -Q1P+1,0,􁪽􋵊J (0 ' v) -1 1 = I: I: b ( 1 , 0) ( beta 1 beta2· 0) v. . r=-2 s=-2 I' "-1􁪽􋵲r,J+s bs ( beta 1, beta2·, v) 1 1 + 􁪽􋵉 I: br(beta1, beta2; 0) Yi+r,j+s b􁪽􋴱1,0)(beta1, beta2; v) r=-2 s=-2 (XIV.8) where beta1 = v alPha1(i+1)_1,j + (1-v) alPha1(i+1)_1,j_1 and beta2 = v alpha 1 .. + (1-v) alpha 1. . 1J 1,J- 1 = v alPha?(i+1)_1,j + (1-v) alPha2(i+1)_1,j_1 = v alPha21·J· + (1-v) alpha2 .. 1,J- 1 for 0 < u < and 0 < v < 1. From equation (VII.6), it can be seen that the first term of equation (XIV.8) is equal to equation (XIV.7) scaled by a factor of beta 1, whereas the second term of each equation is identical. Thus, QPIO􁪽􋴨(O v) is not a multiple of QP,0)(1 v) and hence the unit -1+1,J ' -1j " tangent vector is not continuous. Thus the surface with continuous shape parameters does not behave analogously to the corresponding curve. Noting that the lack of continuity occurs with the continuous shape parameters, perhaps this problem could be overcome by a higher order blending of the shape parameters. The present piecewise bilinear blending could be replaced by piecewise bicubic blending for unit tangent vector continuity, or by bicubic spline blending, following Gordon's [11] spline-blended surface interpolation, for curvature vector continuity.
Reference URL	https://collections.lib.utah.edu/ark:/87278/s64f49w4/105730