By Wolfgang Dahmen, Mariano Gasca, Charles A. Micchelli
Assembled here's a number of articles awarded at a NATO complex STU DY INSTITUTE held at Puerto de los angeles Cruz, Tenerife, Spain throughout the interval of July tenth to twenty first, 1989. as well as the editors of those court cases Professor Larry L. Schumaker from Vanderbilt collage, Nashville, Tennessee, served as a member of the overseas organizing committee. The contents of the contribu tions fall in the heading of COMPUTATION OF CURVES AND SURFACES and for that reason tackle mathematical and computational matters touching on the dis play, modeling, interrogation and illustration of advanced geometrical items in numerous clinical and technical environments. As is the motive of the NATO ASI application the assembly was once weeks in size and the physique of the clinical actions was once equipped round favorite specialists. every one of them awarded lectures on his present learn task. We have been lucky to have 16 distinctive invited audio system representing 9 NATO nations: W. Bohm (Federal Republic of Germany), C. de Boor (USA), C.K. Chui (USA), W. Dahmen (Federal Republic of Germany), F. Fontanella (Italy), M. Gasca (Spain), R. Goldman (Canada), T.N.T. Goodman (UK), J.A. Gregory (UK), C. Hoffman (USA), J. Hoschek (Federal Republic of Germany), A. Le Mehaute (France), T. Lyche (Norway), C.A. Micchelli (USA), 1.1. Schumaker (USA), C. Traas (The Netherlands). The viewers consisted of either younger researchers in addition to proven scientists from twelve NATO international locations and several other non-NATO countries.
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Additional info for Computation of Curves and Surfaces
We list some of their most important features in Table 5. Most of these properties can be verified directly from the de Boor algorithm. For further details and explanations see [16,39]. Piecewise Polynomial-- Local Control Affine Invariant Convex Hull Property* Recursive Evaluation Algorithm -- In-Out Property Subdivision Algorithm -- Knot Insertion Techniques Two Term Differentiation Formula Interpolation of Control Points With Special Choice of Knots (tK+J =tK for J=I, ... :::;t2N"') Table 5: Properties of B-Spline Curves A few words about two of the properties this table: local control and knot insertion.
AN,O[AN,OPO + (t-tN)/(t2N-t)(AN,1 P1 + ... +(t2N-t)/(t-tN)AN,OPO)···)· The formula employed will depend upon the values of the shape parameters t1, ... ,t2N and the curve parameter t. Suppose, as is often the case, that the shape parameters are nondecreasing. If t is near tN' then the first formula will be preferred to avoid values near zero in the denominator, while if t is near tN+ 1, then the second formula will be preferred for the same reason. This Homer's method is faster than recursive evaluation because nes ed multiplication requires only O(N) multiplications whereas triangular recursion requires O(N ) multiplications.
N), j,n E Z was given in . From our point of view the central facts read as follows. 2) and generates a multiresolution analysis then > is a refinable function with respect to a finite mask a and the subdivision scheme Sa converges. 9) kEZ and the function ~(x) = 2:( -1tal_ n nEZ is an orthonormal wavelet of compact support. 9) forms a compactly supported orthonormal wavelet. For the proof of this result the reader is referred to . 8) which induce convergent subdivision schemes one may follow the construction given in .
Computation of Curves and Surfaces by Wolfgang Dahmen, Mariano Gasca, Charles A. Micchelli