ISSN:
1572-9265
Keywords:
AMS(MOS)
;
65D07
;
sec. class. 65D10 and 65D05
;
Splines
;
weighted splines
;
interpolation
;
smoothing
;
Bézier representation
;
B-splines
;
geometric continuity
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract The classical weighted spline introduced by Ph. Cinquin (1981), (see also K. Salkauskas (1984) and T.A. Foley (1986)) consists in minimizing ∫ a b w(t)(x″(t))2 dt under the conditionsx(t i )=y i ,i=1,...,n, where the functionw is piecewise constant on the subdivisiona〈t 1〈t 2〈...〈t n 〈b. The solution is a cubic spline, but it is notC 2. We consider here the minimization of $$\int_a^b {\frac{{\left( {x''\left( t \right)} \right)^2 }}{{q\left( t \right)}}} dt$$ whereq is a piecewise polynomial function. We shall study in detail the case whenq is continuous and piecewise linear on the subdivision. The valuesq i =q(t i ) act as shape parameters. The solution is aC 2 quartic spline, but surprisingly it has, in fact, all the advantages of the cubic spline, namely: - computing the solution leads to a symmetric tri-diagonal linear system, - computing the corresponding smoothing spline leads to a block 2×2 tri-diagonal system, - the associatedB-spline is based on 4 intervals of the subdivision (as in the case of the classical cubicB-spline). The properties of this new weighted spline will be developed and its efficiency for interpolating, smoothing or designing will be illustrated on a selection of examples. Finally, a weighted spline withG 2 continuity is described, which has three shape parameters at each knot.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02145582
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