Interpolation with interval and point tension controls using cubic weighted v-splines. (English) Zbl 0626.65008

Consider the partition P: a\(=t_ 1<...<t_ n=b\) of the interval [a,b] and given sequences \(x=\{x\}^ n_{i=1}\), \(y=\{y_ i\}^ n_{i=1}\), \(v=\{v\}^ n_{i=1}\) and \(w=\{w_ i\}^{n-1}_{i=1}\) such that \(v_ i\geq 0\), \(w_ i>0\) for relevant values of i. Let us define a functional V on the space of \(C^ 1\)-cubic piecewise polynomials (pp) over [a,b] by \[ V(f)=\sum^{n-1}_{i=1}(w_ i\int^{t_ i+1}_{t_ i}[f''(t)]^ 2dt)+\sum^{n}_{i=1}v_ i(f'(t_ i))^ 2. \] The author introduces the weighted v-spline interpolant X(t) and realizes it as the minimizing function corresponding to the functional V, over all \(C^ 1\)-cubic pp-functions f satisfying the interpolatory conditions \(f(t_ i)=x_ i\) \((i=1,2,...,n)\), and certain end conditions [cf. K. Salkauskas, Rocky Mt. J. Math. 14, 239-250 (1984; Zbl 0551.41017); G. M. Nielson in: Computer Aided Geometric Design, 209-235 (1974; Zbl 0316.68006)]. The weighted v-spline interpolant corresponding to y will be denoted by Y(t).
Certain properties of the interpolant X are studied. Considering the unit parametrization: \(t_ i=i\) and the cord-length parametrization: \(t_ 1=0\), \(t_{i+1}=t_ i+d_ i\), where \(d_ i\) is the distance between \((x_ i,y_ i)\) and \((x_{i+1},y_{i+1})\), the author discusses how the parametrization affects the weighted v-spline curve (X(t),Y(t)) and how the parameters \(w_ i\) and \(v_ i\) control the shape of the curve relative to the parametrization used.
Algorithms are also developed to recover the parametric weighted v- spline. Several examples are given to see the effect on parametric weighted v-spline for different choices of w and v. Finally, the author shows the effect of the shape control parameters w and v under different parametrizations used.
Reviewer: H.P.Dikshit


65D07 Numerical computation using splines
65D05 Numerical interpolation
65D10 Numerical smoothing, curve fitting
41A15 Spline approximation


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