## 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

### MSC:

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

### Citations:

Zbl 0551.41017; Zbl 0316.68006

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