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A convexity-preserving ${C}^{2}$ parametric rational cubic interpolation. (English) Zbl 0763.41001
A ${C}^{2}$ parametric rational cubic interpolant $r\left(t\right)=x\left(t\right)i+y\left(t\right)j$, $t\in \left[{t}_{1},{t}_{n}\right]$ to data $S=\left\{\left({x}_{j},{y}_{j}\right)\mid$ $1,\cdots ,n\right\}$ is defined in terms of non-negative tension parameters ${\tau }_{j}$, $j=1,\cdots ,n-1$. Let $P$ be the polygonal line defined by the directed line segments joining the points $\left({x}_{j},{y}_{j}\right)$, $j=1,\cdots ,n$. Sufficient conditions are derived which ensure that $r\left(t\right)$ is a strictly convex function on strictly left/right winding polygonal line segments $P$. It is then proved that there always exist ${\tau }_{j}$, $j=1,\cdots ,n-1$ for which $r\left(t\right)$ preserves the local left/right winding properties of any polygonal line $P$. An example application is discussed.
MSC:
 41A05 Interpolation (approximations and expansions)
References:
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