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A new kind of interpolation problem. Rough shape curve and fine shape curves of a range of discrete points. (Chinese. English summary) Zbl 0595.65011
Summary: Given a set of points in the plane: \(P_ i(x_ i,y_ i)\) \((i=0,1,...,n\); \(x_ 0<x_ 1<...<x_ n)\). Suppose that any three consecutive points \(P_{i-1}\), \(P_ i\), \(P_{i+1}\) lie on a minor arc of a circle and any four consecutive points \(P_{i-1}\), \(P_ i\), \(P_{i+1}\), \(P_{i+2}\) are not concyclic. A curve is said to be a fine shape curve of the given set of points \(\{P_ i\}\), if it is a curve of class \(C^ 2\) passing through these points, the number \(\nu\) of its points of inflection is minimal, the number \(\mu\) of its points of extreme curvature is minimal and it satisfies certain additional conditions which will fix the intervals to which certain points of inflection or points of extreme curvature will belong. How can we interpolate points into the given set of points so that every fine shape curve of the new set of points will always be a fine shape curve of the original set of points ? The present paper gives a solution to the case \(\mu =\nu =0\) and indicates how to get an heuristic algorithm in general to make the new set of points to have a fine shape curve with \(\nu\) ’ points of inflection and with \(\mu\) ’ points of extreme curvature so that \(\nu '=\nu\) and the difference \(\mu\) ’-\(\mu\) is small. At the end of this paper some remarks are made to account for why this problem was raised and how it is related to the method of sequence of circular rates which is applied to the Hudon Hull Construction System.
65D05 Numerical interpolation
41A05 Interpolation in approximation theory