A shape-preserving approximation by weighted cubic splines

Authors:
Tae-Wan Kim;Boris Kvasov
Affiliations:
Department of Naval Architecture and Ocean Engineering, Research Institute of Marine Systems Engineering, Seoul National University, Seoul 151-744, Republic of Korea;Department of Mathematical Modeling, Institute of Computational Technologies, Russian Academy of Sciences, Novosibirsk 630090, Russia
Venue:
Journal of Computational and Applied Mathematics
Year:
2012

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Abstract

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.