Summary
AC 2 parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t∈[t 1,t n] to data S={(xj, yj)|j=1,...,n} is defined in terms of non-negative tension parametersτ j ,j=1,...,n−1. LetP be the polygonal line defined by the directed line segments joining the points (x j ,y j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always existτ j ,j=1,...,n−1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.
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References
Clements, J.C. (1990): Convexity-preserving piecewise rational cubic interpolation. SIAM J. Numer. Anal., Vol.27, No. 4, pp. 1016–1023
Farin, G. (1989): Rational Curves and Surfaces. In: T. Lyche, L.L. Schumaker, eds., Mathematical Methods in Computer Aided Geometric Design. Academic Press, Boston, pp. 215–238
Foley, T.A., Goodman, T.N.T., Unsworth, K. (1989): An Algorithm for Shape Preserving Parametric Interpolating Curves with C2 Continuity. In: T. Lyche, L.L. Schumaker, eds., Mathematical Methods in Computer Aided Geometric Design. Academic Press, Boston, pp. 249–259
Fritsch, F.N., Carlson, R.E. (1980): Monotone piecewise cubic interpolation. SIAM J. Numer. Anal.17, 238–246
Gregory, J.A., Delbourgo, R. (1982): Piecewise rational quadratic interpolation to monotonic data. IMA J. Numer. Anal.2, 123–130
Preparata, F.P., Shamos, M.I. (1985): Computational Geometry, An Introduction. Springer, Berlin Heidelberg New York
Millman, R.S., Parker, G.D. (1977): Elements of Differential Geometry. Prentice Hall, Englewood Cliffs, NJ
Sakai, M., Lopez de Silanes, M.C. (1986): A simple rational spline and its application to monotonic interpolation to monotonic data. Numer. Math.50, 171–182
Sapidis, N.S., Kaklis, P.D., Loukakis, T.A. (1988): A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation. Numer. Math.54, 179–192
Sapidis, N.S., Kaklis, P.D. (1988): An Algorithm for Constructing Convexity and Monotonicity-Preserving Splines in Tension. Computer Aided Geometric Design. Vol. 5. Elsevier North-Holland, Amsterdam New York, pp. 127–137
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This research was supported in part by the natural Sciences and Engineering Research Council of Canada.
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Clements, J.C. A convexity-preservingC 2 parametric rational cubic interpolation. Numer. Math. 63, 165–171 (1992). https://doi.org/10.1007/BF01385853
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DOI: https://doi.org/10.1007/BF01385853