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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(t)=x(t)i+y(t)j, t[t 1 ,t n ] to data S={(x j ,y j ) 1,,n} is defined in terms of non-negative tension parameters τ j , j=1,,n-1. Let P be the polygonal line defined by the directed line segments joining the points (x j ,y j ), j=1,,n. Sufficient conditions are derived which ensure that r(t) is a strictly convex function on strictly left/right winding polygonal line segments P. It is then proved that there always exist τ j , j=1,,n-1 for which r(t) preserves the local left/right winding properties of any polygonal line P. An example application is discussed.
MSC:
41A05Interpolation (approximations and expansions)
References:
[1]Clements, J.C. (1990): Convexity-preserving piecewise rational cubic interpolation. SIAM J. Numer. Anal., Vol.27, No. 4, pp. 1016-1023 · Zbl 0702.65010 · doi:10.1137/0727059
[2]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
[3]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
[4]Fritsch, F.N., Carlson, R.E. (1980): Monotone piecewise cubic interpolation. SIAM J. Numer. Anal.17, 238-246 · Zbl 0423.65011 · doi:10.1137/0717021
[5]Gregory, J.A., Delbourgo, R. (1982): Piecewise rational quadratic interpolation to monotonic data. IMA J. Numer. Anal.2, 123-130 · Zbl 0481.65004 · doi:10.1093/imanum/2.2.123
[6]Preparata, F.P., Shamos, M.I. (1985): Computational Geometry, An Introduction. Springer, Berlin Heidelberg New York
[7]Millman, R.S., Parker, G.D. (1977): Elements of Differential Geometry. Prentice Hall, Englewood Cliffs, NJ
[8]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 · Zbl 0632.65006 · doi:10.1007/BF01390428
[9]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 · Zbl 0636.65010 · doi:10.1007/BF01396973
[10]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