Abstract
In this work, we use a kind of \(C^1\) rational interpolation splines in one and two dimensions to generate curves and surfaces with region control. Simple data-dependent sufficient constraints are derived on the local control parameters to generate \(C^1\) interpolation curves lying strictly between two given piecewise linear curves and \(C^1\) interpolation surfaces lying strictly between two given piecewise bi-cubic blending linear interpolation surfaces.
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References
Abbas M, Majid AA, Ali JM (2014) Positivity-preserving rational bi-cubic spline interpolation for 3D positive data. Appl Math Comput 234:460–476
Brodlie KW, Asim MR, Unsworth K (2005) Constrained visualization using the shepard interpolation family. Comput Graph Forum 24:809–820
Chan ES, Ong BH (1999) Range restricted scattered data interpolation using convex combination of cubic Bézier triangles. J Comput Appl Math 136:135–147
Duan Q, Wang X, Cheng F (1999) Constrained interpolation using rational cubic spline curve with linear denominators. Korean J Comput Appl Math 6:203–215
Duan Q, Djidjeli K, Price WG, Twizell EH (2000) Weighted rational cubic spline interpolation and its application. J Comput Appl Math 117:121–135
Duan Q, Wang L, Twizell EH (2005) A new \(C^2\) rational interpolation based on function values and constrained control of the interpolant curves. Appl Math Comput 161:311–322
Duan Q, Zhang Y, Wang L, Twizell EH (2006) Region control and approximation of a weighted rational interpolating curves. Commun Numer Methods Eng 22:41–53
Hussain MZ, Sarfraz M (2008) Positivity-preserving interpolation of positive data by rational cubics. J Comput Appl Math 218:446–458
Hussain MZ, Hussain M, Aqeel B (2014) Shape preserving surfaces with constraints on tension parameters. Appl Math Comput 247:442–464
Qin XB, Qin L, Xu QS (2016) \(C^1\) positive-preserving interpolation schemes with local free parameters. IAENG Int J Comput Sci 43:1–9
Qin XB, Qin L, Xu QS (2017) \(C^1\) rational interpolation schemes for modelling positive and/or monotonic 3D data. SCIENCEASIA 43:186–194
Sarfraz M, Hussain MZ, Nisar A (2010) Positive data modeling using spline function. Appl Math Comput 216:2036–2049
Walther M, Schmidt JW (1999) Range restricted interpolation using Gregory’s rational cubic splines. J Comput Appl Math 103:221–237
Zhu YP (2018) \(C^2\) Rational quartic/cubic spline interpolant with shape constraints. Res Math 73:73–127
Acknowledgements
The authors thank the anonymous referees for their insightful comments and constructive suggestions. The research is supported by the National Natural Science Foundation of China (no. 61802129), the Postdoctoral Science Foundation of China (no. 2015M571931), the Fundamental Research Funds for the Central Universities (no. 2017MS121) and the Natural Science Foundation Guangdong Province, China (no. 2018A030310381).
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Communicated by Antonio José Silva Neto.
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Zhu, Y., Wang, M. A class of \(C^1\) rational interpolation splines in one and two dimensions with region control. Comp. Appl. Math. 39, 69 (2020). https://doi.org/10.1007/s40314-020-1067-2
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DOI: https://doi.org/10.1007/s40314-020-1067-2