A C 1 piecewise rational trigonometric cubic function with four shape parameters has been constructed to address the problem of visualizing positive data. Simple data-dependent constraints on shape parameters are derived to preserve positivity and assure smoothness. The method is then extended to positive surface data by rational trigonometric bicubic function. The order of approximation of developed interpolant is O h 3 i. 2 2h i δ 2 i,j F xy i,j 1 α i,j − α i,j 1 0 if α i,j α i,j 1 ,
Rational cubic and bicubic trigonometric schemes are developed to conserve monotonicity of curve and surface data, respectively. The rational cubic function has four parameters in each subinterval, while the rational bicubic partially blended function has eight parameters in each rectangular patch. The monotonicity of curve and surface data is retained by developing constraints on some of these parameters in description of rational cubic and bicubic trigonometric functions. The remaining parameters are kept free to modify the shape of curve and surface if required. The developed algorithm is verified mathematically and demonstrated graphically.
The aim of this paper is to develop a local positivity preserving scheme when the data amassed from different sources is positioned at sparse points. The proposed algorithm first triangulates the irregular data using Delauny triangulation method, therewith interpolates each boundary and radial curve of the triangle by C¹ rational trigonometric cubic function. Half of the parameters in the description of the interpolant are constrained to keep up the positive shape of data while the remaining half are set free for users’ requirement. Orthogonality of trigonometric function assures much smoother surface as compared to polynomial functions. The proposed scheme can be of great use in areas of surface reconstruction and deformation, signal processing, CAD/CAM design, solving differential equations, and image restoration.
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