Positivity-preserving rational bi-cubic spline interpolation for 3D positive data

This paper deals with the shape preserving interpolation problem for visualization of 3D positive data. A required display of 3D data looks smooth and pleasant. A rational bi-cubic function involving six shape parameters is presented for this objective which is an extension of piecewise rational function in the form of cubic/quadratic involving three shape parameters. Simple data dependent constraints for shape parameters are derived to conserve the inherited shape feature (positivity) of 3D data. Remaining shape parameters are left free for designer to modify the shape of positive surface as per industrial needs. The interpolant is not only local, C 1 but also it is a computationally economical in comparison with existing schemes. Several numerical examples are supplied to support the worth of proposed interpolant. Shape preserving interpolation problem for visualization of 3D positive data is one of the basic problem in computer graphics, computer aided geometric design, data visualization and engineering. It also arises frequently in many fields including military, education, art, medicine, advertising, transport, etc. Curve and surface design plays a significant role not only in these fields but also in manufacturing different products such as ship design, car modeling and airplane fuselages and wings. In many interpolation problems, it is essential that the interpolant conserves some inherited shape features of data like positivity, monotonicity and convexity. The goal of this paper is to conserve the hereditary characteristic (positivity) of 3D data. Positivity-preserving problem occurs in visualizing a physical quantity that cannot be negative which may arise if the data is taken from some scientific, social or business environments. Depreciation of the price of computers in the market is an important example of positive data. Ordinary spline methods usually ignore these characteristics thus exhibiting undesirable inflections or oscillations in resulting curves and surfaces. Due to this reason, many investigations during the past years have been directed towards shape preserving interpolation schemes which are quoted as: A rational cubic [1], bi-cubic interpolants [2,7–8] and rational bi-cubic partially blended function [10–12] have a common feature in a way that no extra knots are used for shape preservation of positive 2D and 3D data. In contrast, the piecewise cubic Hermite interpolation [3–4] and piecewise bi-cubic function [5] conserved the shape of data by inserting one or two extra knots in the interval where the interpolants do not conserve the desired shape characteristics of data. Shape preserving interpolants problem