Abstract
The construction of a range restricted bivariate C1 ( or G1 ) interpolant to scattered data is considered in which the interpolant is positive everywhere if the original data are positive. This study is motivated by earlier work in which sufficient conditions are derived on Bézier points in order to ensure that surfaces comprising cubic Bézier triangular patches are always positive and satisfy C1 continuity conditions. In the current work, simpler and more relaxed conditions are derived on the Bézier points. The gradients at the data sites are then calculated (and modified if necessary) to ensure that these conditions are satisfied. Each triangular patch of the interpolating surface is formed as a convex combination of three quartic Bézier triangular patches. Its construction is local and easily extended to include as upper and lower constraints to the interpolant surfaces of the form z = P(x,y) where P is a polynomial of degree less or equal to 4. Moreover, C1 ( or G1 ) piecewise polynomial surfaces consisting of polynomial pieces of the form z = P(x,y) on the triangulation of the data sites are also admissible constraints. A number of examples are presented.
Highlights
The properties that are most often used to quantify “shape” in shape preserving interpolation are positivity, convexity and monotonicity
This paper will propose sufficient conditions on the bivariate quartic function upon triangulation of the data in order to visualize the positive scattered data which may come from certain scientific phenomena
The sufficient conditions on the remaining Bézier ordinates will be derived to ensure the positivity of the entire patch
Summary
The properties that are most often used to quantify “shape” in shape preserving interpolation are positivity, convexity and monotonicity. [2] describes the construction of range restricted bivariate C1 interpolants to scattered data where sufficient non-negativity condition on the Bézier ordinates are derived to ensure the non-negativity of a cubic Bézier triangular patch. We will construct a range restricted positivity-preserving bivariate C1 (or G1) quartic interpolants to scattered data using similar approach adopted in [13]. Initial values of Bezier ordinates except for the values determined by gradients at vertices of each triangle are computed using the method of minimized sum of squares of principles curvartures ([7], [9]) with respect to the G1 continuity conditions [5] on each non-boundary edge over the triangular mesh using the quadratic form of an objective function [15]
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