Shape Control Of Order (4,2)~1Rational Interpolating Spline Surface

This paper mainly studies the construction of a kind of rational interpolation spline surface andits shape control. The research in the paper is the extension and improvement of the method ofrational interpolation spline based on function values. The main results are as follows:The shape control of a kind of weighted rational interpolation with shape parameters of order(4,2)1is discussed in the given region. Considering the interpolating curves to be above, below orbetween the given broken lines or piecewise quadratic curves respectively, and by falling theproblems into constraints of shape parameters, the corresponding exhibit inequations are derived andthe sufficient conditions are got. Thus, by choosing parameters and weight coefficient shape control ofrational interpolation are realized.Using the method of a kind of rational interpolation with shape parameters of order (4,2)1basedonly on function values, constriction of a kind of rational interpolation of order (4,2)1is given, someproperties are studied for the new rational spline interpolating surfaces, for example, boundaries,limits, analysis and canonical etc..It is proved that the precision of the spline isO (k3)(k is thedimension of the rectangular net), while the precision of the kind of rational interpolation surface isO (k2)at present.By calculating the Gauss curvature, a matrix representation with11-order coefficient matrix and10-degree bibariate polynomial vector function is introduced as identification of the functionalconvexity. The necessary and sufficient condition for the rational interpolation surface to be convex isderived. Combining the theory of zero points for polynomial with real coefficient, several conditionsof judging the convexity of a kind of rational surface of order (4,2)1are deduced. Control of thesurface to local convex is come true. The relative stiffness of the convexity of the exsitinginterpolation surface are solved. Examples are given to verify the validity and effectiveness.The kind of order (4,2)1rational interpolation spline surface proposed in the article has a simpleand explict mathematical representation, good geometry behavior. As a tool of interpolation, it has agood approximation, and the interpolating surface can be modified by selecting suitable parametersunder the condition that the interpolating data are not changed. Especially, the convexity of thesurface can be judged prior, and it can be processed the expected global convexity preserving surfacedesign, achieving the local modification and shape control.