Piecewise Parametric Surfaces Of N×n Degree And Their Convexity Discriminants

This paper analyzes the convexity of piecewise parametric surfaces of n×n degree,and focus on the discriminant condition of the global convexity of the free-form surface.This is a difficult and complicated problem.Several piecewise parametric surfaces commonly used in computational geometry belong to important piecewise parametric surfaces of n×n degree.This article is a generalization and deepening of the literature [30].In this paper,the free-form surface is expressed as a unified piecewise parametric matrix of n×n degree.Using algebraic analytic method,the complex calculation of Gauss curvature is simplified and deduced for the characteristics of the parametric surfaces on the rectangular domain.So(6n-4)×(6n-4)degree discriminant function is obtained,which is used to discriminate the convexity of the surface.Further,using the zero point theory of polynomial equations and the symbolic algorithm,the discriminant function variables are divided and ruled,and several sufficient conditions for the global convexity of the parametric surface are derived.Therefore,the convexity analysis and shape control of the piecewise parametric surfaces of n×n degree can be performed a priori.Using the relationship between the free-form surface basis function and the power-base function,the free-form surface are also represented as a form of the piecewise parametric surface of n×n degree and their global convexity is discussed.Then,the specific convexity discriminant conditions are derived for the bicubic B-spline surface and the double quadruple Coons surface,and examples are given to illustrate the application of these conditions in convexity analysis and discrimination.The research in this paper solves the problem of global convexity discrimination of parametric surfaces on the rectangular domain and overcomes the defects of other methods that rely heavily on the coordinate system.In particular,a double quadruple Coons surface is introduced.Without changing the information of the corner points of the surface,the shape factor can be adjusted to achieve the purpose of surface convexity.