Research On Polynomial Curves And Surfaces With Two Shape Parameters

Bézier method and B-spline method play an important role in traditional geometric modeling,which provides a flexible curve and surface technology for the modern industry.In recent years,with the development of the geometry industry,the traditional Bézier method and B-spline method have been difficult to meet people's diversified needs.Therefore,the construction of the basis function with shape parameters has become a hot issue in the research of curve and surface design.However,the improvement of the Bézier method and B-spline method at present,most of them can't express the conic accurately and neglect the all positive and variation reduction of the basis function.This paper constructs three new basis functions with two shape parameters separately in quasi-cubic algebraic polynomial space,quasi-cubic triangular polynomial space,and space combining quasi-cubic algebraic polynomial space and quasi-cubic triangular polynomial space.To solve the problems of traditional literature.The main research work of this paper is as follows:(1)A set of quasi-cubic rational Bernstein basis functions with full positiveness is proposed in quasi-cubic algebraic polynomial space T1=span{1,3t2-2t3,(1-?t)(1-t)3,(1-?+?t)t3}-bbtt.Based on the new basis,a quasi-cubic non-uniform B-spline basis with two shape parameters is constructed.Besides,the new basis extending to the triangular domain,a quasi-cubic Bernstein-Bézier basis with shape parameters on the triangular domain is constructed.(2)A singular mixed quasi-Bézier basis function with shape parameters is constructed in the quasi-cubic triangular polynomial space,sin,T2=span{1,sin2 t,(1-sint)2(1-?sint),(1-cost)2(1-?cost)} mltttt based on the idea of weight and singular mixing technology.The singular mixed quasi-Bézier curve generated by the new base can not only accurately represent quadratic curves such as elliptical arcs,circular arcs,and parabolic arcs,but also can reach1 G and2G continuity the curve when certain conditions are met.The curve is extended to the surface using a tensor product that can accurately represent ellipsoidal and spherical surfaces.(3)A new quasi cubic Bernstein basis function is obtained by combining the quasi cubic quasi Bernstein basis on quasi cubic polynomial space-span{1,3t2-2t3,(1-t)3,t3} and the quasi cubic quasi Bernstein basis on quasi cubic trigonometric polynomial spacespan {1,sin2 t,(1-sint)2(1-?sint),(1-cost)2(1-?cost)}.The curve constructed by the new basis can not only keep the advantages of algebra polynomial space and trigonometric polynomial space but also achieve G1 and G2 continuity and can accurately represent quadratic curves such as parabolic arc,elliptic arc,and circular arc.