Research On The Construction Method Of Trigonometric Polynomial Curve And Surface

Bezier curves and B-spline curves play an important role in traditional geometric design.In recent years,with the development of the geometric industry,the traditional Bezier curve and the B-spline curve have been difficult to meet people's needs due to their own defects.At the same time,many rational forms of Bezier curves are proposed,which solves the problem of traditional methods.However,rational methods not only have progressive problems,but improper use of weight factors can be destructive to curve and surface design.In view of the above problems,a large number of quasi Bernstein basis or quasi B-like spline-like basis with shape parameters have been proposed.At present,there are many improvements on the classical B-spline method,and the uniform B-spline is the main one,but this method has not been widely used in geometric design.The reason is that there are three main insufficients in this method.?The curve constructed under the polynomial space frame cannot accurately represent the conic curve;? Only retains some basic properties of the classical B-spline method,such as geometric invariance,convex hull,affine invariance,etc.,important properties such as variation diminishing and convexity are often neglected;?Such methods can mostly achieve continuity,which has met the needs of most geometric industries,but for some high-order continuous geometric designs,these methods are difficult to achieve.However,most of the improvement methods only solve one or two problems and fail to consider them comprehensively.The purpose of this paper is to explore the basic functions that can solve the above three types of problems at the same time,to improve the content and method of CAGD,and to provide flexible and adaptable curve and surface design techniques and methods for computer-aided geometric design.The main research function space of this paper are as follows:(1)T1=span{1,sin2 t,(1-sint)(1-?sint),(1-cos t)(1-? cost)};(2)T2=span{1,sin2t,(1-sin t)2e-?sint,(1-cost)2e-?cost};(3)T3=span{1,sin2t,(1-sint)3/[1+(?-3)sint],(1-cost)3/[1+(?-3)cost]};(4)T4=span{1,sin2t,(1-?sint)(1-sint)3,(1-?cost)(1-cost)3}.