Research Of Reparameterization Of Bézier Curves For Reducing Order In CAD

Computer Aided Geometric Design and also called CAGD for short,its core content is:representation,calculation,analysis and summary of curve and surface information in the computer modeling program.Degree reduction of curves is one of the hot issues in CAGD,its goal is reducing the degree of polynomial curves.This paper study the reparameterization in the precision degree reduction,and the application of reparameterization is wider than that in the non-reparametric case.The existing algorithm to realize the exact degree reduction of the curve by reparametricization is realized by recursion.In the form of power base of polynomial curve,the algorithm can get reparametric polynomial and the curve in the form of power base by using the property of the remainder of polynomial.However,recursive algorithms often occupy a large amount of memory in the program,the operation speed is slow,and the operation time is longer.In order to solve this problem,this paper completely abandons the idea of polynomial division and presents a new algorithm.This algorithm does not require recursion and can be used to detect any Bézier curves,and whether the degree can be reduced by polynomial reparameterization.If possible,a polynomial curve with the lowest degree will be obtained accurately.First of all,theoretical analysis of the problem.The relation between base functions in the form of high and low power bases before and after parameterization is expressed by equations,instead of solving the equations,we use the recursive relation between the lower order and the higher order polynomials to give the coefficients for reparametric polynomials directly by pyramid algorithm,and calculate the control points of the curve after reducing degree.Each control point is calculated and substituted into the other equations associated to verify that the correct degree reduction is obtained.In this process,it is guaranteed that if the reparametric polynomial exists,and the polynomial for reparameterization is unique to within a scale factor and a constant.And then the algorithm is given,which input the control points of any Bézier curve.The single-point and straight-line cases are judged first,then converted to a power-base form for judgment and calculation,and finally returned to Bézier form,and the Boolean quantity that indicates whether degree reduction by reparameterization is output.When the order can be reduced,output of the control points of lowest degree of Bézier curve and reparameterized polynomial.Finally,programming verification was carried out with Maple,and examples were given for even reduce to odd,even reduce to even,odd reduce to odd,and non-parametric reduction in plane and space.From which,The special case of parameter interval non-coincidence was found and corrected,In these examples,compared with the existing recursive algorithm,the algorithm in this paper greatly reduces the running time and improves the efficiency.