Research On Optimization Of The NURBS Curve Fitting And Polygon Approximation Of Bézier Curve

In the fields of Computer Aided Geometric Design,the curve fitting and the linear approximation of curve have always been an important research direction of Geometric Modeling.At present,there are many methods of curve fitting and curve linear approximation,however,for the Non-Uniform Rational B-Spline(NURBS)curve fitting scattered data points and the polygon approximation Bézier curve,it is still open and has important research significance because of its wide application in industrial production.When the NURBS curve fits scattered data points,the data point parameters,the control vertices,nodes and weight factors of NURBS curves are important factors that affect the final fitting effect.In order to make the NURBS curve fit scattered data points more accurately,a NURBS curve fitting optimization algorithm based on Least Square Progressive and Iterative Approximation(LSPIA)is proposed.Firstly,determine an initial NURBS and use the LSPIA algorithm to optimize the control vertices;then the data point parameters,the nodes and the weights of the fitting curve are optimized and improved;finally,the fitting NURBS curve with high precision is obtained by iterations.To avoid or reduce the impact of other variables,they are kept unchanged,when a type of variable is optimized.The NURBS curve fitting optimization algorithm based on LSPIA makes full use of the advantages of LSPIA algorithm.In the iterative process,the control points obtained from the previous iteration can be reused,so the operation time is saved.The example of the algorithm shows that the algorithm can preserve the shape of the fitting curve.For the research of polygon approximation Bézier curve,we use the piecewise linear approximation for the Bézier curve based on least square method.Proposed a method to approximate the Bézier curve with Least Square Polygons(LSP),and present an explicit expression for the LSP vertices without further solving the linear system,the expression relates only to the control vertices of the Bézier curve and the number of edges that approximate the polygon.When the number of vertices of an approximation polygon is given,the LSP can be uniquely identified,and the error between the polygon and the curve can also be calculated.