Approximation Of Circular Arcs By Quartic Bezier Curves Based On The Newton Iteration Method

In this paper, an in-depth study of the approximation method for circular arcs by Bezier curves is given, which is frequently used in Computer Aided Geometric Design, and an approximation method for circular arcs by quartic Bezier curves based on Newton iteration method is proposed.Firstly, a brief review and summary of the development history as well as the study object of CAGD is given, and then the study history of parametric curves and surfaces, so that one can understand the importance of circular arcs; secondly, an introduction of the relevant knowledge of the approximation of circular arcs in detail is made, focusing so far both at home and abroad on different methods and properties of approximation for circular arcs by quartic Bezier curves. On the basis of analysis of the disadvantage of the existing methods, the representation of error function and get its relationship with the Hausdorff distance is obtained, which in turn discoves and points out that, using Newton iteration method can minimize the the absolute value of the maximum of the error function, so that making the Hausdorff distance as small as possible. And the smaller the Hausdorff distance is, the closer the Bezier curves to the circular arcs.When using Newton iterative method, the rationality of the iteration as well as the data used in the iterative process is discussed in the first place. For the selection of the initial value, the concept of the Golden section method is used, and the iteration results show that this selection method of initial value can get more accurate results, which is at a faster the speed of iteration. Finally we calculate the Hausdorff distances of different methods, comparing the results of Newton iteration method with the previous methods, find that the approximation curves obtained from the Newton iteration method is better.In the end, we draw a conclusion of the methods mentioned above and make a prospect of this paper. Though the results of Newton iteration method are better, it requires extensive calculation and iteration, and the requirement of the selection of the initial values is even higher, so how to improve the efficiency of computation is very important. Focusing on promoting the Newton iteration method in high efficiency in the application of higher order approximations is worthy of our continuing exploration and research.