Curve Fitting Of Scattered Data Points And Quadratic Curve Fitting, A New Method

Reverse engineering is a new technique developing with the development of Computer Science and the progress of data measuring technology. Its appearance has in fact changed the design mode of producing material objects in CAD system from drawing: It designs and offers a new way for fast production and rapid prototyping. Curve and surface fitting are two important problems in reverse engineering. Reconstructing the geometrical model of the object from the sample points carries on the foundation of analyzing, calculating and drawing of the object. It is also an important way to study the nature of curves and surfaces.This paper first introduces the background and importance of the study. Some existing work and methods about conic fitting are introduced. Conic is one of basic elements in reconstructing a model because of its good geometric characters, low order and flexible parameters. It plays an important role in the area of computer graphics and CAGD.In chapter 2, some methods of curve fitting given data points are explored. And we summarize those methods put forward in the area of fitting a curve to points set in order as well as unorganized points. The problem of fitting a curve from unorganized points is more common in reverse engineering. There are four methods having been studied, which are fitting with least-squares method, mathematical model method, skeleton method and dispersion method. It is usually based on the principle of Least-squares method to solve the problem of conic fitting. So in this chapter, we introduce the concept of Least-squares method.In chapter 3, we analyze the algorithms put forward by the former papers in order to find a fitting conic. For typical algorithms, we provides detailed procedure of solving. Also the objective functions adopted in conic fitting are analyzed and some problems concerned with them are proved. According to the objective function, fittingmethods can be divided into two types: objective function based on the algebraic distance or orthogonal distance.In chapter 4, a new method of conic fitting is put forward. Based on Least-squares method, finally we can obtain a homogeneous simultaneous equations system. It has only zero solution. In the methods based on algebraic distance, we often let a certain algebraic formula be a constant. So the remainder equations are not a homogeneous simultaneous and we can obtain coefficients of a fitting conic. But after the analysis, we find that every method has its own drawbacks. Approximating error for some points are too high or computation is very expensive. The method based on orthogonal distance is nonlinear in nature and finally we can obtain a nonlinear equation(equations). To solve this nonlinear equation(equations), computation will be so expensive that it is not convenient to use in the applications. So still we choose the objective function based on algebraic distance. The new method defines the objective function based on algebraic distance, and obtains six basic conies with six simple and different constraints. According to the principle of Least-squares method, we define an objective function and at last obtain six weights. The method produces the final fitting conic by adding six weights to coefficients of six basic conies. This method has cheap computation and considers all kinds of situations, so it is easily used in the applications.At the last, this paper provides the examples for comparing errors of the curves produced by the new method and former methods. It is proved that this new method is valid and practical.