Research Of The Optimization Problem On Bezier Curve

The so-called optimization problem is to find a set of parameter values to meet some of the best metrics under the certain conditions, and then to make some of the performance index of the system reach the maximum or minimum. The optimization problem is so important that it is widely applied in social, managerial, economical, industrial and other fields. In mathematics, we optimize by seeking the minimum or maximum of a function, In business or project, optimization problem is to minimize costs or maximize profits. Optimization problem can be divided into many types according to the nature of constraint function, objective function and the values of optimal variables. Each type of optimization problem has its specific solution based on their different natures. The optimization problem of Bezier curves is to achieve our desired effects by increasing some constraints or changing the control point of Bezier curves. The Bezier curve is widely used in Computer Aided Geometric Design because of its characteristics. On the basis of analyzing Bezier curves, this paper is to research the problem of shortest Bezier curves about how to change the control point so as to minimize the length of Bezier curves. At the same time, we study the isoperimetric type problem for Bezier curves, which is to find the largest area enclosed by the curves and the corresponding control points in a closed Bezier curves whose perimeter is given and draw the curves at the same time. On this basis, we add tangent vector as a constraint to solve the optimization problem of Bezier curves with multiple constraints.The main structure of the paper is as follows:the first chapter mainly introduces the background and significance and the current situation of optimization problem for Bezier curves, which mainly include shortest Bezier curves problem and the isoperimetric problem of Bezier curves. The second chapter introduces the concept of optimization methods and their development, and also introduces several methods for solving the optimization problems, which gives us a better understanding on the optimization method. The third chapter introduces the mathematical expression of Bezier curves for shortest Bezier curves, and then introduces the basic concept and basic principle of shortest Bezier curves and its general expression of equations. Finally, it introduces two methods for solving the shortest Bezier curves problem: Particle Swarm Optimization and Pattern search method. The method of Particle Swarm Optimization is easy to understand and implement and has strong global search ability. Because of this characteristic, the method is widely used in science and engineering. Pattern search method is mainly to find an objective function in a direction sets. The purpose of this method is to find the descent direction and to solve the problem by comparing the function values. The chapter IV is to research the isoperimetric type problem for Bezier curves of degree n. In this section, we present an efficient algorithm, which solves an isoperimetric type problem in the class of Bezier curves of degree n and gives some examples for this problem. The chapter V researches the isoperimetric problem for Bezier curves with tangent vector constraint, which is to add a constraint on the basis of chapter IV and solve the problem using Lagrange multiplier method.