专业硕士论文代写-《一类三次代数曲线的插值和

一类三次代数曲线的插值和逼近

代写硕士论文-【摘要】 曲线的插值和逼近是计算几何中的一个重要研究课题,它有着重要的理论意义和应用价值。在科学研究和外形设计中,通过测量获得一系列数据点,然后用曲线去插值和逼近数据点,接着进行曲线调节和误差估计,最后在电脑上获得几何图形。尽管这方面已经研究出很多有效的方法,如利用多项式曲线、分段三次埃尔米特曲线、分段代数样条曲线、贝齐尔曲线、B样条曲线、有理B样条曲线、三次代数曲线去插值和逼近等,但是仍有一些欠缺,比如如何保形插值、光滑拼接、调节曲线以及减小误差等。为此,本文研究一类具有几何约束的三次代数曲线的光滑拼接和保凸性问题,得到了三次代数曲线的性质、G~1、G~2光滑拼接定理及保凸性定理,进而给出了这类代数曲线的插值逼近算法,并通过计算实例说明这种算法的优点。主要工作如下:第一章是绪论部分,论述了曲线插值和逼近的重要意义和发展历程,详细分析和总结了国内外各种曲线插值和逼近方法的优点和不足,最后简要介绍了本文的研究内容。第二章介绍隐式代数曲线及切矢的定义,全局凸曲线的定义,基于几何约束的三次代数曲线的构造和三次代数曲线的合理分割以及误差的计算方法,从而为后面章节的理论研究和计算实例奠定基础。第三至六章利用几何与代数相结合的方法,研究此类三次代数曲线的光滑拼接和保凸性问题,分析了曲线在不同曲率下的凹凸性,探讨了光滑拼接的充要条件,得出控制多边形为凸时拼接曲线也为凸的性质,最后分别得到了G~1、G~2光滑拼接定理及保凸性定理。第七章给出这类三次代数曲线插值逼近的算法,并将算法所得到的曲线分别与Pade样条函数、有理二次B样条、有理三次B样条曲线作比较,通过计算实例说明这种算法具有计算简单、几何直观、易于调控、易于实现光滑拼接、保持原曲线的重要几何性质等优点,并且能将误差控制在给定的范围内,对曲线有较好的插值逼近效果。第八章在总结全文的同时,提出了需要进一步研究的问题。

硕士论文代写-【Abstract】 Interpolation and approximation of curves is an important topic of study in the field of CAGD due to its great theoretical significance and application value. In scientific research and shape designing, curves are usually applied to interpolate and approximate practical data which are obtained in measurement, then curves are adjusted and errors are estimated. At last geometric drawings are obtained on computer. Although there are many effective methods in the field, such as interpolation and approximation with polynomial curves, piecewise cubic hermite curves, piecewise algebraic spline curves, Bezier curves, B-spline curves, rational B-spline curves, cubic algebraic curves and so on,there are still some shortcomings, for example, how to conduct shape-preserving interpolation, smooth connection, curve adjustment, error reduction and so on.In order to overcome these shortcomings, we investigate the problems of smooth connection and convex-preserving for a class of cubic algebraic curves with geometric constraints. Thus the G~1 and G~2smooth connection theorems and convex- preserving theorems are obtained. The algorithm of interpolation and approximation for the class of cubic algebraic curves is also given. The advantages of the algorithm are showed by numerical examples. The main achievements are as follows:In the first chapter, the significance and the development of interpolation and approximation of curves are addressed, the advantages and disadvantages of various methods of curve interpolation and approximation at home and abroad are analyzed and summarized in detail, and finally the main contents of the paper are introduced.In the second chapter, the definitions of implicit algebraic curve and tangent vector,the definition of global convexity curves, the conformation of cubic algebraic curves with geometric constraints, the proper segmentation of cubic algebraic curves and the computational method of errors are introduced, thus lain a foundation for theoretical research and numerical examples in subsequent chapters.In the third, fourth, fifth and sixth chapters, the problems of smooth connection and convex-preserving for the class of cubic algebraic curves with geometrical and // algebraic methods are investigated. The concavity and convexity of cubic algebraic curves under different curvature are analyzed, the necessary and sufficient conditions of smooth connection are discussed and that the connected curve is convex while its control-polygon is convex is proposed. At last, G~1and G~2smooth connection theorems and convex-preserving theorems are given.In the seventh chapter, the algorithm of interpolation and approximation for the class of cubic algebraic curves is given. The curves drawn by the algorithm are compared with some other curves respectively, such as Pade spline, rational quadratic B-spline and rational cubic B-spline. Numerical experiments show that the algorithm has many advantages, such as little computation, geometric intuition, being easy to control shape and conduct smooth connection, keeping important geometric features of the original curve and so on. Moreover, the errors can be controlled in the given range. The effect of the algorithm is satisfactory.In the eighth chapter, the summary of the paper is given and the future research work is put forward.