The Interpolation And Approximation With A Class Of Cubic Algebraic Curves

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.