The Connection And Application Of Extended Bézier Curves

The design of complex curves and surfaces always need to use piecewise technique in the field of CAD/CAM, so the splicing of curves and surfaces has been generated. Quartic Bézier curves with shape parameters have not only all advantages of Bézier, excellent shape adjustments, but also much lower computing complexity than some algebraic polynomial curves. H-Bézier curves retain a lot of favorable characteristics of polynomial curves and can accurately represent transcendental curves, such as catenary curves, and so on. QT-Bézier curves not only have many properties of Bézier curves and can better adjust the shapes of the graphics by modifying parameters, but also can represent accurately ellipse and parabolic. TC-Bézier curve structure and splicing are simple and it can present accurately circle, and so on. So, it is very useful to three-dimensional shape modeling. CE-Bézier curves have excellent shape adjustability and their computation complexities are lower than those of non-algebraic polynomial curves, but they can’t represent circle accurately, and so on. Therefore, the connections between quartic Bézier curves with shape parameters and H-Bézier curves, QT-Bézier curves and H-Bézier curves, TC-Bézier curves and CE-Bézier curves are studied in the paper, and subsequent two kinds of the splicing algorithm of the paper are compared with the splicing algorithm of curves with the same type and degree, respectively. It further highlights the advantages of those two kinds of splicing algorithm of the paper.Firstly, according to the excellent properties of quartic Bézier curves with shape parameters, H-Bézier curves, QT-Bézier curves, TC-Bézier curves, CE-Bézier curves and the connection conditions between these curves, the splicing theorems of quartic Bézier curves with shape parameters and H-Bézier curves, QT-Bézier curves and H-Bézier curves, TC-Bézier curves and CE-Bézier curves are given. Secondly, their splicing algorithm and the applications are given. Finally, the splicing algorithm of QT-Bézier curves and H-Bézier curves, TC-Bézier curves and CE-Bézier curves are compared with the splicing algorithm of curves with the same type and degree, respectively. It further highlights the advantages of curves and surfaces modeling for the two kinds of splicing algorithm of the paper. The main contents of this paper are as follows:In the first chapter, the development history of curves and surfaces is introduced. Research status of connection between the curves is also discussed. In addition, the main contents of the paper are summarized.In the second chapter, the definitions and properties of QT-Bézier curves, H-Bézier curves, TC-Bézier curves, CE-Bézier curves and quartic Bézier curves with shape parameters, and conditions of connection between the curves are introduced, which lay a foundation for research of subsequent chapters.In the third chapter, based on excellent properties of quartic Bézier curves with shape parameters and H-Bézier curves, respectively, the theorems of connection between these two kinds of curves and experimental examples are given.In the fourth chapter, based on excellent properties of QT-Bézier curves and H-Bézier curves, respectively, the theorems of connection between these two kinds of curves and experimental examples are given. And we compare this method with algorithm for splicing of two H-Bézier curves. It further highlights the research purpose of the paper.In the fifth chapter, based on excellent properties of TC-Bézier curves and CE-Bézier curves, respectively, the theorems of connection between these two kinds of curves and experimental examples are given. And we compare this method with algorithm for splicing of two CE-Bézier curves. It further highlights the research purpose of the paper.In the sixth chapter, the summary of this paper is given and the problems for further study are put forward.