Two Extensions And Researches Of The Four Bezier Curves And Surfaces

The present dissertation is concerned with two extensions and applications of the four Bezier curves and surfaces. This dissertation is divided into five chapters, of which the first chapter is an introduction that reviews the development history of CAGD briefly and introduces the status quo about curve and surface modeling and shape modification method and the main contents of this article systematically.The second chapter introduces the definition and properties of the Bezier curve and an extension of the n degree Bernstein basis functions that containing an adjustable constant parameter. A class of k trigonometric polynomial Bezier curves constructed in the trigonometric space which has characteristics of Bezier curves is presented in the third chapter. By parameterization again these curves are specified in interval [0,1].What’s more, this thesis briefly introduces quasi-three trigonometric polynomial Bezier curves and quasi-quartic trigonometric polynomial Bezier curves with two shape parameters.In the fourth chapter, the author promotes the Bezier curve on the basis of the second chapter and constructs five polynomial αβ Bezier curves, these curves have the same nature with traditional quarti Bezier curves. Whenα=1,β=(1-λ)/5, these curves degrade into quarti Bezier curves with an adjustable constant parameter in the second chapter. The author also discusses continuity condition of two pieces of curves, the geometric meaning of shape parameters and the effect on curves are discussed by the author. We can get physical modeling of the curves and surfaces by taking advantage of adjustments of the shape parameters.The author presents a class of quasi-quartic TC-Bezier curves with multiple shape parameters in the fourth chapter on the basis of the third chapter, When α=0, the curves degrade into a class of quasi-quartic trigonometric polynomial Bezier curves with two shape parameters proposed by Yang Lian and LI Jun Cheng. The author also discusses continuity condition of two pieces of curves with shape parameters and the effect different shape parameters have on curves and use them to represent accurately some quadratic curves such as the arc of a circle, an ellipse, or a parabola,a heart line and represent analogously some transcendental curves such as the cylindrical spiral curve with high precision. Physical modeling of the curves and surfaces can be presented by taking advantage of adjustments of the shape parameters.The sixth chapter summarizes major work, basic theories and innovations as well as shortcomings in the full text and prospects the research work in the future.This paper contains fifty-eight figures, zero tables and sixty references.