Extention Of Bézier Curves And Its Application

This thesis is composed of five chapters.In the first chapter, the author briefly introduces the background and the maincontent of this thesis; The second chapter, wu's a class of polynomial function ofn+1 degree that containing an adjustable constant parameterλis presented. Theyare an extension of n degree Bernstein basis functions. Properties of this new basisare analyzed, based on which a n+1 degree polynomial curve with a shapeparameterλis defined. The curve, to be calledλ-Bezier curve not only inherits themost properties of n-degree Bezier curve, such as endpoints' properties, symmetry,convex hull property, geometric invariability, affine invariance, convex-preservingproperty, variation diminishing property and so on, but also can be adjusted inshape by changing the value ofλwithout changement of control points. Whenλ=0,the curve degenerates to n-degree Bezier Curve; Based on chapter two, A class of5th degree polynomial base function contains two parametersλ,αare presentedfirst. They are extensions of quartic base functions. Accordingly, we define apolynomial curves contains two parametersλ,α, which named quatricλα-Beziercurves. This kind of curve not only inhert the properties of quatric Bezier curves, butalso is adjustable in shape and more fit close to the control polygon. This curveconverge to quatric Bezier curve whenλ=α=0. Then gives the extensions of n+1-degreeλα-Bezier cruves. The fourth chapter, a new kind of trigonometricpolynomial curve with two parameters, calledλT-Bezier curves. The offeredmethod can represent not only common polynomial curves but also some quadriccurves, elliptic curves, and so on. The enactment of parametersλ₁,λ₂ makes thecurves modulatory. The junction can come to G³ continous when two sectionsjoined together. The examples illustrate the availability of this kind of curves.