【作者】 张元巨；

【导师】 黄有度；

【作者基本信息】 合肥工业大学 ， 计算数学， 2007， 硕士

【摘要】 本文一共包含五章内容第一章，简单的介绍了研究背景以及主要研究内容；第二章，介绍了吴晓勤给出的含有单参数λ的n+1次多项式基函数，其是n次Bernstein基函数的扩展；分析了这组基的性质，基于该组基定义了带有形状参数的n+1次多项式曲线。曲线不仅具有n次Bézier曲线的特性：如端点插值、端边相切、凸包性、变差缩减性、保凸性等，而且具有形状的可调性：在控制顶点不变的情况下，随着参数不同，可产生不同逼近控制多边形的曲线。当λ=0时，曲线可退化为n次Bézier曲线；第三章在第二章的基础上，作者首先给出了含有双参数λ，α的五次多项式基函数，作为四次Bernstein基函数的扩展，相应定义了含双参数λ，α的多项式曲线，称为四次λα-Bézier曲线。曲线不仅具有四次Bézier曲线的一般特性，而且具有形状的可调性和更好的逼近性。当λ=α=0时，曲线退化为四次Bézier曲线。最后将λα-Bézier曲线推广到n(n＞4)次；第四章给出了带有双参数的三角多项式曲线，称为λT-Bézier曲线。其不但可以表示一般多项式曲线，还可以表示二次曲线、超越曲线，对参数的不同设置使得曲线具有较强的可调性，在拼接时可达G~3连续，通过实例给出了该类曲线的有效性。第五章对全文进行总结与展望，提出下一步工作的设想。更多还原

【Abstract】 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.更多还原