双曲混合多项式曲线及其性质

【作者】 胡晴峰；

【导师】 汪国昭；

【作者基本信息】 浙江大学 ， 应用数学， 2003， 硕士

【摘要】 Bézier，B-样条以及它们的有理模型在CAD／CAM系统中有着非常重要的地位，但是这些模型也有着明显的缺陷，譬如不能精确表示很多非代数曲线，如圆弧，摆线，悬链线，双曲螺线等． 本文构造了一类新的曲线，使它既保留Bézier，B-样条等模型的几何性质，又可以弥补一些它们的缺陷．本文利用多项式混合双曲形式在空间中构造了一组新的基，称为H-Bézier基，它具有类似于Bernstein基的端点性质，零点阶数，正性，正规性质，对称性等性质．进一步，文章通过控制多边形的方式定义了H-Bézier曲线．基于H-Bézier基函数的几何性质，H-Bézier曲线具有适合CAD中曲线建模和形状设计的保形性，端点插值性，凸包性，几何不变性和形状控制等很好的几何性质．而且H-Bézier曲线还引入了一个称为形状因子的参数，形状设计者不仅可以像Bézier曲线一样通过调节控制多边形来控制曲线形状，而且还可以调节形状因子来调整曲线对控制多边形的逼近程度．实验表明，对相同的控制多边形，它能够比Bézier曲线更好的保持曲线形状，从而也就能够更方便有效的控制和调整曲线形状． 进一步，本文讨论了空间K_n中的Ball曲线．我们知道Ball曲线除了具有Bézier曲线的几何性质之外，还可以快速的升阶和降阶．本文找出了空间K_n中的Ball基，我们称它为H-Ball基．由这组基通过控制多边形定义的曲线即是空间K_n中的Ball曲线．除了具有多项式空间中的Ball曲线的性质(如保形性，凸包性，快速升降阶等)之外，这类曲线同H-Bézier曲线一样具有形状因子．可以通过调节形状因子调整曲线逼近多项式的程度，从而更便于曲线的形状设计． 本文试图对NURBS等模型的改进做了一些探讨，而且提出了一种一般性构造基函数的方法，使得构造出来的基函数(如H-Bézier)具有良好的几何性质以适合CAD中的应用．更多还原

【Abstract】 Bezier, B-spline, and their rational models play an important role in CAD/CAM systems. But these models show obvious shortcomings, such as no encompassing transcendent (i.e., nonalgebraic) curves, e.g. the circle, the cycloid, the catenary and the hyperbolic helix.This paper presents a new kind of curves which not only inherit good geometric properties of Bezier and B-spline models but also conquer some drawbacks of these models. We construct a new basis, to be called H-Bezier basis, of the space Kn =span{1,t,t2,…,tn-2,sht,cht} through mixing algebraic polynomial and hyperbolicfunctions. This new basis provides properties analogous to Berstein polynomials, including symmetry, zeros of the basis functions, positivity, normalization, etc. Based on this new basis, we define a new kind of curves, to be called H-Bezier curves, with control polygon. According to properties of H-Bezier basis, H-Bezier curves are endowed with wonderful geometric properties including interpolating at endpoints, convex hull, affine invariance and optimal shape preserving. Furthermore, a shape factor of the H-Bezier curve is introduced to control the shape of the designing curves. Hence designers can adjust the shape of curves by changing not only control points but also shape factor. Our experiments show that H-Bezier model approximate to the control polygon more closely than Bezier model. So they are suitable to shape design and modeling in CAD systems.Another task of this paper is to discuss Ball curves inKn. It is well known that Ball curves preserve geometric proerties of Bezier curves and can be degree-elevated and degree-reduced more rapidly. We find out the Ball basis of the space Kn and define the H-Ball curves with this Ball basis. H-Ball curves can be well applied for curves design and shape modeling in CAD systems and related fields in term of their geometric properties.This paper try to explore an alternative model of NURBS and bring forth a general method to construct basis functions which have geometric properties analogous to Bernstein functions.更多还原