【作者】 江平；

【导师】 邬弘毅； 檀结庆；

【作者基本信息】 合肥工业大学 ， 计算机辅助几何设计， 2003， 硕士

【摘要】 本文共分为四章。 第一章介绍Said-Ball曲线的定义及其性质。利用Said-Ball基函数的对偶(泛函)基，得到幂基函数在Said-Ball基下的Marsden恒等式，及从Bézier曲线到Said-Ball曲线的转换．并给出Said-Ball曲线的降阶赋值算法。 第二章介绍Wang-Ball曲线的定义及性质，以及Wang-Ball曲线的递归求值算法。作者在这里首次给出Wang-Ball基函数的对偶泛函，并由此导出从Bernstein基到Wang-Ball基的显式表示。 第三章介绍Said-Bézier型广义Ball曲线(简称为SBGB型曲线)定义，这类曲线包括Said-Ball曲线、Bézier曲线及若干条介乎二者之间的中间曲线。给出它们的递归算法，SBGB基函数的对偶泛函，并由此得到SBGB基函数与Bernstein基函数转换公式。此外，还讨论了SBGB曲线的包络及细分算法。 第四章介绍Wang-Said型广义Ball曲线(WSGB曲线)的定义，此类曲线包括Wang-Ball曲线、Said-Ball曲线及另外若干条中间曲线，以及它们的递归算法。作者在这一章里通过对WSGB曲线的深入研究，得到了本文的两个主要结果： 1．利用特殊多项式的分解与展开，推导出从Bernstein基函数到WSGB基函数的转换公式。 2．构造出WSGB基函数的对偶泛函，并应用WSGB基函数的对偶泛函，得到另一种方法实现WSGB曲线与Bézier曲线的互相转化。更多还原

【Abstract】 The thesis is composed of four chapters.In Chapter one the author gives the definition of the Said-Ball curves and their properties. By means of the dual (functional) of Said-Ball basis, the Marsden identity is got in terms of Said-Ball basis, and a degree reduction algorithm for Said-Ball curves is presented through the transform from Bezier curves to Said-Ball curves.Chapter two is focused on the Wang-Ball curves, including definition, properties and recursion algorithms. The author derives the dual functional of the Wang-Ball basis for the first time and then obtains the explicit expression of Wang-Ball basis in terms of the Bernstein basis.The emphasis of Chapter three is laid upon the generalized Ball curves of Said-Bezier type (SBGB curves) which includes Said-Ball curves, Bezier curves and some of their intermediate curves as special cases. The contents involve the recursion algorithms, the dual functional of SBGB basis and the transformation formula between SBGB basis and Bernstein basis. The convex hull of SBGB and subdivision algorithms are also discussed.Last but not least, generalized Ball curves of the Wang-Said type (WSGB curves) are introduced in Chapter four. It is pointed out that Wang-Ball curves, Said-Ball curves and some other intermediate curves are all members of the WSGB family. By making a thorough study of WSGB curves, the author acquires two main results, i.e., the transformation formula from Bernstein basis to WSGB basis and the construction of the dual functionals for WSGB basis functions.更多还原