【作者】 赵前进；

【导师】 朱功勤； 檀结庆；

【作者基本信息】 合肥工业大学 ， 计算数学， 2002， 硕士

【摘要】 近年来，国际上的学者开始强调“几何计算”。他们认为仅有“几何设计”的概念还不足以刻画CAGD这样的一个既有严格的数学基础，又有重要应用背景的学科，而“几何设计与计算”的概念不仅拓宽了该领域研究的覆盖面，也为该学科赋予了新的生命力。 本文第一章为绪论，简要回顾几何设计与计算的研究概况。第二章研究Bézier曲线曲面的降阶逼近问题，在这一章中，提出了Bézier曲线的点约束降阶逼近新问题，并给出了约束优化方法；讨论了张量积Bézier曲面的逐步降阶逼近新方法，此方法的优点是直接利用Bézier曲线的降阶逼近方法解决张量积Bézier曲面的降阶逼近问题。第三章研究NURBS曲面和有理Bézier曲线的几何约束修改问题，讨论了基于权的NURBS曲面几何约束修改和基于控制顶点或基于权的有理Bézier曲线几何约束修改。第四章研究B-样条曲线的延伸算法，提出了逐步累加弧长参数化方法。最后，利用齐次坐标讨论了NURBS曲线的延伸算法。更多还原

【Abstract】 In resent years,international scholars begin to emphasize "geometric calculation". They think that "geometric design" is not able to describe such a branch of learning such as CAGD,which has a rigorous mathematical foundation as well as important applied background. And yet,"geometric design and calculation " not only widens the studying range in the field,but also gives it new vigor.In this paper,the first chapter is introduction,in which we briefly review some know-how of geometric design and calculation. In the second chapter some problems for degree reduction of Bezier curves and surfacesare studied. In this chapter,degree reduction of Bezier curves with points constraints is proposed and the new approach of constrained optimization is brought in light. The stepwise method for degree reduction of rectangular Bezier surfaces is further discussed. The advantage of this method is that the method of curves can be directly used on rectangular surfaces for degree reduction. In the third chapter,modifying the shape of NURBS surfaces and rational Bezier curves with geometric constraints is studied. The problems based on weights are discussed. In the chapter four,extension algorithms for B-spline curves are investigated. The new step-iterative method of parameterization by accumulative arc length is proposed. Finally,extension algorithms for NURBS curves are discussed by means of homogeneous coordinates.更多还原