【作者】 刘志平；

【导师】 王仁宏；

【作者基本信息】 大连理工大学 ， 计算数学， 2005， 硕士

【摘要】 本文主要讨论Polar形式在计算机辅助几何设计(CACD)中的应用。我们利用Polar形式实现了三角Bézier曲面和四边Bézier曲面之间的相互转化,同时通过应用,我们对Polar形式也有了一个统一、全面的认识;论文的后半部分考虑了参数曲线曲面间的几何连续性问题。具体说,我们的主要工作如下: 第一章介绍了曲面造型的发展历程、发展现状以及它的发展趋势,这是本文选题的主要依据。在1.3小节,我们给出了论文中要用到的一些基础知识。 第二章主要介绍了Polar形式的一些基本知识,包括Polar形式的定义、计算方法以及如何用Polar来表示Bézier曲线曲面。在2.5小节,我们简要地介绍了一下Bézier曲线曲面的位移算子表示形式,这一部分包含我们对常庚哲工作的一个简单推广。 在第三章,我们通过混合使用Polar形式和Bernstein基形式,给出了三角Bézier曲面和四边Bézier曲面之间的相互转化公式。我们利用函数复合的思想去处理这个问题,被复合的两个函数一个用Polar形式表示,另外一个用Bernstein基形式表示,这样就可以充分利用Polar形式的多元仿射性,直接生成新的控制顶点,简化了证明过程。在这一章的后半部分,我们进一步考虑了旨在突破参数域几何拓扑结构限制的广义Bézier曲面问题。 第四章是关于几何连续性的工作,复杂曲面造型中经常要用到分片的思想,片与片之间的光滑拼接就成为一个重要的研究课题。几何连续是一种与具体参数无关的光滑性度量,除了要考虑两片曲面间的几何连续性,更重要的是要考虑N面角点处的几何连续性问题。更多还原

【Abstract】 This paper mainly discuss some application of polar form in Computer Aided Geometry Design. Specifically, we apply the polar form in the conversion between triangular Bezier surfaces and rectangular Bezier surfaces. This problem helps us to understand the theory of polar form in a unified way. At the last half of this paper, we consider the problem of geometric continuity of parametric curves and parametric surfaces.In the first chapter, we introduce the progress of surface modeling, its current status and its future development, which are the most important reasons to choose the topic of this passage. In chapter 1.3, we present some basis knowledge which will be used in this paper.In the second chapter, we introduce the definition of polar form, the three representations of it and how to represent Bezier curves and surfaces by polar form. In chapter 2.5, we brief introduce how to represent Bezier curves and surfaces by shifting operator. In the third chapter, we use functional composition to deal with the conversion between triangular Bezier surfaces and rectangular Bezier surfaces. The polar form is used as the representation for one of the composite functions, whereas the Bernstein representation is used for another function. At the last part of this chapter, we consider the problem about generalized Bezier surface, which is defined in arbitrary convex polygon.In the chapter four, we consider the problem of geometric continuity. The idea of piecewise is often used in complex surface modeling, so the smoothness conditions become an important question for discussion. Geometric continuity is a kind of smooth measurement which is independence of parametric. We should think about geometric continuity at nodes, besides think about geometric continuity between two patches.更多还原