隐式代数曲线在CAGD中的性质及应用研究

【作者】 秦洪元；

【导师】 彭国华；

【作者基本信息】 西北工业大学 ， 计算数学， 2004， 硕士

【摘要】 在计算机辅助几何设计（Computer aided geometrid design）领域中，熟知有两种定义曲线曲面的方法，参数形式及隐式形式。参数形式以其构造简单，计算容易等特点而流行于世并成为几何设计的主流，然而近20年的研究与使用经验表明隐式形式也有参数形式无法比拟的优点，本文仅就隐式曲线在计算机辅助几何设计中应用及性质进行了分析和研究。其主要结果如下： 本文首先对隐式曲线研究的现状、主流和趋势等作了总的概述。在此基础上对近20年来隐式曲线在计算机辅助几何设计中的应用及各种隐式形式的曲线特点进行了总结和评价。 本文在对插值一类几何约束的隐式代数曲线的构造基础上，给出了这样的隐式三次代数曲线二阶几何连续光滑拼接的条件，并给出了实验结果。理论分析及实验结果表明这样的拼接仍有一个相应的自由度，能对曲线形状进行调节，以达到较好的设计效果。 根据G^n-1函数样条(G^n-1-functional spline)曲线的思想，本文对正方形四顶点采用三次隐式代数曲线进行插值构造出二阶几何连续光滑封闭曲线，利用这种方法，通过适当地选择基曲线，可以在不引入任何控制点的情况下，使设计出的三次隐式曲线在插值节点处具有要求的曲率，并且仍然有一个自由参数可以对曲线形状进行调整，因此具有较好的性质。 本文在参—参曲面交线基础上，详细推导了隐—隐形式两曲面空间交线的微分几何性质，这些性质包括截交及切交下的切矢，曲率矢以及高阶导矢等。这些性质的给出将有利于空间隐式曲线在造型中的进一步研究。更多还原

【Abstract】 In Computer-Aided Geometric Design field, two forms defined curves/surfaces are well known: parameter form and implicit one. Parameter form becomes mainstream of geometric design because of characteristic such as: simple construction, easy computation, etc.. While near 20 years research illustrates implicit form has advantage that parameter one hasn’t. In this paper only the application and properties of implicit curves in CAGD have been researched. The main results are as follows:Firstly the researching status quo, mainstream and tend of implicit curves are summarized in this paper. Based on these near 20 years the applicant of implicit curves in CAGD and the characteristics of all sorts of implicit one are summarized and evaluated.In this paper the G2 -continuous blending conditions of two algebraic curves implicitly defined based on the construction of algebraic curves that interpolate the geometric constraints are presented. The theoretical analysis and the experimental results demonstrate that cubic algebraic curves having G2-continuous blending still have a freedom to fit the shape of the curve.G2-continuous closed curves are constructed through interpolating the vertices of a square using cubic implicit algebraic curves by Gn-1-functional spline idea in this paper. In virtue of this ways, the cubic curves implicitly defined are constructed and a freedom is still had to adjust the curve shapes without introducing any reference points. The experimental results also illustrate the good design effect of this approach.Based on parameter-parameter surfaces intersection curve, the differential geometric properties of implicit-implicit ones are in detail in the paper. These properties include tangents, curvature vector and higher-order derivative vector in cases of transversal intersection and tangent intersection. These results will be useful to the next research of space implicit curve in geometric modeling.更多还原