参数曲面交轨迹的理论与算法研究

参数曲面，即以参数形式表示的曲面，在计算机辅助几何设计与实体造型中的应用极为广泛。作为几何计算研究中的核心问题之一，曲面求交自二十世纪70年代以来就一直备受众多学者的关注。鉴于参数曲面在实际中的广泛应用，参数曲面的交问题也变得尤为重要。本项目主要借助计算机代数中的工具，研究参数曲面交轨迹的相关理论及算法，主要内容包括：(1)通过计算带投影的 Groebner 序列，实现两个有理参数曲面交的最小代数簇到精确参数轨迹的转化；(2)分别基于正则系统、Groebner 基和 Bezoutian 矩阵,计算三个有理参数超曲面在四维实空间中的交轨迹；(3)使用去根式化的技巧，研究可用根式表示的非有理参数曲线或曲面交轨迹的算法。通过本项目的研究，不仅可以进一步明确计算机代数中消元工具之间的各种关系，还能丰富和扩展曲面交理论的研究，促进交理论在实际中更为广泛的应用，具有重要的科学意义和实用价值。

Parametrized surfaces, i.e., the surfaces with parametric representations, have many applications in computer aided geometric design and solid modeling. Since the 1970s, the intersection problem of surfaces has been attracting much more attention of researchers as one of the core research topics in geometric computing. In consideration of the widespread use of parametrized surface, its intersection problem is also very important. This project aims to study some theories and algorithms on the intersection of parametrized surfaces, including (1)transformation from minimal variety to exact parametric locus of intersection of two rational parametrized surfaces by computing the Groebner series with projection; (2)computing the intersection loci of three rational parametrized hypersurfaces in 4-dimensional real space based regular systems, Groebner bases and Bezoutian matrices respectively; (3)studying the algorithm for computing the intersection loci of non-rational curves or surfaces with radical parametric representations by using the technique of removing radical expressions. The work supported by this project has very important scientific significance and application value. It not only makes the relationship between the elimination tools in computer algebra more clear, but also enriches and expands the research of intersection theory and its application in practice.