mu基理论及其在计算几何中的应用

Mu基源于动曲线与动曲面理论，是用以研究曲线和曲面性质的有效代数工具。因其良好的代数与几何性质，成为联结曲线与曲面参数表示与隐式表示之间的桥梁，是Syzygy模理论在计算机辅助设计和几何建模领域中的新应用。我们将把mu基理论及算法推广到一般有理代数曲面上，并由此构造一般有理曲面自交线检测的高效计算方法；我们将研究空间有理曲线隐式化的mu基算法以及空间有理曲线相应的Rees代数结构的分析；此外我们还将进行浮点mu基及近似mu基的推广问题的研究。本项目所研究的问题是计算几何与几何建模学科的重要问题，这些问题的解决将为计算机辅助几何设计在工业领域的应用开辟更广阔的空间。项目申请人在mu基理论及算法方面有很好的基础，有望在所提问题上取得实质性进展。

The technique of mu-bases is originated from the theory of moving curves and moving surfaces, which provides a connection between the parametric forms and implicit forms of curves and surfaces due to their special algebraic and geometric properties. Mu-bases are new applications of the theory of Syzygy module in Computer Aided Geometric Design and Geometric Modeling. We shall extend the existing theory and algorithms of mu-bases to general rational surfaces, and therefore develop symbolic algorithms for the detection of self-intersection loci of general rational surfaces; for rational space curves we shall focus on their implicitization problem as well as study the structure of the Rees algebra of the ideal of the corresponding space curve; we shall also develop floating point mu-basis and approximate mu-basis algorithms. The problems involved in this project are of great significance in computational geometry and geometric modeling, which shall contribute to open up a broader space for the application of Computer Aided Geometric Design in industry. The applicants have solid background in theory and applications of mu-bases and are expected to achieve great progress in this area.