有理曲面的μ基理论及其应用

项目名称： 有理曲面的μ基理论及其应用

项目编号： No.11301116

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 史晓冉

作者单位： 哈尔滨工业大学

项目金额： 23万元

中文摘要： μ基是新近出现在几何造型领域中研究曲线和曲面性质与计算的一种代数工具，它提供了一种联系曲线和曲面的参数表示与隐式表示之间的桥梁。基于曲线μ基的优异成果，我们将把μ基的理论推广到一般的有理代数曲面上。研究一般有理曲面的μ基的高效算法，给出完善的有理曲面μ基的代数与几何性质。完成μ基在曲面造型中的重要应用：计算有理曲面上的特征点（奇异点，自交线，拐点等）；构造快速的有理曲面的隐式化方法；曲面重新恰当参数化等。并且对基于隐式方程表示的曲面，可以利用μ基构造它的参数表示，达到μ基在曲面表示上的二面性。项目申请人已经完成了某些特殊曲面的μ基的研究，有良好的理论基础，有望在μ基理论上有实质性的进展，发掘μ基在计算代数几何中的价值。

中文关键词： mu 基；隐式化；动平面；有理曲面；圆锥曲线

英文摘要： The μ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of rational curves/surfaces. Based on the existing research results of μ-bases for rational curves, we shall extend the theory of μ-bases to general rational surfaces. Develop efficient algorithms to compute μ-bases for general rational surfaces, and get the perfect algebraic and geometric properties. We shall present more applications of μ-bases in rational surfaces, such as: Compute the singular points, intersection curves and inflection points on the surfaces; Construct efficient algorithms to get the implicit equation for the rational surfaces; Get new proper parametric representations and so on. On the other hand, when begin with implicit surfaces, we also can construct the μ-bases then to get parametric forms. μ-bases for surfaces connect the two representations as well as the μ-bases for curves. The applicants have achieved some good theories on μ-bases for some special rational surfaces and are expected to get more great progress in this area, develop the value of μ-bases theory on computing algebraic geometry.

英文关键词： mu-basis；implicitization；moving plane；rational surface；conic section