双有理映射理论及相关自由变形方法

Birational map is a special kind of map. That is, both the map and its inverse can be represented by rational functions. On the other hand, Free-Form Deformation (FFD) is a tool for deforming geometric models and graphical images by warping the ambient space. FFD has numerous applications in geometric modeling, shape optimization, etc. However, the rational representation is needed during the process of deformation for some applications. Based on the advantages of rational functions, birational FFD method is presented. Although there are some discussions on theory of birational map in classical algebraic geometry, how to get the representation of birational map for different applications is still an important and difficult question. Currently, based on the theory of moving lines, birational quadrilateral map and birational 2D free-form deformation of degree (1, n) are presented. However, how to get birational maps of higher degree and three dimensional cases is still open problems. The construction of birational maps shall provide a novel way to solve the problems of free-form deformation, image warping, parametrization of domain for Isogeometric analysis, etc.

双有理映射为一种特殊的映射关系，即映射及其逆映射皆可表示为有理函数形式。另一方面，自由变形方法是一种对几何模型与图形化图像变形的一种重要方法，在几何造型，形状优化等方面有着广泛应用。但在众多应用场景下，需要高效地建立变形前与变形后对象之间的联系。由于有理函数的独特优势，故出现了双有理自由变形的概念。虽然双有理映射理论在经典的代数几何中有了一定的探讨，但如何在相关应用背景中构造出所需的双有理映射是一个困难且很重要的问题。借助于动直线理论，目前已经给出了平面双线性及平面(1, n)次的双有理映射的构造方法。如何推广到一般情形，即高次数与三维情形，有待深入探讨。相关的研究结果将为自由形体变换，图像扭曲(warping)等问题提供一种新的解决思路。

给定区域边界，构造区域间的双有理映射在自由形体变换，等几何分析等领域中具有重要应用。本项目研究基于复有理表示下的双有理映射构造。由于复有理表示的特性，为避开双有理映射构造中涉及的复杂的符号计算提供了新的尝试。主要研究成果是：基于复有理表示的二次双有理映射构造，推广复有理曲线理论，并研究了其合冲的特性与应用，推广复有理表示下的一般双有理映射构造，体上双有理映射的构造方法以及其它与等几何分析相关主题的研究。这些研究成果有助于为等几何分析的区域参数化提供新的方法。经过四年的研究，本项目发表有标注论文15篇，其中SCI/EI收录9篇，主要代表作发表在Computer Methods in Applied Mechanics and Engineering, Computer Aided Geometric Design, Journal of Computational and Applied Mathematics等专业顶级期刊上。培养毕业硕士7名。