四元数在计算机辅助几何设计与计算机图形学中的应用研究

Quaternions are a kind of number system which extend complex numbers to 4-dimensional space, and form four-dimensional associative normed division algebra over real numbers. In the field of computer graphics, quaternions are generally used to express rotation and orientation. They make rotation calculation more simple, convenient and elegant. So unit quaternion curves are often used to express orientation curves for rigid body motion in 3D animation. Quaternions have been being more widely used in many fields. This project intends to carry out the research on quaternions and their applications in computer aided geometric design and computer graphics. The topics include: (A)Construction of fairing unit quaternion interpolation spline curves; (B)Progressive iteration approximation algorithm and convergence analysis for unit quaternion B-spline curves ; (C)Dual quaternion perspective projection method and its applications. These topics are not only of great theoretical significance, but also of high practical values in developing China's animation industry, improving the level of high-tech product design in CAD , enhancing the market competitiveness of the products, etc.

四元数把复平面推广到四维空间，形成实数上的四维结合赋范可除代数。在计算机图形学领域，四元数一般用来表达旋转和朝向，它使得旋转计算更加简洁、方便、优雅。因此单位四元数曲线常用来表示3D动画中刚体运动的朝向曲线。四元数的应用领域正不断拓展。本项目拟开展四元数在计算机辅助几何设计与计算机图形学中的应用研究，具体内容包括：一、光顺的单位四元数插值样条曲线的构造；二、单位四元数B样条曲线的渐进迭代逼近算法及收敛性分析；三、对偶四元数透视投影方法及其应用研究。本课题不仅具有重要的理论意义，而且对发展我国动漫产业，增强产品设计与制造中的高新技术含量，增进产品的市场竞争能力等具有实际价值。

由于四元数表达旋转的简洁性和直观性，单位四元数曲线常用来表示3D动画中刚体运动的朝向曲线。四元数的应用领域正不断拓展。本项目预定开展四元数在计算机辅助几何设计与计算机图形学中的应用研究，具体内容包括：一、光顺的单位四元数插值样条曲线的构造：C2连续的单位四元数插值样条曲线的构造、C3连续的单位四元数插值样条曲线的构造、带参数的G2连续的单位四元数超值样条曲线；二、单位四元数B样条曲线的渐进迭代逼近算法及收敛性分析；三、对偶四元数透视投影方法及其应用研究。本课题不仅具有重要的理论意义，而且对发展我国动漫产业，增强产品设计与制造中的高新技术含量，增进产品的市场竞争能力等具有实际价值。