Blossoming & Successive Linear Combination Algorithms: An Algorithmic Approach to Computer Aided Geometric Designs

这个项目涉及两个理论工具--逐次线性组合算法和开花--在CAGD(计算机辅助几何设计)中的各种曲线和曲面方案中的应用。连续线性组合算法是通过(通常)点或向量对的连续线性组合来进行的任何算法。例如，De Boor算法和Boehm的节点插入算法。开花是将包含有关曲线或曲面的信息的多元多重仿射对称多项式与多项式曲线或曲面相关联的技术。这两个工具是相关的，以前曾成功地应用于Bezier和B-Spline曲线和曲面。在这里，这些工具将应用于CAGD感兴趣的许多其他曲线和曲面类型--例如有理B-Spline、Catmull-Rom、Chebyshev和多元Spline方案。这些工具的成功应用将导致推导出这些格式的新算法和新性质，开发新的曲面格式，简化和统一CAGD理论的部分内容，并最终形成更好的实用工具，用于使用自由曲线和曲面进行设计和制造。

This project concerns the application of two theoretical tools-successive linear combination algorithms and blossoming- to a variety of curve and surface schemes in CAGD (computer aided geometric design). A successive linear combination algorithm is any algorithm that proceeds by taking successive linear combinations of (usually) pairs of points or vectors. Examples include the de Boor algorithm and Boehm's knot insertion algorithm. Blossoming is the technique of associating with a polynomial curve or surface a multivariate multiaffine symmetric polynomial that contains information about the curve or surface. The two tools are related, and have been applied previously to Bezier and B-spline curves and surfaces with great success. Here these tools will be applied to many of the other curve and surface types-such as rational B-spline, Catmull-Rom, Chebyshev, and multivariate spline schemes-that are of interest to CAGD. Successful application of these tools will result in derivation of new algorithms and new properties for these schemes, development of new surface schemes, simplification and unification of parts of the theory of CAGD, and ultimately in better practical tools for design and manufacture using freeform curves and surfaces.