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

该项目涉及连续线性组合算法和开花这两种理论工具在 CAGD（计算机辅助几何设计）中各种曲线和曲面方案的应用。 连续线性组合算法是通过采用（通常）点或向量对的连续线性组合来进行的任何算法。 示例包括 de Boor 算法和 Boehm 结插入算法。 开花是将多项式曲线或曲面与包含有关曲线或曲面的信息的多元多元仿射对称多项式相关联的技术。 这两个工具是相关的，并且之前已应用于贝塞尔曲线和 B 样条曲线和曲面，并取得了巨大成功。 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 样条、Catmull-Rom、Chebyshev 和多元样条方案。 这些工具的成功应用将导致这些方案的新算法和新属性的推导、新曲面方案的开发、CAGD 理论部分的简化和统一，并最终产生使用自由曲线和曲面进行设计和制造的更好的实用工具。

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.