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

该项目涉及两个理论的应用 工具-连续线性组合算法和开花- 对CAGD（计算机辅助设计）中的各种曲线、曲面表示 辅助几何设计）。 连续的线性组合 算法是任何通过取连续的 （通常）成对点或向量的线性组合。 例子包括de Boor算法和Boehm结 插入算法 这是一种技术 将多项式曲线或曲面与多元 包含信息的多重仿射对称多项式 关于曲线或曲面。 这两个工具是相关的， 以前已经应用于贝塞尔曲线和B样条曲线， 表面上的巨大成功。 在这里，这些工具将被应用 许多其他曲线和曲面类型，如有理 B样条、Catmull-Rom、Chebyshev和多元样条 这些都是CAGD感兴趣的。 成功应用 这些工具将导致新算法的推导， 这些方案的新性质，新表面的开发 方案，简化和统一的部分理论 并最终成为更好的实用设计工具 以及使用自由形式的曲线和表面制造。

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.