论文标题
$ c^1 $平滑理性的一般类：应用于确切椭圆和椭圆形的构造
A general class of $C^1$ smooth rational splines: Application to construction of exact ellipses and ellipsoids
论文作者
论文摘要
在本文中，我们描述了$ c^1 $平滑的有理样条的一般类，尤其是对椭圆形和椭圆形的确切描述，这是CAD和CAE的一些最重要的原始方法。单变量有理样条是通过通过所谓的设计 - 直通分析兼容提取矩阵来转换多组NURBS基函数来组装的；允许不同的NURB具有不同的多项式程度和重量功能。单变量样条的张量产物产生多元花键。在双变量环境中，我们描述了张量产生的张力量的相似设计 - 触及分析如何兼容兼容的转换，从而能够构建包含一个或两个极性奇异性的平滑表面。该材料是独立的，并提供了使所有工具都可以由CAD或CAE从业人员轻松实施的现有软件支持NURBS。为此，我们明确地介绍了用NURBS来描述我们花键的矩阵（a），以及（b）通过执行（局部）学位高程和结插入来帮助精炼细条。最后，所有$ c^1 $样条结构产生了局部支持的样条基函数，并形成了统一的凸形分区。
In this paper, we describe a general class of $C^1$ smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids - some of the most important primitives for CAD and CAE. The univariate rational splines are assembled by transforming multiple sets of NURBS basis functions via so-called design-through-analysis compatible extraction matrices; different sets of NURBS are allowed to have different polynomial degrees and weight functions. Tensor products of the univariate splines yield multivariate splines. In the bivariate setting, we describe how similar design-through-analysis compatible transformations of the tensor-product splines enable the construction of smooth surfaces containing one or two polar singularities. The material is self-contained, and is presented such that all tools can be easily implemented by CAD or CAE practitioners within existing software that support NURBS. To this end, we explicitly present the matrices (a) that describe our splines in terms of NURBS, and (b) that help refine the splines by performing (local) degree elevation and knot insertion. Finally, all $C^1$ spline constructions yield spline basis functions that are locally supported and form a convex partition of unity.
