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.

在本文中，我们描述了一个通用类 $C^{\mathrm{1个}}$平滑的有理样条，尤其可以精确描述椭圆和椭球-这是CAD和CAE的一些最重要的原语。通过所谓的“通过分析设计”兼容的提取矩阵来变换多组NURBS基函数，从而组装出单变量有理样条；不同的NURBS集被允许具有不同的多项式度和权函数。单变量样条的张量积产生多变量样条。在双变量设置中，我们描述了张量积样条的类似的“通过分析设计兼容转换”如何使包含一个或两个极性奇点的光滑表面的构造成为可能。材料是独立的，并加以介绍，以便CAD或CAE从业人员可以在支持NURBS的现有软件中轻松实现所有工具。为此，我们明确提出了矩阵（a）以NURBS形式描述样条线，以及（b）通过执行（局部）度升高和结点插入来帮助细化样条线的矩阵。最后，全部$C^{\mathrm{1个}}$ 样条线结构产生局部支持的样条线基函数，并形成一个凸凸的单位分区。