In applications like computer aided design, geometric models are often represented numerically as polynomial splines or NURBS, even when they originate from primitive geometry. For purposes such as redesign and isogeometric analysis, it is of interest to extract information about the underlying geometry through reverse engineering. In this work we develop a novel method to determine these primitive shapes by combining clustering analysis with approximate implicitization. The proposed method is automatic and can recover algebraic hypersurfaces of any degree in any dimension. In exact arithmetic, the algorithm returns exact results. All the required parameters, such as the implicit degree of the patches and the number of clusters of the model, are inferred using numerical approaches in order to obtain an algorithm that requires as little manual input as possible. The effectiveness, efficiency and robustness of the method are shown both in a theoretical analysis and in numerical examples implemented in Python.

在诸如计算机辅助设计之类的应用中，即使几何模型源自原始几何，也经常用数字将其表示为多项式样条或NURBS。出于重新设计和等几何分析之类的目的，通过逆向工程提取有关基础几何的信息是很有意义的。在这项工作中，我们开发了一种通过将聚类分析与近似隐式化相结合来确定这些原始形状的新颖方法。所提出的方法是自动的，并且可以在任何维度上恢复任何程度的代数超曲面。在精确算术中，该算法返回精确结果。所有必需的参数，例如补丁的隐含程度和模型的簇数，为了获得所需的人工输入尽可能少的算法，可以使用数值方法进行推断。在理论分析和用Python实现的数值示例中均显示了该方法的有效性，效率和鲁棒性。