Many algorithms require discriminative boundaries, such as separating hyperplanes or hyperballs, or are specifically designed to work on spherical data. By applying inversive geometry, we show that the two discriminative boundaries can be used interchangeably, and that general Euclidean data can be transformed into spherical data, whenever a change in point distances is acceptable. We provide explicit formulae to embed general Euclidean data into spherical data and to unembed it back. We further show a duality between hyperspherical caps, i.e., the volume created by a separating hyperplane on spherical data, and hyperballs and provide explicit formulae to map between the two. We further provide equations to translate inner products and Euclidean distances between the two spaces, to avoid explicit embedding and unembedding. We also provide a method to enforce projections of the general Euclidean space onto hemi-hyperspheres and propose an intrinsic dimensionality based method to obtain "all-purpose" parameters. To show the usefulness of the cap-ball-duality, we discuss example applications in machine learning and vector similarity search.

利用逆向几何方法，将欧几里得数据转换为球形数据，实现了可互换使用的区分边界，并提出了嵌入和反嵌入的具体公式。研究表明球形数据中的分割超平面体积与球体之间存在对偶关系，并提供了两者之间的映射公式。此外，还提出了一种投影方法和通过内在维度方法获得“通用”参数的途径，并在机器学习和向量相似性搜索领域讨论了实例应用。

使用反演几何互换使用超平面和超球的显式公式