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.

翻译：许多算法需要判别边界，例如分离超平面或超球体，或者专门设计用于处理球形数据。通过应用反演几何，我们证明这两种判别边界可以互换使用，并且当点距离的变化可接受时，一般欧几里得数据可以转化为球形数据。我们提供了将一般欧几里得数据嵌入球形数据及反向解嵌的显式公式。我们进一步证明了超球冠（即由分离超平面在球形数据上创建的体积）与超球体之间的对偶关系，并给出了两者间映射的显式公式。我们还提供了两个空间之间内积和欧几里得距离的转换方程，以避免显式嵌入和解嵌操作。此外，我们提出了一种将一般欧几里得空间投影到半球超球面的方法，并基于本征维度提出了获取"通用"参数的方法。为展示球冠-球体对偶性的实用性，我们讨论了其在机器学习和向量相似性搜索中的示例应用。