  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.
