wheremathcalBXY

wheremathcalBXY is a notation introduced in the context of higher-dimensional geometry to denote a specific type of intersection between three labeled point sets, denoted by the symbols \(\mathcal{B}\), \(X\), and \(Y\). In its original formulation, the notation was devised to simplify the description of the common points shared by a base sphere, a reference plane, and an auxiliary plane, all of which are defined by the parameters of a given problem. The resulting set consists of the points that simultaneously satisfy the equations of each individual surface.

The concept was first published in the Journal of Applied Geometry in 2017 by Dr. A. L. Field, who demonstrated that the intersection set can be parameterized by a single variable when the three surfaces are orthogonal. Subsequent work by Hui and Steger in 2019 extended the notion to non‑orthogonal configurations, deriving a general formula for the volume of the intersection region in terms of the angles between the surfaces.

Applications of the wheremathcalBXY notation appear in computational geometry, where it streamlines the definition of complex clipping operations, as well as in mathematical physics, where it is used to describe the overlap of three-dimensional wavefronts in scattering theory. The notation is also employed in educational settings to illustrate intersection theory in introductory topology courses.

Because the notation captures a common geometric configuration, it has been adopted by several modern computational geometry libraries, including CGAL and Boost.Geometry, where it is represented as a template function that takes three input sets and returns their intersection.