Computing rational bisectors

Bisector construction plays an important role in many geometric computations. This article explains how to compute rational bisectors of point-surface and sphere-surface pairs. This article shows that the bisector of a point and a rational surface in R³ (3D space) is also a rational surface. This result implies that the bisector of a sphere and a surface with a rational offset is also a rational surface. Even a simple rational bisector between two spheres and that between a point and a sphere have many important applications in practice. The bisector between a cube and a sphere consists of various surface patches, some of them are the bisectors between portions of the sphere and the corners of the cube. An application that uses the bisector of two spheres (of different radii) occurs in computing an optimal path for an airplane trying to avoid radar detection. Assuming each radar has different intensity, we can model the influence regions with spheres of different radii. The optimal path must lie on the bisector surface of the spheres