Sphericity and geometric properties of ratios in Möbius and Laguerre geometry

Abstract Motivated by the recent need to work with spherical vertex stars in applications and theory, we contribute to the algebraic description of the sphericity of points and planes. Driven by the well-known characterization of four concyclic points through cross-ratios we extend this notion to sphericity of five points in 3-space by making use of quaternionic ratios which we call diagonal-ratios. We investigate the dual setting and obtain properties of a corresponding diagonal-ratio in terms of dual quaternions for five planes in tangential contact with a sphere.