Focal and Generalized Focal Geometry Associated with Rotational Hypersurfaces in E4

Abstract

This paper investigates focal and generalized focal geometry associated with rotational hypersurfaces in the four-dimensional Euclidean space E4. Explicit parametrizations are derived for several classes of rotational hypersurfaces, including toroidal and special Monge-type structures. The corresponding focal hypersurfaces are constructed and their geometric properties are analyzed through the computation of shape operators, Gaussian curvature, and mean curvature. Conditions for degeneracy, flatness, and minimality are determined. In addition, generalized focal hypersurfaces defined via principal curvature-dependent offsets are introduced and examined. The results provide explicit geometric descriptions and curvature characterizations that extend classical focal constructions from E3 to higher-dimensional hypersurface geometry, while highlighting the role of rotational symmetry in the structure and degeneracy of focal hypersurfaces.