Tubular Hypersurfaces with Respect to the Quasi-frame in Four-dimensional Euclidean Space

This study focuses on the differential geometric analysis of a newly defined class of hypersurfaces within four-dimensional Euclidean space E⁴. These hypersurfaces are intrinsically generated as the envelope of a one-parameter family of spheres with constant radius, whose centers trace a space curve with a quasi-frame a generalized orthonormal frame specifically adapted to the curve. Initially, general parametrizations are systematically established for directional canal hypersurfaces, which encompass directional tubular hypersurfaces as a generalized case, and separately for directional tubular hypersurfaces in the four-dimensional Euclidean space E⁴. Subsequently, the fundamental geometric properties of directional tubular hypersurface are meticulously investigated. Specifically, the first and second fundamental forms, the unit normal vector field, the shape operator, and including Gaussian curvature and mean curvature are explicitly computed. Finally, this theoretical formulation is exemplified and validated through detailed illustrative examples. These examples include comprehensive symbolic computations and visualizations of the directional tubular hypersurfaces rendered in various projection subspaces, thereby elucidating both the construction methodology and the inherent geometric interpretations of the directional tubular hypersurfaces.