On cylindrical directions of regular surfaces in $\mathbb{R}^4$ derived from contact with spherical cylinders

Abstract

We investigate the contact of regular surfaces in $\mathbb{R}^4$ with spherical cylinders $S^2(r) \times \mathbb{R}$. We introduce cylindrical directions of these surfaces, defined by the axes of spherical cylinders having degenerate contact with the surfaces, and show that these directions capture the ridge condition of projected regular surfaces under orthogonal projection along these directions.