Geometry of tubular surfaces and their focal surfaces in Euclidean 3-space

In this study, we examined the focal surfaces of tubular surfaces in Euclidean 3-space E³ . We achieved some significant results for these surfaces in accordance with the modified orthogonal frame. Additionally, we proposed a few geometric invariants that illustrated the geometric characteristics of these surfaces, such as flat, minimal, Weingarten, and linear-Weingarten surfaces, using the traditional methods of differential geometry. Additionally, the asymptotic and geodesic curves of these surfaces have been researched. At last, we presented an example as an instance of use to validate our theoretical findings.