A Surface Family with a Mutual Geodesic Curve in Galilean 3-Space G3

Abstract

This article gives an approach for establishing a surface family with a mutual geodesic curve in Galilean 3-space G3. Given a smooth space curve, we derive the sufficient and necessary condition for the given curve to be geodesic on it. Furthermore, we resolve the conditions when the surface is a ruled surface. Consequently, its ability to develop with the mutual geodesic of the elements of the surface family is researched. Meanwhile, we explain this approach by submitting considerable examples.

Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space

“…where l(s), m(s), A(χ), B(χ), f , and g are C 1 functions. Since there are no constraints joined to the specified curve in Equations ( 18), (20), or (22), the set { M, M} interpolating {β(s), β(s)} as common asymptotic Bertrand curves can permanently be offered by choosing convenient marching-scale functions.…”

“…Almoneef and Abdel-Baky [21] designed a surface family with Bertrand curves to be geodesic curves. AL-Jedani and Abdel-Baky [22] inspected a surface family and developable surface family with a joint geodesic curve.…”

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

“…Two curves are a BC if there exists a bijection between them such that both curves have common principal normals [1,2,28]. BCs have been utilized as private models of offset curves in computer-aided design (CAD) and computer-aided manufacture (CAM) (see [29][30][31]). Different researchers investigated Bertrand curves, which are important for the theory of curves, in different cases with various conditions and spaces, such as in [32][33][34][35][36].…”

Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

“…[31] investigated a similar idea but used the geodesic curves instead of principal ones. In addition, a team of researchers, referred to as Li et al and cited in [32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49],conducted theoretical studies and advancements on soliton theory, submanifold theory, and other related topics. Further motivation can be found in these papers.…”