Hypersurfaces with a common asymptotic curve in the 4D Galilean space 𝔾4

Abstract

In this study, we investigate how to construct hypersurfaces using an asymptotic curve in the 4D Galilean space [Formula: see text]. Then, by utilizing the Serret–Frenet frame, we give the conditions that a curve becomes asymptotic on a hypersurface when the marching-scale functions are three types of functions. After then, an example connected to our method is given for the aim of visibility.

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

“…where l(s), m(s), U(t), V(t), f , and g are C 1 functions. Since there is no restriction attached to the given curve in Equations ( 15), (17), or (19), the surface family with β(s) as the mutual geodesic, can constantly be established by choosing suitable marching-scale functions. Clearly, Equations ( 13), ( 15), ( 17) and ( 19) are more convenient and elegant for implementations than those reported [14].…”

“…They derived necessary and sufficient conditions for this curve to be an iso-geodesic curve on the parametric surfaces [14]. The problem of constructing a hypersurface family with a mutual geodesic curve in the 4D Galilean space G 4 has been addressed in [15][16][17]. Therefore, designing a surface from a given curve in Galilean 3-space G 3 is attractive.…”

Hypersurfaces With a Common Geodesic Curve in 4D Euclidean space E4

“…However, there is little written works on differential geometry of parametric surface family in Euclidean, and non-Euclidean 4-spaces [3,6,10,13,15]. Thus, the current study hopes to serve such a need.…”

Surface Family Pair with Bertrand Pair as Common Geodesic Curves in Galilean 3-Space 𝔾3

“…On the parametric surfaces, they constructed the necessary and sufficient conditions for this curve to be an iso-geodesic curve [20]. The problem of designing a hypersurface family with common geodesic curve in 4D Galilean space G 4 has been addressed in [21][22][23].…”