Hypersurface Family with a Common Isoasymptotic Curve

Abstract

In the present paper, we handle the problem of finding a hypersurface family from a given asymptotic curve in R 4. Using the Frenet frame of the given asymptotic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be asymptotic. We illustrate this method by presenting some examples.

Hypersurface families with Smarandache curves in Galilean 4-space

“…By using these studies, the problem of …nding a surface pencil from a given spatial asymptotic curve has been investigated in [4] and the necessary and su¢ cient condition for the given curve to be the asymptotic curve for the parametric surface has been stated in [1]. Also, the problem of …nding a hypersurface family from a given asymptotic curve in R 4 has been handled in [5].…”

“…The factor-decomposition form possesses an evident advantage: the designer can select di¤erent sets of functions to adjust the shape of the surface until they are grati…ed with the design, and the resulting surface is guaranteed to belong to the isogeodesic surface pencil with the curve as the common geodesic [15]. Also in [3] and [11], the three types of the marching-scale function which have three parameters have been studied in 4-dimensional Galilean and Euclidean spaces, M . ALTIN, A. KAZAN, H.B.…”

Surface Family with a Common Natural Asymptotic Lift of a Timelike Curve in Minkowski 3-space

“…First, we start with fundamentals required for the paper. Minkowski 3space 3 1 is the vector space 3 equipped with the Lorentzian inner product g given by x0  ) and a timelike vector X is said to be positive (resp. negative) if and only if…”

“…where  is the Lorentzian cross product in 3 1 [15]. The binormal vector field B(s) is the unique spacelike unit vector field perpendicular to the timelike plane…”

“…where is the Lorentzian cross product in Ó 1 3 13 . The binormal vector field B s is the unique spacelike unit vector fleld perpendicular to the timelike plane T s , N s at every point s of , such that T, N, B has the same orientation as Ó 1 3 .…”