Spacelike Surfaces with a Common Line of Curvature in Lorentz-Minkowski 3-Space

Abstract

This paper aims to study spacelike surfaces from a given spacelike curve in Minkowski 3–space. Also, we investigate the necessary and sufficient conditions for the given space-like curve to be the line of curvature on the space-like surface. Depending on the causal character of the curve, the necessary and sufficient conditions for the given space-like curve to satisfy the line of curvature and the geodesic (resp. asymptotic) requirements are also analyzed. Furthermore, we give with illustration some computational examples in support of our main results.

Kinematic Geometry of a Timelike Line Trajectory in Hyperbolic Locomotions

“…In this section, we list notations of dual Lorentzian vectors and E. Study map (See [13][14][15][16][17][18][19]): A non-null oriented line L in Minkowski 3-space E 3 1 can be appointed by a point q ∈ L and a normalized vector u of L, that is, u 2 = ±1. To have coordinates for L, one must have the moment vector u * = q × u in E 3 1 .…”

“…The dual hyperbolic, and Lorentzian (de Sitter space) unit spheres with the center 0, respectively, are [13][14][15][16][17][18][19]:…”

“…Then, a regular curve on H 2 + matches a T−like ruled surface in E 3 1 . Also, a regular curve on S 2 1 matches a S−like or T−like ruled surface in E 3 1 [13][14][15][16][17][18][19]. +m and H 2 + f , respectively.…”

“…Conversely, in the Euclidean three-space E 3 , the metric is exclusively positive definite. Therefore, the kinematic and geometrical clarifications hold significant importance in E 3 1 [13][14][15][16][17][18][19]. In this paper, we utilized the E. Study map for investigating the kinematic-geometry of a timelike (T−like) line trajectory in one-parameter hyperbolic spatial locomotions.…”

On the Timelike Circular Surface and Singularities in Minkowski 3-Space

“…A canonical CI surface is a subset of surfaces that are relatively easier to systematically and kinematically characterize. There exist two distinct categories of canonical CI surfaces (see, e.g., [14][15][16][17][18][19][20]). The first surface is referred to as the canonical CI surface, whereas the second surface is known as the non-canonical CI surface.…”