Spacelike Sweeping Surfaces and Singularities in Minkowski 3-Space

Fatemah Mofarreh,Rashad A Abdel-Baky,Nadia Alluhaibi,Alessandro Gasparetto

doi:10.1155/2021/5130941

Abstract

We considered the spacelike sweeping surface with rotation minimizing frames at Minkowski 3-space E 1 3 . We presented the new geometric invariant to demonstrate geometric properties and local singularities for this surface. Then, we derived sufficient and necessary conditions of the surface to become developable ruled surfaces. Additionally, its singularities are studied. Finally, examples are illustrated to explain the applications of the theoretical results.

Highlights

Mathematical Problems in Engineering and geometric modeling as it is used for motion analysis or designing cars and ships [15,16,17,18,19]
Based on Serret–Frenet formulas, the singularity of those functions can be studied from the view of extrinsic differential geometry
Analogous to Bishop frame in Euclidean 3-space, there is a similar Lorentzian frame which named Lorentzian Bishop frame, constructed along the curve at Lorentzian space, and it is the analog of the Bishop type frame as applied to Lorentzian geometry

Summary

Preliminaries

Some definitions and basic concepts are given which will be used (for instance [8, 24, 25]). Ζ3 where ψ(s) τζ1 − κζ is the Darboux vector of the Serret–Frenet frame. E moving pseudoorthogonal frame 􏼈ξ1, ξ3, ξ3}, along the nonnull space curve α(s), is rotation minimizing frame (RMF) respecting to ξ1 in case its angular velocity ω insures 〈ω, ξ1〉 0 or, the derivatives of ξ2 and ξ3 are both parallel to ξ1. Using Definition 1, it is observed that the Serret–Frenet frame is RMF respecting to the principal normal ξ2 but not respecting to the tangent ζ1 and the binormal ζ3. E surface at Minkowski 3-space E31 named the timelike surface in case the induced metric at the surface is the Lorentz metric and it named the spacelike surface in case the induced metric at the surface is a positive definite Riemannian metric, which means the normal vector on spacelike (timelike) surface is the timelike (spacelike) vector Definition 3. e surface at Minkowski 3-space E31 named the timelike surface in case the induced metric at the surface is the Lorentz metric and it named the spacelike surface in case the induced metric at the surface is a positive definite Riemannian metric, which means the normal vector on spacelike (timelike) surface is the timelike (spacelike) vector

Spacelike Sweeping Surface and Its Singularities

Conclusion