Singularities for One-Parameter Null Hypersurfaces of Anti-de Sitter Spacelike Curves in Semi-Euclidean Space

Wang, Yongqiao; Pei, Donghe; Gao, Ruimei

doi:10.1155/2016/2319741

Cited by 5 publications

(1 citation statement)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the present paper, we investigate one-parameter developable surfaces, which are related to the space curves, as a fundamental case for the research of the highest dimensional manifolds in Euclidean 3-space. We investigated the singularities of hypersurfaces in semi-Euclidean space [7][8][9][10]. However, at least to the best of our knowledge, there exists little literature concerning the singularities of one-parameter developable surfaces related to regular space curves in Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

Singularities for One-Parameter Developable Surfaces of Curves

2019

Self Cite

View full text Add to dashboard Cite

Developable surfaces, which are important objects of study, have attracted a lot of attention from many mathematicians. In this paper, we study the geometric properties of one-parameter developable surfaces associated with regular curves. According to singularity theory, the generic singularities of these developable surfaces are classified-they are swallowtails and cuspidal edges. In addition, we give some examples of developable surfaces which have symmetric singularity models.Symmetry 2019, 11, 108 2 of 15 surfaces are sections of one-parameter developable surfaces. We also define the one-parameter support functions on regular space curves, which can be used to study the geometric properties of one-parameter developable surfaces. In fact, one-parameter developable surfaces are the discriminant sets of these functions. The main result, Theorem 2, shows that the singularities of developable surfaces are A k -singularities (k = 2, 3) of these functions.The organization of this paper is as follows: We review the concepts of ruled surfaces in Euclidean space in Section 2. In Section 3, the one-parameter developable surfaces of a space curve are defined, and we obtain two geometric invariants of the curve. We also get singularities of one-parameter developable surfaces (Theorem 1), and Theorem 2 gives the classification of these singularities in this section. The preparations for the proof of Theorem 2 are in Sections 4 and 5. In the last section, we give some examples to illustrate the main results in this paper. Basic NotationLet R 3 be 3-dimensional Euclidean space and x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 ) ∈ R 3 . We denote their standard inner product by x · y, and the norm of x is denoted by x . Let γ : I → R 3 be a curve and the tangent vector respect to t isγ(t) = dγ/dt(t). The arc-length is s(t) = t t 0

show abstract

Section: Introductionmentioning

confidence: 99%

Singularities for One-Parameter Developable Surfaces of Curves

2019

Self Cite

View full text Add to dashboard Cite

show abstract

An Investigation of Security Approaches in Industrial Robots

Bai

Lei

Zhang

et al. 2019

2019 5th International Conference on Control, Automation and Robotics (ICCAR)

View full text Add to dashboard Cite

Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

Song

Wang

2018

Journal of Function Spaces

View full text Add to dashboard Cite

In this paper, a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves located in nullcone in 4-dimensional semi-Euclidean space with index 2 are discussed. Using the unfolding theory in singularity theory, the singularities of the indefinite hypersurfaces and the indefinite surfaces are classified and the different kinds of singularities are estimated by means of a geometric invariant . Meanwhile, the definition of osculating nullcone is presented; the study shows that the differential geometric invariant of timelike curves measured also the order of the contact between a timelike curve and a osculating nullcone ONC * V 0. Finally, some relevant counterexamples are indicated.

show abstract

Singularities for One-Parameter Null Hypersurfaces of Anti-de Sitter Spacelike Curves in Semi-Euclidean Space

Cited by 5 publications

References 18 publications

Singularities for One-Parameter Developable Surfaces of Curves

Singularities for One-Parameter Developable Surfaces of Curves

An Investigation of Security Approaches in Industrial Robots

Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

Contact Info

Product

Resources

About