scholarly journals Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space

2016 ◽  
Vol 14 (1) ◽  
pp. 889-896 ◽  
Author(s):  
Liang Chen
Keyword(s):  
Singularity Theory ◽  
Smooth Functions ◽  
Geometric Properties ◽  
Timelike Surface

AbstractIn this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.

scholarly journals Generic affine differential geometry of plane curves

1998 ◽  
Vol 41 (2) ◽  
pp. 315-324 ◽  
Author(s):  
Shyuichi Izumiya ◽  
Takasi Sano
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Affine Differential Geometry ◽  
Plane Curves ◽  
View Point ◽  
Smooth Functions ◽  
Affine Invariants

We study affine invariants of plane curves from the view point of the singularity theory of smooth functions

scholarly journals Generic affine differential geometry of space curves

1998 ◽  
Vol 128 (2) ◽  
pp. 301-314 ◽  
Author(s):  
Shyuichi Izumiya ◽  
Takasi Sano
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Affine Differential Geometry ◽  
Smooth Functions ◽  
Space Curves ◽  
Affine Invariants

We study affine invariants of space curves from the viewpoint of singularity theory of smooth functions. With the aid of singularity theory, we define a new equi-affine frame for space curves. We also introduce two surfaces associated with this equi-affine frame and give a generic classification of the singularities of those surfaces.

On binary differential equations and umbilics

1989 ◽  
Vol 111 (1-2) ◽  
pp. 147-168 ◽  
Author(s):  
J.W. Bruce ◽  
D.L. Fidal
Keyword(s):  
Principal Direction ◽  
Singularity Theory ◽  
Smooth Functions ◽  
Additional Information ◽  
Structure Of Flow ◽  
Surfaces In Euclidean Space ◽  
Blowing Up ◽  
Direction Fields ◽  
Image Position

SynopsisIn this paper we give the local classification of solution curves of bivalued direction fields determined by the equationwhere a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.

BIFURCATION ANALYSIS ON A SELF-EXCITED HYSTERETIC SYSTEM

2004 ◽  
Vol 14 (08) ◽  
pp. 2825-2842 ◽  
Author(s):  
ZHIQIANG WU ◽  
PEI YU ◽  
KEQI WANG
Keyword(s):  
Mechanical System ◽  
Singularity Theory ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Smooth Functions ◽  
Restoring Force ◽  
Symbolic Notation ◽  
Hysteretic Damper ◽  
Bifurcation Problems ◽  
Unconstrained Problems

This paper investigates periodic bifurcation solutions of a mechanical system which involves a van der Pol type damping and a hysteretic damper representing restoring force. This system has recently been studied based on the singularity theory for bifurcations of smooth functions. However, the results do not actually take into account the property of nonsmoothness involved in the system. In particular, the transition varieties due to constraint boundaries were ignored, resulting in failure in finding some important bifurcation solutions. To reveal all possible bifurcation patterns for such systems, a new method is developed in this paper. With this method, a continuous, piecewise smooth bifurcation problem can be transformed into several subbifurcation problems with either single-sided or double-sided constraints. Further, the constrained bifurcation problems are converted to unconstrained problems and then singularity theory is employed to find transition varieties. Explicit formulas are applied to reconsider the mechanical system. Numerical simulations are carried out to verify analytical predictions. Moreover, symbolic notation for a sequence of bifurcations is introduced to easily show the characteristics of bifurcations, and also the comparison of different bifurcations. The method developed in this paper can be easily extended to study bifurcation problems with other types of nonsmoothness.

Hyperbolic worldsheets and worldlines of null Cartan curves in de Sitter 3-space

2021 ◽  
pp. 2150026
Author(s):  
Qingxin Zhou ◽  
Jingbo Xu ◽  
Zhigang Wang
Keyword(s):  
Singularity Theory ◽  
Close Relation ◽  
Dual Relationships ◽  
Normal Curve ◽  
De Sitter ◽  
Geometric Invariant ◽  
Geometric Properties ◽  
Hyperbolic Quadric ◽  
Cuspidal Edge

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

scholarly journals Singularities for One-Parameter Developable Surfaces of Curves

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 108 ◽  
Author(s):  
Qiming Zhao ◽  
Donghe Pei ◽  
Yongqiao Wang
Keyword(s):  
Singularity Theory ◽  
Geometric Properties ◽  
Developable Surfaces ◽  
Generic Singularities

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.

Mannheim curves and spherical curves

2020 ◽  
Vol 17 (07) ◽  
pp. 2050101 ◽  
Author(s):  
Yongqiao Wang ◽  
Yuan Chang
Keyword(s):  
Singularity Theory ◽  
Geometric Properties ◽  
Developable Surfaces ◽  
Classification Of Singularities

In this paper, we study cylindrical helices, Mannheim curves and Darboux developable surfaces associated to Mannheim curves. We give a method for constructing Mannheim curves from spherical curves and demonstrate that all Mannheim curves can be constructed by such a way. We also introduce the spherical Darboux images of Mannheim curves, and we give a classification of singularities of Darboux developable surfaces and spherical Darboux images. Moreover, we study the geometric properties of singularities as applications of singularity theory for spherical curves.

scholarly journals On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type

2020 ◽  
Author(s):  
Joe Kamimoto
Keyword(s):  
Infinite Order ◽  
Singularity Theory ◽  
Holomorphic Curves ◽  
Real Hypersurfaces ◽  
Sufficient Condition ◽  
Holomorphic Curve ◽  
Geometric Properties ◽  
Important Concept ◽  
Infinite Type ◽  
Newton Polyhedra

AbstractThe purpose of this paper is to investigate the geometric properties of real hypersurfaces of D’Angelo infinite type in $${{\mathbb {C}}}^n$$ C n . In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra, which is an important concept in singularity theory. More precisely, equivalence conditions are given in the case of some model hypersurfaces.

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