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

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.

2020 ◽  
Vol 17 (07) ◽  
pp. 2050101 ◽  
Author(s):  
Yongqiao Wang ◽  
Yuan Chang

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.


2016 ◽  
Vol 14 (1) ◽  
pp. 889-896 ◽  
Author(s):  
Liang Chen

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.


2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yongqiao Wang ◽  
Donghe Pei ◽  
Ruimei Gao

We consider one-parameter null hypersurfaces associated with spacelike curves. The spacelike curves are in anti-de Sitter 3-space while one-parameter null hypersurfaces lie in 4-dimensional semi-Euclidean space with index 2. We classify the generic singularities of the hypersurfaces, which are cuspidal edges and swallowtails. And we reveal the geometric meanings of the singularities of such hypersurfaces by the singularity theory.


Author(s):  
Qingxin Zhou ◽  
Jingbo Xu ◽  
Zhigang Wang

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.


Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1106
Author(s):  
Jie Huang ◽  
Donghe Pei

We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By using structure functions, we can characterize the ruled surface generated by special curves.


Author(s):  
Joe Kamimoto

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.


2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.


2020 ◽  
pp. 026010602097524
Author(s):  
Darren G Candow ◽  
Philip D Chilibeck ◽  
Julianne Gordon ◽  
Emelie Vogt ◽  
Tim Landeryou ◽  
...  

Background: The combination of creatine supplementation and resistance training (10–12 weeks) has been shown to increase bone mineral content and reduce a urinary indicator of bone resorption in older males compared with placebo. However, the longer-term effects (12 months) of creatine and resistance training on bone mineral density and bone geometric properties in older males is unknown. Aim: To assess the effects of 12 months of creatine supplementation and supervised, whole-body resistance training on bone mineral density, bone geometric properties, muscle accretion, and strength in older males. Methods: Participants were randomized to supplement with creatine ( n = 18, 49–69 years, 0.1 g·kg-1·d-1) or placebo ( n = 20, 49–67 years, 0.1 g·kg-1·d-1) during 12 months of supervised, whole-body resistance training. Results: After 12 months of training, both groups experienced similar changes in bone mineral density and geometry, bone speed of sound, lean tissue and fat mass, muscle thickness, and muscle strength. There was a trend ( p = 0.061) for creatine to increase the section modulus of the narrow part of the femoral neck, an indicator of bone bending strength, compared with placebo. Adverse events did not differ between creatine and placebo. Conclusions: Twelve months of creatine supplementation and supervised, whole-body resistance training had no greater effect on measures of bone, muscle, or strength in older males compared with placebo.


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