Ruled surfaces with vanishing second Gaussian curvature

1992 ◽  
Vol 113 (3) ◽  
pp. 177-181 ◽  
Author(s):  
D. E. Blair ◽  
Th. Koufogiorgos
Keyword(s):  
Gaussian Curvature ◽  
Ruled Surfaces ◽  
Second Gaussian Curvature

scholarly journals On the second Gaussian curvature of ruled surfaces in Euclidean $3$-space

2006 ◽  
Vol 37 (3) ◽  
pp. 221-226 ◽  
Author(s):  
Dae Won Yoon
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Mean ◽  
Second Gaussian Curvature

In this paper, we mainly investigate non developable ruled surface in a 3-dimensional Euclidean space satisfying the equation $K_{II} = KH$ along each ruling, where $K$ is the Gaussian curvature, $H$ is the mean curvature and $K_{II}$ is the second Gaussian curvature.

scholarly journals Some Characterizations of Generalized Null Scrolls

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 931 ◽  
Author(s):  
Jinhua Qian ◽  
Xueshan Fu ◽  
Seoung Dal Jung
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Structure Functions ◽  
Ruled Surfaces ◽  
Base Curve ◽  
Second Gaussian Curvature

In this work, a family of ruled surfaces named generalized null scrolls in Minkowski 3-space are investigated via the defined structure functions. The relations between the base curve and the ruling flow of the generalized null scroll are revealed. The Gaussian curvature, mean curvature, second Gaussian curvature and the second mean curvature are given and related to each other. Last but not least, the generalized null scrolls whose base curves are k-type null helices are discussed and several examples are presented.

scholarly journals Weingarten and Linear Weingarten Type Tubular Surfaces in

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Yılmaz Tunçer ◽  
Dae Won Yoon ◽  
Murat Kemal Karacan
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
The Mean ◽  
Second Gaussian Curvature ◽  
Tubular Surfaces

We study tubular surfaces in Euclidean 3-space satisfying some equations in terms of the Gaussian curvature, the mean curvature, the second Gaussian curvature, and the second mean curvature. This paper is a completion of Weingarten and linear Weingarten tubular surfaces in Euclidean 3-space.

scholarly journals Offset Ruled Surface in Euclidean Space with Density

2021 ◽  
Vol 29 (1) ◽  
pp. 219-233
Author(s):  
Neslihan Ulucan ◽  
Mahmut Akyigit
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Surfaces In Euclidean Space ◽  
The Mean

Abstract In this paper, offset ruled surfaces in these spaces are defined by using the geometry of ruled surfaces in Euclidean space with density. The mean curvature and Gaussian curvature of these surfaces are studied. In addition, the relationships between the mean curvature and mean curvature with density, and the Gaussian curvature and the Gaussian curvature with density of the offset ruled surfaces in E 3 with density e z and e − x 2− y 2 are given.

scholarly journals Null scrolls with prescribed curvatures in Lorentz-Minkowski 3-space

2020 ◽  
Vol 28 (3) ◽  
pp. 229-240
Author(s):  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Nonlinear Differential Equation ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Ruled Surfaces ◽  
First Order ◽  
Order Nonlinear Differential Equation ◽  
Base Curve

AbstractIn Lorentz-Minkowski 3-space, null scrolls are ruled surfaces with a null base curve and null rulings. Their mean, as well as their Gaussian curvature, depends only on a parameter of a base curve. In the present paper, we obtain the first-order nonlinear differential equation (Riccati equation) which relates curvatures of a base curve to curvatures of a null scroll. Conditioned by this equation, we can determine a family of null scrolls with a given null base curve and prescribed curvatures, in particular, a family of minimal and constant mean curvature null scrolls.

scholarly journals RULED SURFACES AND TANGENT BUNDLE OF PSEUDO-SPHERE OF NATURAL LIFT CURVES

2020 ◽  
Vol 20 (3) ◽  
pp. 573-586
Author(s):  
EMEL KARACA ◽  
MUSTAFA CALISKAN
Keyword(s):  
Unit Sphere ◽  
Gaussian Curvature ◽  
Tangent Bundle ◽  
Mean Curvature ◽  
Shape Operator ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Curve Shape ◽  
Lift Curve

In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, TM, and Lorentzian unit sphere, S12 is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, TM, and hyperbolic unit sphere, H2 is given. According to E. Study mapping, any curve on S12 or H2 corresponds to a ruled surface in R13. By constructing these isomorphisms, we correspond to any natural lift curve on TM or TM a unique ruled surface in R13. Then we calculate striction curve, shape operator, Gaussian curvature and mean curvature of these ruled surfaces. We give developability condition of these ruled surfaces. Finally, we give examples to support the main results.

scholarly journals Universally bistable shells with nonzero Gaussian curvature for two-way transition waves

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Nikolaos Vasios ◽  
Bolei Deng ◽  
Benjamin Gorissen ◽  
Katia Bertoldi
Keyword(s):  
Gaussian Curvature ◽  
Energy Landscape ◽  
Propagation Distance ◽  
Energy Landscapes ◽  
Bending Energy ◽  
Nonlinear Dynamic Response ◽  
One Dimensional ◽  
Doubly Curved Shells ◽  
Doubly Curved ◽  
The Waves

AbstractMulti-welled energy landscapes arising in shells with nonzero Gaussian curvature typically fade away as their thickness becomes larger because of the increased bending energy required for inversion. Motivated by this limitation, we propose a strategy to realize doubly curved shells that are bistable for any thickness. We then study the nonlinear dynamic response of one-dimensional (1D) arrays of our universally bistable shells when coupled by compressible fluid cavities. We find that the system supports the propagation of bidirectional transition waves whose characteristics can be tuned by varying both geometric parameters as well as the amount of energy supplied to initiate the waves. However, since our bistable shells have equal energy minima, the distance traveled by such waves is limited by dissipation. To overcome this limitation, we identify a strategy to realize thick bistable shells with tunable energy landscape and show that their strategic placement within the 1D array can extend the propagation distance of the supported bidirectional transition waves.

scholarly journals Comment on ,,On the integrability of 2D Hamiltonian systems with variable Gaussian curvature” by A. A. Elmandouh

2021 ◽  
Author(s):  
Wojciech Szumiński ◽  
Andrzej J. Maciejewski
Keyword(s):  
Hamiltonian Systems ◽  
Gaussian Curvature ◽  
Necessary Conditions ◽  
Constant Gaussian Curvature ◽  
Local Coordinates ◽  
Hamiltonian Functions

AbstractIn the paper [1], the author formulates in Theorem 2 necessary conditions for integrability of a certain class of Hamiltonian systems with non-constant Gaussian curvature, which depends on local coordinates. We give a counterexample to show that this theorem is not correct in general. This contradiction is explained in some extent. However, the main result of this note is our theorem that gives new simple and easy to check necessary conditions to integrability of the system considered in [1]. We present several examples, which show that the obtained conditions are effective. Moreover, we justify that our criterion can be extended to wider class of systems, which are given by non-meromorphic Hamiltonian functions.

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