scholarly journals A note on the ruled surface

1937 ◽  
Vol 30 ◽  
pp. i-ii
Author(s):  
R. Wilson
Keyword(s):  
Mean Curvature ◽  
Geodesic Curvature ◽  
Ruled Surface ◽  
Local Geometry ◽  
Parametric Curves ◽  
The Mean ◽  
Curvature And Torsion ◽  
Asymptotic Lines ◽  
Special Case ◽  
Elegant Form

The generators and their orthogonal trajectories form, perhaps, the most useful set of parametric curves for the study of the local geometry of a ruled surface. It is not generally realised, however, that the fundamental quantities of the surface can be expressed quite simply in terms of the geodesic curvature, the geodesic torsion and the normal curvature of the directrix, that particular orthogonal trajectory which is chosen as base curve. Certain of the results are similar in form to those arising in the special case of a surface which is generated by the principal normals to a given curve, except that the curvature and torsion are geodetic. In addition it is possible to obtain in an elegant form the differential equation of the curved asymptotic lines and the expression for the mean 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 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 Pansu–Wulff shapes in ℍ1

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián Pozuelo ◽  
Manuel Ritoré
Keyword(s):  
Mean Curvature ◽  
Isoperimetric Problem ◽  
Variational Formula ◽  
Geodesic Curvature ◽  
Vertical Axis ◽  
Singular Part ◽  
Tangent Plane ◽  
Curvature Function ◽  
Invariant Norm ◽  
The First Variation

Abstract We consider an asymmetric left-invariant norm ∥ ⋅ ∥ K {\|\cdot\|_{K}} in the first Heisenberg group ℍ 1 {\mathbb{H}^{1}} induced by a convex body K ⊂ ℝ 2 {K\subset\mathbb{R}^{2}} containing the origin in its interior. Associated to ∥ ⋅ ∥ K {\|\cdot\|_{K}} there is a perimeter functional, that coincides with the classical sub-Riemannian perimeter in case K is the closed unit disk centered at the origin of ℝ 2 {{\mathbb{R}}^{2}} . Under the assumption that K has C 2 {C^{2}} boundary with strictly positive geodesic curvature we compute the first variation formula of perimeter for sets with C 2 {C^{2}} boundary. The localization of the variational formula in the non-singular part of the boundary, composed of the points where the tangent plane is not horizontal, allows us to define a mean curvature function H K {H_{K}} out of the singular set. In the case of non-vanishing mean curvature, the condition that H K {H_{K}} be constant implies that the non-singular portion of the boundary is foliated by horizontal liftings of translations of ∂ ⁡ K {\partial K} dilated by a factor of 1 H K {\frac{1}{H_{K}}} . Based on this we can define a sphere 𝕊 K {\mathbb{S}_{K}} with constant mean curvature 1 by considering the union of all horizontal liftings of ∂ ⁡ K {\partial K} starting from ( 0 , 0 , 0 ) {(0,0,0)} until they meet again in a point of the vertical axis. We give some geometric properties of this sphere and, moreover, we prove that, up to non-homogeneous dilations and left-translations, they are the only solutions of the sub-Finsler isoperimetric problem in a restricted class of sets.

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

The focusing of radiation by a random surface when the source is at a finite distance

1960 ◽  
Vol 56 (1) ◽  
pp. 27-40 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Mean Curvature ◽  
Joint Distribution ◽  
Total Curvature ◽  
Statistical Distribution ◽  
Constant Parameter ◽  
Random Surface ◽  
Second Derivatives ◽  
Finite Distance ◽  
The Mean ◽  
Image Position

Imagine a nearly horizontal, statistically uniform, random surface ζ(x, y), Gaussian in the sense that the second derivatives , , have a normal joint distribution. The problem considered is the statistical distribution of the quantitywhere J and Ω denote the mean curvature and total curvature of the surface, respectively, and ν is a constant parameter.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

scholarly journals Finite topology self-translating surfaces for the mean curvature flow in R3

2017 ◽  
Vol 320 ◽  
pp. 674-729 ◽  
Author(s):  
Juan Dávila ◽  
Manuel del Pino ◽  
Xuan Hien Nguyen
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Finite Topology ◽  
The Mean
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