scholarly journals The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

Symmetry ◽  
2018 ◽  
Vol 10 (9) ◽  
pp. 398 ◽  
Author(s):  
Erhan Güler ◽  
Hasan Hacısalihoğlu ◽  
Young Kim
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Gauss Map ◽  
Beltrami Operator ◽  
The Third ◽  
The Mean

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space E 4 . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in E 4 . We also draw some figures of the rotational hypersurface.

scholarly journals The Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space

2018 ◽  
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Beltrami Operator ◽  
Dimensional Euclidean Space ◽  
The Third ◽  
The Mean

We consider rotational hypersurface in the four dimensional Euclidean space. We calculate the mean curvature and the Gaussian curvature, and some relations of the rotational hypersurface. Moreover, we define the third Laplace-Beltrami operator and apply it to the rotational hypersurface.

scholarly journals Umbilical points on surfaces in RN

1985 ◽  
Vol 100 ◽  
pp. 135-143 ◽  
Author(s):  
Kazuyuki Enomoto
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Normal Connection ◽  
Round Sphere ◽  
Positive Gaussian Curvature ◽  
Connected Surface ◽  
Umbilical Points ◽  
The Mean

Let ϕ: M → RN be an isometric imbedding of a compact, connected surface M into a Euclidean space RN. ψ is said to be umbilical at a point p of M if all principal curvatures are equal for any normal direction. It is known that if the Euler characteristic of M is not zero and N = 3, then ψ is umbilical at some point on M. In this paper we study umbilical points of surfaces of higher codimension. In Theorem 1, we show that if M is homeomorphic to either a 2-sphere or a 2-dimensional projective space and if the normal connection of ψ is flat, then ψ is umbilical at some point on M. In Section 2, we consider a surface M whose Gaussian curvature is positive constant. If the surface is compact and N = 3, Liebmann’s theorem says that it must be a round sphere. However, if N ≥ 4, the surface is not rigid: For any isometric imbedding Φ of R3 into R4 Φ(S2(r)) is a compact surface of constant positive Gaussian curvature 1/r2. We use Theorem 1 to show that if the normal connection of ψ is flat and the length of the mean curvature vector of ψ is constant, then ψ(M) is a round sphere in some R3 ⊂ RN. When N = 4, our conditions on ψ is satisfied if the mean curvature vector is parallel with respect to the normal connection. Our theorem fails if the surface is not compact, while the corresponding theorem holds locally for a surface with parallel mean curvature vector (See Remark (i) in Section 3).

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 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 A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces

2019 ◽  
Vol 12 (1) ◽  
pp. 70-78
Author(s):  
Sudip Kumar Das ◽  
Mirza Cenanovic ◽  
Junfeng Zhang
Keyword(s):  
Mean Curvature ◽  
Beltrami Operator ◽  
Force Balance ◽  
Normal Vector ◽  
Triangulated Surface ◽  
Balance Principle ◽  
Alternative Expression ◽  
Triangulated Surfaces ◽  
The Mean ◽  
Mesh Structures

In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.

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 Ruled Surfaces of Generalized Self-Similar Solutions of the Mean Curvature Flow

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Rafael López
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Position Vector ◽  
Translation Surfaces ◽  
Self Similar ◽  
The Mean ◽  
Similar Solutions

AbstractIn Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation $$H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda $$ H = α ⟨ N , x ⟩ + λ , where N is the Gauss map, $${\mathbf {x}}$$ x is the position vector, and $$\alpha $$ α and $$\lambda $$ λ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.

SURFACES IN Sn WITH PRESCRIBED GAUSS MAP AND MEAN CURVATURE VECTOR FIELD

2006 ◽  
Vol 17 (10) ◽  
pp. 1127-1143
Author(s):  
AYAKO TANAKA
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Gauss Map ◽  
Necessary And Sufficient Conditions ◽  
Mean Curvature Vector ◽  
The Mean ◽  
Necessary And Sufficient

We give relations between the Gauss map and the mean curvature vector field of a surface in the Euclidean unit n-sphere Sn. These relations are necessary and sufficient conditions for the existence of a surface in Sn with prescribed Gauss map and mean curvature vector field. We show that such surfaces can be expressed explicitly by using given data.

scholarly journals 3D Palmprint Recognition Using Dempster-Shafer Fusion Theory

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Jianyun Ni ◽  
Jing Luo ◽  
Wubin Liu
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Imaging Technique ◽  
Recognition Performance ◽  
Belief Function ◽  
Recognition Algorithm ◽  
Palmprint Recognition ◽  
Equal Error Rate ◽  
The Mean ◽  
Fusion Theory

This paper proposed a novel 3D palmprint recognition algorithm by combining 3D palmprint features using D-S fusion theory. Firstly, the structured light imaging is used to acquire the 3D palmprint data. Secondly, two types of unique features, including mean curvature feature and Gaussian curvature feature, are extracted. Thirdly, the belief function of the mean curvature recognition and the Gaussian curvature recognition was assigned, respectively. Fourthly, the fusion belief function from the proposed method was determined by the Dempster-shafer (D-S) fusion theory. Finally, palmprint recognition was accomplished according to the classification criteria. A 3D palmprint database with 1000 range images from 100 individuals was established, on which extensive experiments were performed. The results show that the proposed method 3D palmprint recognition is much more robust to illumination variations and condition changes of palmprint than MCR and GCR. Meanwhile, by fusing mean curvature and Gaussian curvature feature, the experimental results are promising (the average equal error rate of 0.404%). In the future, imaging technique needs further improvement for a better recognition performance.

scholarly journals SMYTH SURFACES AND THE DREHRISS

2005 ◽  
Vol 48 (3) ◽  
pp. 549-555
Author(s):  
John M. Burns ◽  
Michael J. Clancy
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Immersed Surfaces ◽  
Trivial Group ◽  
The Mean ◽  
Isometric Deformations

AbstractIsometric deformations of immersed surfaces in Euclidean 3-space are studied by means of the drehriss. When the immersion is of constant mean curvature and the deformation preserves the mean curvature, we determine the drehriss explicitly in terms of the immersion and its Gauss map. These methods are applied to obtain an alternative classification of the Smyth surfaces, i.e. constant mean curvature immersions of the plane into Euclidean 3-space which admit the action of $S^1$ as a non-trivial group of internal isometries.

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