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 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 TUBULAR SURFACES WITH DARBOUX FRAME IN GALILEAN 3-SPACE

2019 ◽  
pp. 253
Author(s):  
Sezai Kızıltuğ ◽  
Mustafa Dede ◽  
Cumali Ekici
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Frenet Frame ◽  
Darboux Frame ◽  
Tubular Surface ◽  
The Mean ◽  
Tubular Surfaces

In this paper, we define tubular surface by using a Darboux frame instead of a Frenet frame. Subsequently, we compute the Gaussian curvature and the mean curvature of the tubular surface with a Darboux frame. Moreover, we obtain some characterizations for special curves on this tubular surface in a Galilean 3-space.

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 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 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 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 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 Geometric Characterizations of Canal Surfaces in Minkowski 3-Space II

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 703 ◽  
Author(s):  
Jinhua Qian ◽  
Mengfei Su ◽  
Xueshan Fu ◽  
Seoung Dal Jung
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Space Curve ◽  
Timelike Curve ◽  
Geometric Properties ◽  
Spacelike Curve ◽  
Space Curves ◽  
Canal Surfaces ◽  
The Mean ◽  
The Relationship

Canal surfaces are defined and divided into nine types in Minkowski 3-space E 1 3 , which are obtained as the envelope of a family of pseudospheres S 1 2 , pseudohyperbolic spheres H 0 2 , or lightlike cones Q 2 , whose centers lie on a space curve (resp. spacelike curve, timelike curve, or null curve). This paper focuses on canal surfaces foliated by pseudohyperbolic spheres H 0 2 along three kinds of space curves in E 1 3 . The geometric properties of such surfaces are presented by classifying the linear Weingarten canal surfaces, especially the relationship between the Gaussian curvature and the mean curvature of canal surfaces. Last but not least, two examples are shown to illustrate the construction of such surfaces.

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 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.

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