scholarly journals Modified Roller Coaster Surface in Space

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 195 ◽  
Author(s):  
Selçuk BAŞ ◽  
Talat KÖRPINAR
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Roller Coaster ◽  
Orthogonal Frame

In this paper, a new modified roller coaster surface according to a modified orthogonal frame is investigated in Euclidean 3-space. In this method, a new modified roller coaster surface is modeled. Both the Gaussian curvature and mean curvature of roller coaster surfaces are investigated. Subsequently, we obtain several characterizations in Euclidean 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 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 Totally real surfaces inCP2with parallel mean curvature vector

1992 ◽  
Vol 15 (3) ◽  
pp. 589-592
Author(s):  
M. A. Al-Gwaiz ◽  
Sharief Deshmukh
Keyword(s):  
Gaussian Curvature ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Real Surface ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Constant Gaussian Curvature ◽  
Totally Geodesic ◽  
Parallel Mean Curvature ◽  
Totally Real

It has been shown that a totally real surface inCP2with parallel mean curvature vector and constant Gaussian curvature is either flat or totally geodesic.

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 Self-assembly of convex particles on spherocylindrical surfaces

Soft Matter ◽  
2018 ◽  
Vol 14 (28) ◽  
pp. 5728-5740 ◽  
Author(s):  
Guillermo R. Lázaro ◽  
Bogdan Dragnea ◽  
Michael F. Hagan
Keyword(s):  
Gaussian Curvature ◽  
Self Assembly ◽  
Mean Curvature ◽  
Continuum Theory

Simulations and continuum theory of self-assembly of conical subunits around a spherocylindrical template show the tuning the template mean curvature, Gaussian curvature, and curvature anisotropy enables the controlled formation of a rich array of assembly geometries.

HELICOIDAL SURFACES WITH PRESCRIBED CURVATURES IN Nil3

2013 ◽  
Vol 24 (14) ◽  
pp. 1350107 ◽  
Author(s):  
DAE WON YOON ◽  
DONG-SOO KIM ◽  
YOUNG HO KIM ◽  
JAE WON LEE
Keyword(s):  
Heisenberg Group ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Smooth Functions ◽  
Constant Gaussian Curvature ◽  
3 Dimensional ◽  
Helicoidal Surfaces ◽  
Prescribed Gaussian Curvature

In the present paper, we study helicoidal surfaces in the 3-dimensional Heisenberg group Nil3. Also, we construct helicoidal surfaces in Nil3 with prescribed Gaussian curvature or mean curvature given by smooth functions. As the results, we classify helicoidal surfaces with constant Gaussian curvature or constant mean curvature.

scholarly journals SPACELIKE TRANSLATION SURFACES IN MINKOWSKI 4-SPACE E_1^4

2020 ◽  
pp. 789
Author(s):  
Sezgin Büyükkütük ◽  
Günay Öztürk
Keyword(s):  
Minkowski Space ◽  
Gaussian Curvature ◽  
Mean Curvature ◽  
Translation Surfaces

In the present paper, we consider spacelike translation surfaces in $4$-dimensio\-nal Minkowski space. We characterize such surfaces in terms of their Gaussian curvature and mean curvature functions. We classify flat and minimal spacelike translation surfaces in $\mathbb{E}_{1}^{4}$. 

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.

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