scholarly journals Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 226 ◽  
Author(s):  
Erhan Güler
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Beltrami Operator ◽  
Dimensional Euclidean Space ◽  
Natural Numbers ◽  
Rotational Surface ◽  
Isometric Deformation ◽  
Helicoidal Surface

We consider a new kind of helicoidal surface for natural numbers ( m , n ) in the three-dimensional Euclidean space. We study a helicoidal surface of value ( m , n ) , which is locally isometric to a rotational surface of value ( m , n ) . In addition, we calculate the Laplace–Beltrami operator of the rotational surface of value ( 0 , 1 ) .

scholarly journals Bour’s theorem and helicoidal surfaces with constant mean curvature in the Bianchi–Cartan–Vranceanu spaces

2021 ◽  
Author(s):  
Renzo Caddeo ◽  
Irene I. Onnis ◽  
Paola Piu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Classical Result ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Helicoidal Surfaces ◽  
Helicoidal Surface ◽  
Groups Of Isometries ◽  
Two Parameter

AbstractIn this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space $${\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.

scholarly journals On a Question of Bourgain about Geometric Incidences

2008 ◽  
Vol 17 (4) ◽  
pp. 619-625 ◽  
Author(s):  
JÓZSEF SOLYMOSI ◽  
CSABA D. TÓTH
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Geometric Incidences

Given a set of s points and a set of n2 lines in three-dimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, we show that s = Ω(n11/4). This is the first non-trivial answer to a question recently posed by Jean Bourgain.

scholarly journals Orthogonal Isomorphic Representations Of Free Groups

1956 ◽  
Vol 8 ◽  
pp. 256-262 ◽  
Author(s):  
J. De Groot
Keyword(s):  
Euclidean Space ◽  
Free Groups ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Orthogonal Matrices ◽  
Abelian Subgroups ◽  
Proper Orthogonal

1. Introduction. We consider the group of proper orthogonal transformations (rotations) in three-dimensional Euclidean space, represented by real orthogonal matrices (aik) (i, k = 1,2,3) with determinant + 1 . It is known that this rotation group contains free (non-abelian) subgroups; in fact Hausdorff (5) showed how to find two rotations P and Q generating a group with only two non-trivial relationsP2 = Q3 = I.

Gröbner basis and resultant method for the forward displacement of 3-DoF planar parallel manipulators in seven-dimensional kinematic space

Robotica ◽  
2015 ◽  
Vol 34 (11) ◽  
pp. 2610-2628 ◽  
Author(s):  
Davood Naderi ◽  
Mehdi Tale-Masouleh ◽  
Payam Varshovi-Jaghargh
Keyword(s):  
Euclidean Space ◽  
Gröbner Basis ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Parallel Robots ◽  
Dimensional Euclidean Space ◽  
Grobner Basis ◽  
Kinematic Space ◽  
Forward Kinematic Problem ◽  
Forward Kinematic

SUMMARYIn this paper, the forward kinematic analysis of 3-degree-of-freedom planar parallel robots with identical limb structures is presented. The proposed algorithm is based on Study's kinematic mapping (E. Study, “von den Bewegungen und Umlegungen,” Math. Ann.39, 441–565 (1891)), resultant method, and the Gröbner basis in seven-dimensional kinematic space. The obtained solution in seven-dimensional kinematic space of the forward kinematic problem is mapped into three-dimensional Euclidean space. An alternative solution of the forward kinematic problem is obtained using resultant method in three-dimensional Euclidean space, and the result is compared with the obtained mapping result from seven-dimensional kinematic space. Both approaches lead to the same maximum number of solutions: 2, 6, 6, 6, 2, 2, 2, 6, 2, and 2 for the forward kinematic problem of planar parallel robots; 3-RPR, 3-RPR, 3-RRR, 3-RRR, 3-RRP, 3-RPP, 3-RPP, 3-PRR, 3-PRR, and 3-PRP, respectively.

The Evaluation of Moments of Regions With Piecewise-Linearly Approximated Boundaries via Line Integration

1989 ◽  
Author(s):  
J. Angeles ◽  
M. J. Al-Daccak
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Inertia Tensor ◽  
Divergence Theorem ◽  
Dimensional Euclidean Space ◽  
Repeated Application ◽  
The Subject ◽  
First Moment

Abstract The subject of this paper is the computation of the first three moments of bounded regions imbedded in the three-dimensional Euclidean space. The method adopted here is based upon a repeated application of Gauss’s Divergence Theorem to reduce the computation of the said moments — volume, vector first moment and inertia tensor — to line integration. Explicit, readily implementable formulae are developed to evaluate the said moments for arbitrary solids, given their piecewise-linearly approximated boundary. An example is included that illustrates the applicability of the formulae.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

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