Slant Helix Curves and Acceleration Centers in Minkowski 3-Space Ε31

2017 ◽  
Vol 6 (1) ◽  
pp. 133-141 ◽  
Author(s):  
Murat Bekar ◽  
Yusuf Yayli
Keyword(s):  
Vector Field ◽  
Coordinate Frame ◽  
Normal Vector ◽  
Angular Velocity Vector ◽  
Normal Vector Field ◽  
3 Dimensional ◽  
Slant Helix ◽  
Dimensional Minkowski Space ◽  
Unit Speed ◽  
Instant Screw Axis

In this study, some basic concepts (e.g., instant screw axis (ISA), instantaneous pole points, acceleration pole points) will be given and analyzed about an alternative one-parameter motion of a rigid-body in 3-dimensional Minkowski space Ε31 obtained by moving coordinate frame {N, C, W} along a non-null unit speed curve α = α(t), where N, C and W correspond to unit principal normal vector field, derivative vector field of unit principal normal vector field and Darboux vector field (or angular-velocity vector field) of the non-null unit speed curve α, respectively.

The slant helices according to N-Bishop frame of the spacelike curve with spacelike principal normal in Minkowski 3-space

2019 ◽  
Vol 12 (06) ◽  
pp. 2040009
Author(s):  
Hatce Kusak Samanci ◽  
Ayhan Yildiz
Keyword(s):  
Vector Field ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Spacelike Curve ◽  
Bishop Frame ◽  
Constant Angle ◽  
Slant Helix ◽  
Constant Direction ◽  
Definition Of

If the principal normal vector field of a curve makes a constant angle with constant direction, this curve is called as slant helix. In this paper, a slant helix is defined according to N-Bishop frame of the spacelike curve with a spacelike principal normal. Some characterizations of the slant helices are obtained according to spacelike curve N-Bishop frame with a spacelike principal normal, benefiting from the definition of the slant helices.

A Note on the Representation of Clifford Algebras

2021 ◽  
Vol 62 ◽  
pp. 29-52
Author(s):  
Ying-Qiu Gu ◽  
Keyword(s):  
Clifford Algebras ◽  
Dimensional Space ◽  
Number System ◽  
Space Time ◽  
Coordinate Frame ◽  
Practical Calculation ◽  
3 Dimensional ◽  
Covariant Derivatives ◽  
Dimensional Minkowski Space ◽  
Dimensional Space Time

In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $\Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $\mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.

scholarly journals Constant angle spacelike surface in de Sitter space S₁³

2017 ◽  
Vol 35 (3) ◽  
pp. 79-93
Author(s):  
Tugba Mert ◽  
Baki Karlıga
Keyword(s):  
Vector Field ◽  
De Sitter Space ◽  
Normal Vector ◽  
Spacelike Surface ◽  
De Sitter ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Constant Angle ◽  
Unit Normal Vector Field ◽  
Unit Normal

In this paper; using the angle between unit normal vector field of surfaces and a fixed spacelike axis in R₁⁴, we develop two class of spacelike surface which are called constant timelike angle surfaces with timelike and spacelike axis in de Sitter space S₁³.

scholarly journals Extension of the unit normal vector field from a hypersurface

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Roland Duduchava ◽  
Eugene Shargorodsky ◽  
George Tephnadze
Keyword(s):  
Vector Field ◽  
Existence And Uniqueness ◽  
Elementary Proof ◽  
Gradient Field ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field ◽  
Outer Unit ◽  
Unit Normal

AbstractIn many applications it is important to be able to extend the (outer) unit normal vector field from a hypersurface to its neighborhood in such a way that the result is a unit gradient field. The aim of this paper is to provide an elementary proof of the existence and uniqueness of such an extension.

scholarly journals An integral formula for compact hypersurfaces in a Euclidean space and its applications

1992 ◽  
Vol 34 (3) ◽  
pp. 309-311 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Vector Field ◽  
Euclidean Space ◽  
Smooth Function ◽  
Support Function ◽  
Integral Formula ◽  
Position Vector ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Normal Vector Field ◽  
Unit Normal Vector Field

Let M be a compact hypersurface in a Euclidena space ℝn+1. The support function p of M is the component of the position vector field of Min ℝn+1 along the unit normal vector field to M, which is a smooth function defined on M. Let S be the scalar curvature of M. The object of the present paper is to prove the following theorems.

scholarly journals On compact minimal hypersurfaces in a sphere with constant scalar curvature

1980 ◽  
Vol 78 ◽  
pp. 177-188 ◽  
Author(s):  
Naoya Doi
Keyword(s):  
Vector Field ◽  
Constant Scalar Curvature ◽  
Arbitrary Point ◽  
Second Fundamental Form ◽  
Normal Vector ◽  
Unit Normal Vector ◽  
Covariant Differentiation ◽  
Normal Vector Field ◽  
Unit Normal Vector Field ◽  
Unit Normal

Let M be an n-dimensional hypersurface immersed in the (n + 1)-dimensional unit sphere Sn+1 with the standard metric by an immersion f. We denote by A the second fundamental form of the immersion / which is considered as a symmetric linear transformation of each tangent space TXM, i.e. for an arbitrary point x of M and the unit normal vector field ξ defined in a neighborhood of x, A is given by where is the covariant differentiation in Sn+i and Thus, A depends on the orientation of the unit normal vector field ξ and, in general, it is locally defined on M.

scholarly journals Piecewise smooth reconstruction of normal vector field on digital data

2016 ◽  
Vol 35 (7) ◽  
pp. 157-167 ◽  
Author(s):  
David Coeurjolly ◽  
Marion Foare ◽  
Pierre Gueth ◽  
Jacques-Olivier Lachaud
Keyword(s):  
Vector Field ◽  
Piecewise Smooth ◽  
Digital Data ◽  
Normal Vector ◽  
Normal Vector Field
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