Gauge freedom, anholonomy, and Hopf index of a three-dimensional unit vector field

1993 ◽  
Vol 47 (9) ◽  
pp. 5438-5441 ◽  
Author(s):  
Radha Balakrishnan ◽  
A. R. Bishop ◽  
R. Dandoloff
Keyword(s):  
Vector Field ◽  
Three Dimensional ◽  
Unit Vector ◽  
Gauge Freedom ◽  
Unit Vector Field ◽  
Dimensional Unit

scholarly journals Estimation of 6D Object Pose Using a 2D Bounding Box

Sensors ◽  
2021 ◽  
Vol 21 (9) ◽  
pp. 2939
Author(s):  
Yong Hong ◽  
Jin Liu ◽  
Zahid Jahangir ◽  
Sheng He ◽  
Qing Zhang
Keyword(s):  
Neural Network ◽  
Loss Function ◽  
Three Dimensional ◽  
Unit Vector ◽  
Prediction Algorithm ◽  
Computational Time ◽  
Bounding Box ◽  
Dimensional Unit ◽  
Bounding Boxes ◽  
Rgb Image

This paper provides an efficient way of addressing the problem of detecting or estimating the 6-Dimensional (6D) pose of objects from an RGB image. A quaternion is used to define an object′s three-dimensional pose, but the pose represented by q and the pose represented by -q are equivalent, and the L2 loss between them is very large. Therefore, we define a new quaternion pose loss function to solve this problem. Based on this, we designed a new convolutional neural network named Q-Net to estimate an object’s pose. Considering that the quaternion′s output is a unit vector, a normalization layer is added in Q-Net to hold the output of pose on a four-dimensional unit sphere. We propose a new algorithm, called the Bounding Box Equation, to obtain 3D translation quickly and effectively from 2D bounding boxes. The algorithm uses an entirely new way of assessing the 3D rotation (R) and 3D translation rotation (t) in only one RGB image. This method can upgrade any traditional 2D-box prediction algorithm to a 3D prediction model. We evaluated our model using the LineMod dataset, and experiments have shown that our methodology is more acceptable and efficient in terms of L2 loss and computational time.

scholarly journals Knots connected by wide ribbons

2019 ◽  
Vol 28 (12) ◽  
pp. 1950071
Author(s):  
Susan C. Brooks ◽  
Oguz Durumeric ◽  
Jonathan Simon
Keyword(s):  
Vector Field ◽  
Constant Width ◽  
Unit Vector ◽  
Outer Edge ◽  
Smooth Mapping ◽  
Unit Vector Field ◽  
Finite Set

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus [Formula: see text] in 3-space having constant width [Formula: see text]. Given a regular parametrization [Formula: see text], and a smooth unit vector field [Formula: see text] based along [Formula: see text], for a knot [Formula: see text], we may define a ribbon of width [Formula: see text] associated to [Formula: see text] and [Formula: see text] as the set of all points [Formula: see text], [Formula: see text]. For large [Formula: see text], ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge [Formula: see text] relates to that of the original knot [Formula: see text]. Generically, as [Formula: see text], there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field [Formula: see text]. The particular knot type within the finite set depends on the parametrized curves [Formula: see text], [Formula: see text], and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types [Formula: see text] and [Formula: see text], we can find a smooth ribbon of constant width connecting curves of these two knot types.

scholarly journals A new quaternion valued frame of curves with an application

Filomat ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 315-330
Author(s):  
Gizem Cansu ◽  
Yusuf Yaylı ◽  
İsmail Gök
Keyword(s):  
Vector Field ◽  
Unit Vector ◽  
Space Curve ◽  
Unit Vector Field ◽  
The Right

The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

scholarly journals Slant Curves in the Unit Tangent Bundles of Surfaces

ISRN Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zhong Hua Hou ◽  
Lei Sun
Keyword(s):  
Vector Field ◽  
Tangent Bundle ◽  
Unit Vector ◽  
Sasaki Metric ◽  
Geometric Properties ◽  
Unit Vector Field ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Slant Curves

Let (M,g) be a surface and let (U(TM),G) be the unit tangent bundle of M endowed with the Sasaki metric. We know that any curve Γ(s) in U(TM) consist of a curve γ(s) in M and as unit vector field X(s) along γ(s). In this paper we study the geometric properties γ(s) and X(s) satisfying when Γ(s) is a slant geodesic.

scholarly journals A Ginzburg–Landau Type Energy with Weight and with Convex Potential Near Zero

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 997
Author(s):  
Rejeb Hadiji ◽  
Carmen Perugia
Keyword(s):  
Asymptotic Behavior ◽  
Vector Field ◽  
Lower Bound ◽  
Unit Vector ◽  
Ginzburg Landau ◽  
Positive Weight ◽  
Unit Vector Field

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière.

On the volume of a unit vector field on the three-sphere

1986 ◽  
Vol 61 (1) ◽  
pp. 177-192 ◽  
Author(s):  
Herman Gluck ◽  
Wolfgang Ziller
Keyword(s):  
Vector Field ◽  
Unit Vector ◽  
Three Sphere ◽  
Unit Vector Field

scholarly journals A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

2020 ◽  
Vol 125 (3) ◽  
pp. 203-213
Author(s):  
Fabiano G. B. Brito ◽  
Icaro Gonçalves ◽  
Adriana V. Nicoli
Keyword(s):  
Vector Field ◽  
Lower Bound ◽  
Unit Vector ◽  
Unit Vector Field ◽  
Euclidean Hypersurface

scholarly journals Stripe patterns and a projection-valued formulation of the eikonal equation

2012 ◽  
Vol 370 (1965) ◽  
pp. 1730-1739 ◽  
Author(s):  
Mark A. Peletier ◽  
Marco Veneroni
Keyword(s):  
Vector Field ◽  
Block Copolymers ◽  
Recent Work ◽  
Eikonal Equation ◽  
Unit Vector ◽  
Unit Vector Field ◽  
New Formulation ◽  
Projection Matrices

We describe recent work on striped patterns in a system of block copolymers. A by-product of the characterization of such patterns is a new formulation of the eikonal equation. In this formulation, the unknown is a field of projection matrices of the form P = e ⊗ e , where e is a unit vector field. We describe how this formulation is better adapted to the description of striped patterns than the classical eikonal equation, and illustrate this with examples.

An improved gravitational formalism

1978 ◽  
Vol 56 (9) ◽  
pp. 1202-1203 ◽  
Author(s):  
Peter Rastall
Keyword(s):  
Vector Field ◽  
Scalar Potential ◽  
Unit Vector ◽  
Space Time ◽  
Unit Vector Field ◽  
Theory Of Gravity ◽  
Special Case

In earlier papers, a theory of gravity was developed in which the space-time metric depends on a unit vector field and a scalar potential. The formalism becomes much simpler if one uses just a vector field. The theory is equivalent to a special case of the earlier one.

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