Bishop Frame Research Articles

Optical fiber shape reconstruction algorithm based on 3D Euler spiral model

Abstract This paper presents a reconstruction algorithm that uses a 3D Euler spiral model to construct the microsegment of a 3D space curve to improve the reconstruction efficiency. Euler spiral is a curve whose shape information changes linearly with arc length, which improves the current assumption of the reconstruction algorithms that the shape information of the micro-segment is constant, and effectively reduces the number of interpolations required, and thus improves the efficiency of the shape algorithm. To verify the effectiveness of this algorithm, a method for constructing random space curves is proposed. Three random space curves of different lengths are reconstructed and the results are compared with the reconstruction algorithms based on the homogeneous transformation matrix and Bishop frame. The results show that this model can significantly improve the efficiency of the algorithm. Under the random space curves of 1 m, 10 m, and 100 m, the efficiency is enhanced by about 15, 27, and 30 times respectively. Prepare a shape sensor with a length of 1465mm to verify the highest reconstruction accuracy of 3D Euler spiral model, which is consistent with the simulation results. The results provide a solid foundation for further research in the field of shape sensing, and show potential for promoting the development of applications that rely on real-time shape measurements.

Γ-convergence of a discrete Kirchhoff rod energy

This work is motivated by the classical discrete elastic rod model by Audoly et al. We derive a discrete version of the Kirchhoff elastic energy for rods undergoing bending and torsion and prove Γ-convergence to the continuous model. This discrete energy is given by the bending and torsion energy of an interpolating conforming polynomial curve and provides a simple formula for the bending energy depending in each discrete segment only on angle and adjacent edge lengths. For the liminf-inequality, we need to introduce penalty terms to ensure arc-length parametrization in the limit. For the recovery sequence a discretization with equal Euclidean distance between consecutive points is constructed. Particular care is taken to treat the interaction between bending and torsion by employing a discrete version of the Bishop frame.

A new approach to curve couples with Bishop frame

This paper presents a detailed study of a new generation of the Bishop frame with components including three orthogonal unit vectors, which are tangent vector, normal vector and binormal vector. It is a frame field described on a curve in Euclidean space, which is an alternative to the Frenet frame. It is useful for curves for which the second derivative is not available. Moreover, the conditions which the Bishop frame of one curve coincides with the Bishop frame of another curve are defined. It would be valuable to replicate similar approaches in the Bishop frame of one curve coincides with the Bishop frame of another curve.

Osculating Type Ruled Surfaces with Type-2 Bishop Frame in E3

Osculating Type Ruled Surfaces with Type-2 Bishop Frame in E3

ASSOCIATED CURVES ACCORDING TO BISHOP FRAME IN 4-DIMENSIONAL EUCLIDEAN SPACE

Many studies have been doneaccording to different frames of the theory of curves in Euclidean Space. Many scientists have studied frames such as the Frenet frame, Bishop frame, and Adapted frame in this theory. These frames help us in the characterization of curves.In this study, associated curves with the Frenet curve according to the Bishop frame in 4-dimensional Euclidean space are investigated. Direction and rectifying curves of the Frenet curve according to this frame are given.

Conformable modeling of normalization and recursional electromagnetic fields of spacelike magnetic curves

In this paper, we investigate spacelike magnetic curves according to Bishop frame. Firstly, we present conformable derivatives of Lorentz magnetic fields of these magnetic curves. Moreover, we calculate the conformable derivatives of the normalization and recursional electromagnetic vector fields. Finally, we give conformable energies of normalization and recursional electromagnetic fields related to spacelike magnetic curves.

Bishop Frames of Salkowski Curves in E3

In this study, alternative, type-1 Bishop, type-2 Bishop and N-Bishop frames of Salkowski curves in E3 are calculated. Moreover, curvatures, Darboux and pol vectors of these frames are found. Also, relationships between the Bishop frames, Darboux vectors and pole vectors are given.

OFDR shape sensor based on a femtosecond-laser-inscribed weak fiber Bragg grating array in a multicore fiber.

An optical frequency domain reflectometry (OFDR) shape sensor was demonstrated based on a femtosecond-laser-inscribed weak fiber Bragg grating (WFBG) array in a multicore fiber (MCF). A WFBG array consisting of 60 identical WFBGs was successfully inscribed in each core along a 60 cm long MCF using the femtosecond-laser point-by-point technology, where the length and space of each WFBG were 2 and 8 mm, respectively. The strain distribution of each core in two-dimensional (2D) and three-dimensional (3D) shape sensing was successfully demodulated using the traditional cross correlation algorithm, attributed to the accurate localization of each WFBG. The minimum reconstruction error per unit length of the 2D and 3D shape sensors has been improved to 1.08% and 1.07%, respectively, using the apparent curvature vector method based on the Bishop frame.

A new glimpse into the directional curves associated with a pseudo-null curve via Bishop frame

In this study, the directional associated curves of a pseudo-null curve have been investigated according to Bishop frame in Lorentz-Minkowski 3-space. The relational equations have been obtained between the curvature functions of the pair donor-associated curves and the distance function. Furthermore, an answer has been sought to the question: does a self-associated curve of a pseudo-null curve via Bishop frame exist or not in Lorentz-Minkowski 3-space?

Generalized Bishop frames of regular curves in $\mathbb{E}^{4}$

Generalized Bishop frames of regular curves in $\mathbb{E}^{4}$

Singularities of swept surfaces in Euclidean 3-space

<p>This study examines the local singularities of tube surfaces, especially those of swept surfaces $ (SS) $ in Euclidean 3-space $ \mathcal{E}^{3} $. $ SS $ is created by moving a planar curve through space such that the trajectory of any point on the surface remains perpendicular to the plane. The Type-2 Bishop frame is considered, and the singularities of these $ SS $ are analyzed. Examples are offered and illustrated.</p>

About Bertrand curves and type-1 Bishop frame in three-dimensional Weyl space W3

About Bertrand curves and type-1 Bishop frame in three-dimensional Weyl space W3

Timelike spherical curves according to equiform Bishop frame in 3-dimensional Minkowski space

In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermore, we give certain characterizations of equiform spherical curves in 3-dimensional Minkowski space, which are timelike with an equiform spacelike principal normal vector.

Optical quantum recursive vortex filament flows and energy with the bishop frame

Optical quantum recursive vortex filament flows and energy with the bishop frame

Surface family with a common Mannheim B-pair asymptotic curve

In this paper, we construct a surface family possessing a Mannheim B-pair of a given curve as an asymptotic curve. Using the Bishop frame of the given Mannheim B curves, we present the surface as a linear combination of this frame and analyze the necessary and sufficient condition for a given curve such that its Mannheim B-pairs are both isoparametric and asymptotic on a parametric surface. The extension to ruled surfaces is also outlined. In addition, necessary and sufficient conditions have been given for the development of this ruled surface family. Finally, we present some interesting examples to show the validity of this study.

High-spatial-resolution φ-OFDR shape sensor based on multicore optical fiber with femtosecond-laser-induced permanent scatter arrays.

An optical fiber φ-OFDR shape sensor with a submillimeter spatial resolution of 200 µm was demonstrated by using femtosecond-laser-induced permanent scatter array (PS array) multicore fiber (MCF). A PS array was successfully inscribed in each slightly twisted core of the 400-mm-long MCF. The two-dimensional (2D) and three-dimensional (3D) shapes of the PS-array-inscribed MCF were successfully reconstructed by using PS-assisted φ-OFDR, vector projections, and the Bishop frame based on the PS-array-inscribed MCF. The minimum reconstruction error per unit length of the 2D and 3D shape sensor was 2.21% and 1.45%, respectively.

Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group

In this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group G3. With the help of this frame, we derive a parametric representation for a sweeping surface and show that the parametric curves on this surface are curvature lines. We then examine the local singularities and convexity of this sweeping surface and establish the sufficient and necessary conditions for it to be a developable ruled surface. Additionally, we provide detailed explanations and examples of its applications.

Sweeping Surfaces according to Type-3 Bishop Frames in Euclidean 3-Space

The aim of this work is to investigate sweeping surfaces and their local singularities due to type-3 Bishop frames in Euclidean 3-space, E3. A sweeping surface a is surface that is designed from a section curve positioned along a path, which acts as the vertebral column or spine curve, and it has symmetrical characteristics. In this work, we have specified a sweeping surface and have examined its geometry and singularity. Thereafter, we deduced the circumstances required for this surface to be a developable surface. In great detail, we concentrated on the fundamental discussion on whether the resulting developable surface is a cylindrical, cone or tangent surface. Meanwhile, examples are detailed to explain the applications of the notional outcomes.

Investigation on Isotropic Bezier Sweeping Surface "IBSS" with Bishop Frame

Investigation on Isotropic Bezier Sweeping Surface "IBSS" with Bishop Frame

Optical electromotive microscale with first type Schrödinger frame

In this paper, we define the first frame associated with the non-linear Schrödinger system. Also, we present Lorentz force in the first type of the non-linear Schrödinger frame associated with the Bishop frame in the Minkowski 3-space. We can define timelike pseudo-solitonic first class NLS electromotive microscale. Hence, we design first class NLS optical optimistic pseudo-solitonic density. Finally, we obtain electroosmotic solitonic NLS electromotive microscale is modeled by the first type of the non-linear Schrödinger frame vectors.