Binormal Research Articles

A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space

The main outcome of this work is the construction of a surface pencil with a similarity to Bertrand curves in Euclidean 3-space E3. Then, by exploiting the Serret–Frenet frame, we deduce the sufficient and necessary conditions for a surface pencil with Bertrand curves as joint curvature lines. Consequently, the expansion to the ruled surface pencil is also designed. As demonstrations of our essential findings, we illustrate some models to emphasize the process.

On Darboux Frames of Indicatrices of Spacelike Salkowski Curve with Spacelike Binormal in E13

The aim of this study is to examine Darboux frames and some other geometric properties (geodesic curvatures, geodesic torsions, normal curvatures, Darboux derivative formulas, Darboux vectors, angles, etc.) of the spherical indicatrices on Lorentzian unit sphere S_1^2 and hyperbolic unit sphere H_0^2 of the spacelike Salkowski curve with spacelike binormal in Lorentzian 3-space E_1^3. In this context, new and interesting results have been obtained for this curve. Thus, relationships between the newly obtained curvatures and torsions and the curvature and torsion of the original curve are given. Moreover, the matrix relationship between the Darboux and Frenet frames of these indicatrices is shown. Finally, the Darboux vectors belong to the Darboux frames and the Darboux vectors belong to the Frenet frames of these curves are compared.

3-Boyutlu Öklid Uzayında Adjoint Eğrilerin Binormal Yüzeyleri

In this study, we make a specific contribution to the subject of surfaces with wide application areas defined with the help of associated curves. In particular, we examine the binormal surfaces generated via adjoint curves which are one of the important examples of associated curves. By defining the binormal surfaces of adjoint curves under the Frenet-Serret (FS) frame in 3D Euclidean space, we characterize such surfaces and give some results with the aid of these characterizations.

An Alternative Method for Determination of the Position Vector of a Slant Helix

In this paper, we provide an alternative method to determine the position vector of a slant helix with the help of an alternative moving frame. We then construct a vector differential equation in terms of the principal normal vector of a slant helix using an alternative moving frame. By solving this vector differential equation, we determine the position vector of the slant helix. Afterward, we obtain parametric representations of some examples of slant helices for chosen curvature and torsion functions as an application of the proposed method. Finally, we discuss the method and whether further research should be conducted or not.

Surface Family with Bertrand Curves as Joint Asymptotic Curves in 3D Galilean Space

The primary objective of this work is to discuss a surface family with the similarity of Bertrand curves in 3D Galilean space. Subsequently, by applying the Serret–Frenet frame, we estimate the sufficient and necessary statuses of a surface family with Bertrand curves as joint asymptotic curves. The dilation to ruled surfaces is also summarized. Meanwhile, the epitomes are illustrated to provide an explanation of the theoretical results.

Hierarchical learning of robotic contact policies

The paper addresses the issue of learning tasks where a robot maintains permanent contact with the environment. We propose a new methodology based on a hierarchical learning scheme coupled with task representation through directed graphs. These graphs are constituted of nodes and branches that correspond to the states and robotic actions, respectively. The upper level of the hierarchy essentially operates as a decision-making algorithm. It leverages reinforcement learning (RL) techniques to facilitate optimal decision-making. The actions are generated by a constraint-space following (CSF) controller that autonomously identifies feasible directions for motion. The controller generates robot motion by adjusting its stiffness in the direction defined by the Frenet–Serret frame, which is aligned with the robot path. The proposed framework was experimentally verified through a series of challenging robotic tasks such as maze learning, door opening, learning to shift the manual car gear, and learning car license plate light assembly by disassembly.

Involutive sweeping surfaces with Frenet frame in Euclidean 3-space

In this paper we first define the involutive sweeping surfaces as a new surface form. We then investigate singularity, Gaussian and mean curvatures of these surfaces. By calculating the Gaussian and mean curvatures of the involutive sweeping surfaces, we find necessary conditions of being flat or minimal of these surfaces. Also, we analyze the necessary and sufficient conditions for parameter curves on the surface to be asymptotic, geodesic. Then we investigate the special case that the parameter curves are lines of curvature on the surface. Finally, we illustrate our method of calculation by presenting an example.

The non-orthogonal Serret–Frenet parametrization applied to the path following problem of a manipulator with partially known dynamics

The non-orthogonal Serret–Frenet parametrization applied to the path following problem of a manipulator with partially known dynamics

New Applications on the Flows of Space-like curves specified by Normal acceleration in Minkowski Space ℝ2,1

Abstract This paper investigates the kinematic motions of space-like and time-like curves specified by acceleration fields in Minkowski space ℝ2,1. Through the motion, the relationship between the acceleration fields and velocity fields is determined. In this study, we focus on studying the flows of inextensible space-like curves with a space-like principal normal vector specified by a normal acceleration that equals the curvature of the curve. Through the motion of the inextensible space-like curve with normal acceleration, we prove that the position vector of the curve satisfies a one-dimensional wave equation. We present some novel applications and visualize the flows of curves and their curvatures.

A Finite Element Model for a 6 × K31WS + FC Wire Rope and a Study on Its Mechanical Responses with or without Wire Breakage

The special spatial structure and the long-term harsh working environment make the transient response of wire rope complicated, so accurate assessment and prediction of its mechanical characteristics are of great significance to ensure the safe and stable operation of related equipment. In this paper, a method is proposed for the analysis of the mechanical characteristics of a 6 × K31WS + FC wire rope with or without wire breakage. Firstly, an accurate parametric geometric model of the wire rope is established based on the Frenet frame method, the material properties of the wire rope are acquired by experiments, and the finite element model of the wire rope is built. Then, a mechanical model of the wire rope is proposed to verify the validity of the finite element model. Finally, the influences of the number, distribution, and location of the broken wires on the mechanical characteristics of the wire rope are thoroughly analyzed. This work proposes a comprehensive framework that can quantitatively analyze the mechanical characteristics of the complex wire rope with or without wire breakage, which provides a practical method for its reliability and maintenance evaluation and has guiding significance for the shape design and structural optimization of the wire rope.

Surfaces Family with Bertrand Curves as Common Asymptotic Curves in Euclidean 3–Space E3

The main result of this paper is constructing a surfaces family with the similarity of Bertrand curves in Euclidean 3–space E3. Then, by utilizing the Serret–Frenet frame, we conclude the sufficient and necessary conditions of surfaces family interpolating Bertrand curves as common asymptotic curves. Consequently, the expansion to the ruled surfaces family is also depicted. As implementations of our main results, we demonstrate some examples to confirm the method.

Hypersurfaces With a Common Geodesic Curve in 4D Euclidean space E4

In this paper, we attain the problem of constructing hypersurfaces from a given geodesic curve in 4D Euclidean space E4. Using the Serret–Frenet frame of the given geodesic curve, we express the hypersurface as a linear combination of this frame and analyze the necessary and sufficient conditions for that curve to be geodesic. We illustrate this method by presenting some examples.

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

In this study, we first define the Smarandache curves derived from the Frenet vectors and the Darboux vector of any curve. Then, we construct new ruled surfaces along these Smarandache curves with the direction vectors obtained from the Frenet vectors and the Darboux vector, and give the equations of these surfaces. In addition, we calculate the Gaussian and mean curvatures of these surfaces separately and present the conditions to be minimal and developable for these surfaces. Finally, as an example, we obtain ruled surfaces whose base curves are Viviani’s curves and plot the graphics of these surfaces.

Radar-based path planning of autonomous surface vehicle with static and dynamic obstacles in a Frenet Frame

AbstractNavigation safety at sea is vital for each autonomous surface vehicle (ASV), which involves the problem of motion planning in dynamic environments and their robust tracking through feedback control. We present a practical path-planning method that generates smooth trajectories for a marine vehicle traveling in an unknown environment, where obstacles are detected in real time by millimetre wave (mmWave) radar. Our approach introduces a polynomial curve to describe the lateral and longitudinal trajectories in the Frenet frame, known as the ‘motion primitives’, whose combination ensures that the planning area is properly covered. In addition, we can select a feasible, optimal and collision-free trajectory from such a set of motion primitives that is generated by considering the vehicle dynamics and comfort. The capabilities of proposed algorithm are demonstrated in the experiment with static and dynamic obstacles.

Whole mandibular canal segmentation using transformed dental CBCT volume in Frenet frame

Accurate segmentation of the mandibular canal is essential in dental implant and maxillofacial surgery, which can help prevent nerve or vascular damage inside the mandibular canal. Achieving this is challenging because of the low contrast in CBCT scans and the small scales of mandibular canal areas. Several innovative methods have been proposed for mandibular canal segmentation with positive performance. However, most of these methods segment the mandibular canal based on sliding patches, which may adversely affect the morphological integrity of the tubular structure. In this study, we propose whole mandibular canal segmentation using transformed dental CBCT volume in the Frenet frame. Considering the connectivity of the mandibular canal, we propose to transform the CBCT volume to obtain a sub-volume containing the whole mandibular canal based on the Frenet frame to ensure complete 3D structural information. Moreover, to further improve the performance of mandibular canal segmentation, we use clDice to guarantee the integrity of the mandibular canal structure and segment the mandibular canal. Experimental results on our CBCT dataset show that integrating the proposed transformed volume in the Frenet frame into other state-of-the-art methods achieves a 0.5%∼12.1% improvement in Dice performance. Our proposed method can achieve impressive results with a Dice value of 0.865 (±0.035), and a clDice value of 0.971 (±0.020), suggesting that our method can segment the mandibular canal with superior performance.

Equivalently analytical solution for the large deformation of slender beams under follower loads: a second-order ANCF approach

Follower loads interact with the object's configuration, and the two are unknown variables in geometrically nonlinear analysis. This work aims to solve the large deformation of slender beams under follower loads via the Bernoulli–Euler beam element of absolute nodal coordinate formulation. The second-order approximate function of the beam centerline is utilized to model the beam configuration and the Frenet frame. The elastic force vector and its tangent stiffness matrix are formulated following the decoupling elastic line approach. The constraint of adjacent elements is imposed using Lagrange multipliers. The exact kinematics characteristics of typical follower loads are demonstrated; sequentially, the external force vectors and the load stiffness matrices are rigorously derived based on the configuration and the Frenet frame modelled by the second-order approximate function. Benchmark problems of straight and curved beams under different follower loads are solved, wherein the accuracy and efficiency of the proposed approach are validated as compared to relevant analytical and numerical solutions. It is found that the proposed element yields equivalently analytical nodal solutions, including position, rotation and curvature; considerably fewer proposed elements are used as compared with other finite element methods; the load stiffness matrix of the bending moment raises computational efficiency of the proposed element; the obtained solutions are more accurate than that of the B22 element of ABAQUS when solving pressure loading cases. By further extending the proposed approach, it can enable the unveiling of the spatio-temporal evolution characteristic of slender structures and follower loads from the equivalently analytical perspective.

Geodesic nets on non-compact Riemannian manifolds

Abstract A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.

Magnetic spherical indicatricies in Minkowski 3-space

In this paper, we examine curves drawn by a timelike or spacelike curve[Formula: see text] on the pseudo-hyperbolic sphere [Formula: see text] and the unit Lorentzian sphere [Formula: see text]in three-dimensional Minkowski space [Formula: see text] We present Lorentz forces and magnetic curves produced by the Frenet frame [Formula: see text] on spherical indicatricies in a magnetic field of the curve [Formula: see text]. We calculate magnetic vector fields of magnetic curves for [Formula: see text]. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic spherical indicatricies. We obtain developability conditions for these surfaces. Finally, we illustrate magnetic flux surfaces with some interesting examples.

A New Approach on Some Special Curves

In this paper, we obtain some characterizations for a Frenet curve with the help of an alternative frame different from Frenet frame. Also, in the present study we consider weak biharmonic and harmonic 1-type curves by using the mean curvature vector field of the curve. We also study 1-type and biharmonic curves whose mean curvature vector field is in the kernel of Laplacian. We give some theorems for them in the Euclidean 3-space. Moreover, we give the classifications of these type curves.

The Spinor Expressions of Mannheim Curves in Euclidean 3-Space

In this paper, the spinor formulations of Mannheim curve pair are investigated. First of all, two spinors matching to Mannheim curve pair are given and by considering the relationships between the Frenet frames of Mannheim curve pair, the relationship between two spinors matching to this curve pair are gotten. Therefore, a geometric interpretations of spinors are obtained using the Mannheim curve pair and considering Mannheim curve as helix the spinor formulations of Mannheim curve pair are given. Moreover, the spinor formulations are also obtained for the curvatures of the Mannheim curve pair. Consequently, an example of these spinors is obtained. Therefore, it is thought that this study will make an important contribution to the mathematical analysis and geometric interpretation of spinors, which have many uses in physics.