Binormal Research Articles

Rapid prediction of molecular crystal structures using simple topological and physical descriptors.

Organic molecular crystals constitute a class of materials of critical importance in numerous industries. Despite the ubiquity of these systems, our ability to predict molecular crystal structures starting only from a two-dimensional diagram of the constituent compound(s) remains a significant challenge. Most structure-prediction protocols require a customized interatomic interaction model on which the quality of the results can depend sensitively. To overcome this problem, we introduce a new topological approach to molecular crystal structure prediction. The approach posits that in a stable structure, molecules are oriented such that principal axes and normal ring plane vectors are aligned with specific crystallographic directions and that heavy atoms occupy positions that correspond to minima of a set of geometric order parameters. By minimizing an objective function that encodes these orientations and atomic positions, and filtering based on the vdW free volume and intermolecular close contact distributions derived from the Cambridge Structural Database, stable structures and polymorphs for a given crystal can be predicted entirely mathematically without reliance on an interaction model.

Binormal Curves

We give the definition of binormal curves. The curvature, the torsion and Frenet frame of this curve are calculated. Also, the examples of binormal curves are given.

A non-newtonian approach to geometric phase through optic fiber via multiplicative quaternions

In this paper, we researched magnetic and electromagnetic curves in multiplicative Euclidean 3-space via multiplicative quaternion algebra. Firstly, we examined the geometric phase representation of the polarized light wave in the optic fiber by multiplicative Frenet frame. Using the quaternionic approaches, we were able to derive the magnetic curve equations and theorems. Then, three particular instances have been illustrated with examples of electromagnetic curves and magnetic field equations. Lastly, we provided an interpretation of the findings. With the help of the results, we were able to present an alternative viewpoint on the construction of trajectories (such as circular or spiral-like ones) that do not exist uniquely in the realm of physics.

Enlarged Curvature, Torsion and Torque in Helical Conformations and the Stability and Growth of α-Peptide under the Isochoric and Isobaric Conditions: Variatonal Optimization

The torsional deformation behavior of an elastic bar with a circular cross-section was investigated by applying invariant dyadic analysis, where the small finite displacement functions advocated by Saint-Venant (1855) were fully employed. It was found that the previously overlooked circumferential shear force field generated by pure torsion on the side walls of a bar produces an unusual torque term induced by the skew-symmetric part of the deformation tensor and exhibits quadratic length dependence along the z-axis of the bar. The adaptation of this torque term for a helical conformation of α-peptides creates moments acting on the circular cross-sections and is directed along the surface normal of circular cross-sections, which coincides with the tangent vector of the helix. The projection of this torque along the z-axis of the helix varies quadratically with the azimuthal angle. The radial component of the unusual torque, which also lies along the principal normal vector of the helix, starts to perform a precession motion by tracking a spiral orbit around the z-axis, whereas its apex angle decreases asymptotically with the azimuthal angle and finally reaches a finite value depending on the height of the helix along the z-axis. The ordinary torque terms, which are also deduced from the self- and anti-self-conjugate parts of the deformation tensor, have magnitudes half that of the full torque term reported in the literature. The present results were applied to the helical conformation of α-peptides designated by {3.611} to show that the mechanical stability of strained open-ended helical conformations can be successfully achieved by spontaneous readjustments of the surface and bulk Helmholtz free energies under isothermal isochoric conditions. It has been demonstrated that the main contribution to the mechanical stability of α-peptide 3.611 cannot come alone from the electrostatic dipole-dipole interaction potential of the anti-align excess dipole pairs but also from the surface Helmholtz free energy, which is characterized by a binding free energy of -15.5 eV/molecule (-32.56 Kcal/mole) for an alpha-peptide composed of 11 amino acid residues with a critical arc length of approximately 10 nm, assuming that the shear modulus is G = 1GPa and the surface Helmholtz specific free energy density is fs = 800 erg/cm2. This result was in excellent agreement with the experimental observations of the AH-1 conformation of (Glu)n Cys at pH 8. The present theory indicates that only two excess permanent anti-align dipole pairs for one α-Helical peptide molecule is requirement to stabilize the whole secondary structure of the protein that is exposed to heavy torsional deformation during the folding processes which amounts to 7.75 eV/molecule stored electrostatic energy compared to the interfacial Helmholtz free energy of -23.25 eV/molecule, which is exposed to hydrophobic environments.

A Comparative Study on Autonomous Vehicle Local Path Planning Through Model Predictive Control and Frenet Frame Method

<div>In recent years, autonomous vehicles (AVs) have been receiving increasing attention from investors, automakers, and academia due to the envisioned potentials of AVs in enhancing safety, reducing emissions, and improving comfort. The crucial task in AV development boils down to perception and navigation. The research is underway, in both academia and industry, to improve AV’s perception and navigation and reduce the underlying computation and costs. This article proposes a model predictive control (MPC)-based local path-planning method in the Cartesian framework to overcome the long computation time and lack of smoothness of the Frenet method. A new equation is proposed in the MPC cost function to improve the safety in path planning. In this regard, an AV is built based on a 2015 Nissan Leaf S by modifying the drive-by-wire function and installing environment perception sensors and computation units. The custom-made AV then collected data in Norman, Oklahoma, and assisted in the performance evaluation of the two control algorithms in this work. Both straight roads and curved roads are considered in the evaluation. For the purpose of saving costs and raising real-world implementation potential, the vision-only solution is applied in object detection and bird’s-eye-view coordinate data generation. MPC and Frenet coordinate system approaches are independently employed to generate a safe and smooth path for the AV using the collected data. The two methods are compared in terms of smoothness, safety, and computation time. Compared with the Frenet-based method, the proposed MPC method reduces the computation time by 80%, and the path smoothness is significantly improved.</div>

Darboux Frame of Pancake Curve

In this paper, we focus on such a rare curve: “Pancake curve” which is named by Robert Ferréol. We investigate the Darboux frame of the curve and we calculate the curvatures: geodesic curvature, normal curvature, and geodesic torsion. We discuss two cases, for instance pancake curve as an intersection of Plücker’s conoid and unit cylinder, and intersection of hyperbolic paraboloid and unit cylinder. Moreover, we give the relations between the Frenet and Darboux curvatures. Finally, we give some of examples related with Frenet and Darboux frame and curvatures.

Non-Orthogonal Serret–Frenet Parametrization Applied to Path Following of B-Spline Curves by a Mobile Manipulator

Non-Orthogonal Serret–Frenet Parametrization Applied to Path Following of B-Spline Curves by a Mobile Manipulator

Analytical connection between the Frenet trihedron of a direct curve and the Darboux trihedron of the same curve on the surface

Analytical connection between the Frenet trihedron of a direct curve and the Darboux trihedron of the same curve on the surface

Smarandache Curves of Natural Curves Pair According to Frenet Frame

Smarandache Curves of Natural Curves Pair According to Frenet Frame

Coil Design for Strongly Curved Magnets Based on the Differential Geometry of the Frenet Frame

Coil Design for Strongly Curved Magnets Based on the Differential Geometry of the Frenet Frame

Quasi-position vector curves in Galilean 4-space

The Frenet frame is not suitable for describing the behavior of the curve in the Galilean space since it is not defined everywhere. In this study, an alternative frame, the so-called quasi-frame, is investigated in Galilean 4-space. Furthermore, the quasi-formulas in Galilean 4-space are deduced and quasi-curvatures are obtained in terms of the quasi-frame and its derivatives. Quasi-rectifying, quasi-normal, and quasi-osculating curves are studied in Galilean 4-space. We prove that there is no quasi-normal and accordingly normal curve in Galilean 4-space.

From virtual to actual assisted tape placement − application of the Frenet frame to robotic steering trajectories

A novel computational tool is presented that integrates the design and manufacturing of variable-angle tow composites for automated tape placement, facilitating the creation of tow-steered trajectories for cut-outs. Steering movements of the robot are planned and executed within a MATLAB virtual environment via short paths and linear commands. Leveraging piecewise-continuous clothoid splines and the Frenet frame, the tool furnishes crucial details on trajectory vectors, steering angles, and curvatures. Via four integrated modules, the user can define input parameters such as the robot’s actuators and structural geometry, allowing for the subsequent visualisation of the steered trajectories as well as the generation of the manufacturing code. Finally, to assess the potential and limitations of this methodology, the file containing all details is integrated into the Kuka KR L240-2 robot model to continuously steer a carbon fibre/PEEK tape around an elliptical cut-out in a composite panel using laser-assisted tape placement, representative of an access hole in an aircraft wing.

Analyzing the Ricci Tensor for Slant Submanifolds in Locally Metallic Product Space Forms with a Semi-Symmetric Metric Connection

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with n=2, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely.

Timelike surfaces with Bertrand geodesic curves in Minkowski 3–space

Geodesic curves on a surface play an essential role in reasonable implementation. A curve on a surface is a geodesic curve if its principal normal vector is aligned with the surface normal. Using the Serret–Frenet frame, the timelike (TL) surfaces can be specified as linear combinations of the components of the local frames in Minkowski 3–space E13. With these parametric representations, we obtained the indispensable and required events for the specified Bertrand (B) curves to be the geodesic curves on these surfaces. Afterword, the conclusion regarding the TL ruled surface is also made. Finally, the models are declared to assure that the suggested methods are effective in outcome manufacturing by modifying the styles of the surface pair.

Shape Sensing of Resin Pipe with a Helical-winding Optical Fiber in Bending Test

In this study, we developed a shape-sensing technique for a pipe. An optical fiber was helically bonded on the outer surface of a resin pipe whose length, outer diameter and inner diameter are 3200 mm, 34 mm, and 26.6 mm, respectively. Firstly, four grooves were made on the surface along the pipe in a helical pattern with the same pitch of 360 mm. At the one pipe end, adjacent grooves were offset by 90 degrees and the offset was maintained along the pipe. The single-mode fiber was put into a groove and fixed by epoxy and then it was embedded into the adjacent groove. Consequently, the optical fiber was going back and forth twice along the pipe. At the pipe ends, there were stress-free fibers between the grooves. Shape sensing for the pipe is conducted as follows. 1. Strain distributions along the grooves are measured by a distributed fiber-optic strain sensing system. 2. The bending strain is extracted from the measured strain to calculate the curvature and bending angle. 3. The shape of the pipe is reconstructed based on the curvature and bending angle using the Frenet-Serret formulas. Three-point bending tests were conducted. In the tests, the pipe ends were clamped to be the fixed ends and a point load was applied to the pipe center. Strain distributions were measured by tunable wavelength coherent optical time domain reflectometry (TW-COTDR) or optical frequency domain reflectometry (OFDR). Then the pipe shape was reconstructed as described above. We compared the reconstructed shape with the deformation calculated by finite element analysis and the agreement was excellent. If we insert the developed pipe into a bedrock and reconstruct the pipe shape continuously, it is expected that we can detect cracks or damage in the bedrock. This approach is expected to make rock excavation constructions safer.

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

A curve on a surface is a geodesic curve if its principal normal vector is anywhere aligned with the surface normal. Using the Serret–Frenet frame, a timelike surface couple (TLSC) with the symmetry of a Bertrand couple (BC) can be specified in terms of linear combinations of the components of the local frames in Minkowski 3-space E13. With these parametric representations, the necessary and sufficient conditions for the specified BC are derived to be the geodesic curves defining these surfaces. Afterward, the definition of a TL ruled surface (RS) is also provided. Furthermore, the application of the method to some significant models is given.

Structural Analysis of Glazed Tubular Tiles of Oriental Architectures Based on 3D Point Clouds for Cultural Heritage

Abstract. Laser scanning, along with its resultant 3D point clouds, constitutes a prevalent method for the documentation of cultural heritage. This paper introduces a novel workflow for the structural analysis of glazed tubular tiles that adorn the roofs of historical buildings in the Orient, utilizing 3D point clouds. The workflow integrates a robust segmentation algorithm utilizing the maximum principal curvature and normal vectors. Moreover, clustering algorithms, including DBSCAN, are incorporated to refine the clusters and thus increase segmentation accuracy. Structural analysis is enabled by cylindrical model fitting, which allows for the estimation of parameters and residuals. While the results exhibit commendable performance in individual tile segmentation, it is imperative to address the impact of substantial variations in scanning range and incident angles before engaging in the structural analysis fitting process. The results of experiment demonstrate that under conditions of significantly large scanning angles, the root mean square error (RMSE) for inadequately fitted tiles can extend to 0.066 m, surpassing twice the RMSE observed for well-fitted tiles. The proposed workflow proves to be applicable and exhibits significant potential to advance practices in cultural heritage documentation.

Alternative planes and the curves on them

In this study, the planes formed by Frenet elements are defined with a U vector chosen different from T, N and B, which are the elements of the Frenet frame, and the curves on these planes are also characterized. As it is known, the planes formed by Frenet elements between themselves have been defined and investigated many times. In this present article, the plane formed by an arbitrary chosen vector U with T, N and B is defined and the curveslying in this plane are characterized

On local geometric properties of a tokamak equilibrium

To separate the complexities in plasma physics and geometric effects, compact formulas for local geometric properties of a tokamak equilibrium are presented in this paper. They are written in a form similar to the Frenet formulas. All of the geometric quantities are expressed in terms of curvature and torsion of the three spatial curves for the moving local frame of reference, i.e., local orthogonal vector basis. In this representation, the local magnetic shear and the normalized parallel current are just the differences between two torsions of the vector basis. All of the geometric properties are coordinate invariants and form a prime set of quantities for describing tokamak plasma equilibrium. This prime set can be evaluated in both flux coordinates with closed flux surfaces and cylindrical coordinates including areas with open field lines, which may allow the extension of some analysis on the open field lines outside the last closed surface. Fundamental differential operators for stability and transport studies can be expressed explicitly in terms of these geometric properties. It can also be used to simplify analytic studies.

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.