Principal Curvatures Research Articles

New Characterizations of Hyperspheres and Spherical Hypercylinders in Euclidean Space

Let x be an isometric immersion of a Riemannian n‐manifold M into a Euclidean (n + 1)‐space which does not pass through the origin of . Then, the tangential part of the position vector field x of x is called the canonical vector field, and the normal part gives rise to a scalar function called the support function. Using the canonical vector field, support function, and mean curvature, we establish three new characterizations of hyperspheres. In addition, we prove that if the energy function of M satisfies the static perfect fluid equation, then M has at most two distinct principal curvatures. As an application, we prove that a complete noncompact hypersurface M is a spherical hypercylinder if the energy function of M satisfies the static perfect fluid equation, and it has exactly two distinct nonsimple principal curvatures.

Characterization of imbricate-ruled surfaces via rotation-minimizing Darboux frame in Minkowski 3-space $ \mathrm{E}_1^3 $

<abstract><p>Using a common tangent vector field to a surface along a curve, in this study we discussed a new Darboux frame that we referred to as the rotation-minimizing Darboux frame (RMDF) in Minkowski 3-space. The parametric equation resulting from the RMDF frame for an imbricate-ruled surface was then provided. As a result, minimal (or maximal for timelike surfaces) ruled surfaces were derived, along with the necessary and sufficient criteria for imbricate-ruled surfaces to be developable. The surfaces also described the parameter curves of these surfaces' asymptotic, geodesic, and curvature lines. We also gave an example to emphasize the most significant results.</p></abstract>

FINDING THE RELATIVE TRAJECTORY OF THE CARGO IN THE BODY OF A VEHICLE WHICH IS GOING DOWN OR UP ON A CURVED SECTION OF THE ROAD

With a certain assumption, the load, which during relative movement along the bottom of the body makes translational movement, can be taken as a material point. In this case, finding the relative movement of the load in the body when the car moves along the road with a spatial axis is reduced to finding the relative trajectory of the material point. In known works, it is proposed to use the accompanying trihedron of the trajectory of the transfer motion and Frenet's formula to find the absolute acceleration vector of a point. The independent variable is not time, as in traditional problems, but the arc coordinate of the transfer trajectory, since only under this condition can Frenet's formulas be applied. The vector of absolute acceleration is obtained in projections onto the vertices of the moving accompanying trihedron, without looking for individual components (transfer, relative, and Coriolis acceleration), as is done in traditional approaches. The purpose of the study is to investigate the relative movement of the load, taken as a material point, along the bottom of the body of a truck moving along a curvilinear section of the road with a spatial axis during its ascent or descent. The spatial curve is characterized by two parameters that depend on the kinematics of the accompanying Frenet trihedron. These parameters are the curvature k and the twist σ of the curve. Their values at any point of the curve will be determined if the dependencies k=k(s) and σ=σ(s) are known, where s is the arc coordinate (the arc length of the curve). It is assumed that such a spatial curve is the axis of a curved section of the road that goes uphill or downhill. The application of the accompanying trihedron of the curve and Frenet's formulas made it possible to describe the relative motion of the cargo in the body of a car moving along a road having a spatial axis with constant and variable speeds. The regularities of the relative movement of the cargo when the car is moving on the road, the axis of which is a helical line with constant curvature and twist, have been found. It was found that there is no significant difference between the value of the relative speed of the cargo when the car moves at a constant speed down or uphill at the initial stage. When braking on a descent, a significant relative movement of the cargo is observed, which increases with an increase in the braking acceleration, and when climbing, on the contrary, the relative movement decreases with an increase in the braking acceleration. The regularity of the relative movement of the cargo when the car moves along the road with variable curvature and torsion of the axis line depends on the type of these dependencies as a function of the length of the arc of the axis line.

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.

A Novel Symmetry in Nanocarbons: Pre-Constant Discrete Principal Curvature Structure

A Novel Symmetry in Nanocarbons: Pre-Constant Discrete Principal Curvature Structure

A Survey of Magnetic Field Line Curvature in Jovian Dawn Magnetodisc

AbstractThe Jovian magnetosphere is highly dynamic, influenced by both solar wind and internal processes associated with the rapid planetary rotation and Io's volcanic activities. Accompanying the mass and energy circulations driven by the magnetospheric dynamics, the magnetic configuration also changes dramatically. One of the crucial parameters to characterize the magnetic configuration is magnetic field line curvature (FLC), which generally describes how stretched the field line is. The curvature is pivotal to influence particle behaviors, for example, pitch angle scattering which may lead to auroral particle precipitation. In this work, a method is proposed to investigate the real‐time magnetic FLC in Jovian current sheet using the magnetic field data from the Juno spacecraft. The results indicate that the FLC scattering of ions and relativistic electrons are common in Jovian magnetosphere, providing a crucial insight to understand the particle behaviors.

Accurate tissue flap reconstruction method based on the quadratic surface developability for head and neck soft tissue defects

Soft tissue defects resulting from head and neck tumor resection seriously impact the physical appearance and psychological well-being of patients. The complex curvature of the human head and neck poses a formidable challenge for maxillofacial surgeons to achieve precise aesthetic and functional restoration after surgery. To this end, a normal head and neck volunteer was selected as the subject of investigation. Employing Gaussian curvature analysis, combined with mechanical constraints and principal curvature analysis methods of soft tissue clinical treatment, a precise developable/non-developable area partition map of the head and neck surface was obtained, and a non-developable surface was constructed. Subsequently, a digital design method was proposed for the repair of head and neck soft tissue defects, and an in vitro simulated surgery experiment was conducted. Clinical verification was performed on a patient with tonsil tumor, and the results demonstrated that digital technology-designed flaps improved the accuracy and aesthetic outcome of head and neck soft tissue defect repair surgery. This study validates the feasibility of digital precision repair technology for soft tissue defects after head and neck tumor resection, which effectively assists surgeons in achieving precise flap transplantation reconstruction and improves patients' postoperative satisfaction.

Bifurcations of robust features on surfaces in the Minkowski 3-space

We obtain the bifurcation of some special curves on generic 1-parameter families of surfaces in the Minkowski 3-space. The curves treated here are the locus of points where the induced pseudo metric is degenerate, the discriminant of the lines principal curvature, the parabolic curve and the locus of points where the mean curvature vanishes.

A new spatial data structure for triangular mesh with toroidal patches

We introduce a novel spatial data structure for a triangular mesh consisting of osculating toroidal patches. A toroidal patch is generated using the differential geometry of a vertex by approximating a local surface to estimate the principal curvatures and directions. Within a given tolerance, the adjacent faces of a vertex can effectively be approximated by the toroidal patch. For efficient usage, toroidal patches are constructed in a hierarchical manner using various tolerance values ϵ=10−3,10−4, and 10−5 and combined with the conventional Bounding Volume Hierarchy (BVH). To demonstrate the effectiveness of our approach, we employ our spatial data structure for filling holes in triangular meshes. After generating a hole-filled patch, we update the hole-filled patch’s vertices using the normals and positions of the toroidal patches. To the best of our knowledge, this is the first application of toroidal patches to triangular mesh.

Construction of a mathematical model for approximating the sphere by strips of unfolding surfaces

Approximating non-expandable surfaces by compartments of expandable ones makes it possible to simplify the process of obtaining the required shape without loss of operational properties. There is a known approximation of a sphere using the example of a ball when its surface can consist of polygons. However, this list does not exhaust the possible options for approximating the sphere. Its approximation by truncated cones tangent to parallels or by congruent cylindrical petals tangent to meridians is known. Any line on the surface of a sphere is a line of curvature. This means that the common line of contact of the expanded surface with the sphere will be a line of curvature for the expanded surface as well (rectilinear generatrices of the expanded surface will cross this line at a right angle). When building a sweep of such a surface, the line of contact will be transformed but the rectilinear generatrices will remain perpendicular to it, which simplifies the construction of the sweep. The approximation of the sphere by congruent strips, the number of which can be different, starting from one, is considered. A necessary condition for such an approximation is a common line of contact of adjacent strips. To this end, the line of contact on the sphere or the guide curve must have an appropriate shape. Such a curve is taken as a slope line (a curve whose tangents form a constant angle of inclination to the horizontal plane). The study results are the parametric equations of the strip touching the sphere and its corresponding equations on the sweep. The construction of the strip on the sweep is explained by the invariance of the geodesic curvature of the guide curve when the strip is bent until it aligns with the plane. This explains the difference between the proposed approach and conventional methods of sphere approximation. Approximating the sphere by strips of unfolding surfaces has a practical application in architecture with spherical elements, as well as in religious buildings with domes in the form of a part of the sphere

Enhancing optical character recognition: Efficient techniques for document layout analysis and text line detection

AbstractIn recent years, automatic document and text analysis has gained significant importance, driven by advancements in optical character recognition (OCR) technology and the need for efficient processing of large volumes of printed or handwritten documents. This article specifically focuses on document layout analysis (DLA) and text line detection (TLD), both of which are crucial components of OCR systems. Our objective is to develop an effective method for extracting both textual and non‐textual regions, addressing challenges unique to the Persian (and Persian‐like) language(s). In the DLA stage, we employ deep learning models and a voting system to accurately determine the regions of interest. Additionally, we introduce methods such as optimum font size concepts, angle correction, and a line curvature elimination algorithm in the TLD process to enhance OCR accuracy. Comparative evaluations against state‐of‐the‐art methods demonstrate the superiority of our approach, showcasing a 2.8% improvement in the accuracy of Tesseract‐OCR 5.1.0 (a well‐established commercial OCR system) on the official Iranian newspapers dataset. These findings underscore the importance of addressing DLA and TLD challenges to advance OCR technology for Persian language documents and provide a solid foundation for future research in this domain.

Topology-Mediated Molecule Nucleation Anchoring Enables Inkjet Printing of Organic Semiconducting Single Crystals for High-Performance Printed Electronics.

Printable organic semiconducting single crystals (OSSCs) offer tantalizing opportunities for next-generation wearable electronics, but their development has been plagued by a long-standing yet inherent problem─spatially uncontrolled and stochastic nucleation events─which usually causes the formation of polycrystalline films and hence limited performance. Here, we report a convenient approach to precisely manipulate the elusive molecule nucleation process for high-throughput inkjet printing of OSSCs with record-high mobility. By engineering curvature of the contact line with a teardrop-shaped micropattern, molecule nucleation is elegantly anchored at the vertex of the topological structure, enabling formation of a single nucleus for the subsequent growth of OSSCs. Using this approach, we achieve patterned growth of 2,7-dioctyl[1]benzothieno[3,2-b][1]benzothiophene single crystals, yielding a breakthrough for an organic field-effect transistor array with a high average mobility of 12.5 cm2 V-1 s-1. These findings not only provide keen insights into controlling molecule nucleation kinetics but also offer opportunities for high-performance printed electronics.

Trajectories associated with parabolic and hyperbolic periodic points of piecewise linear area-preserving map

This paper investigates the characteristics of trajectories and invariant sets associated with parabolic and hyperbolic periodic points of a piecewise linear area-preserving map. When the fixed points in both partitions are parabolic, the trajectories may diverge or rotate repeatedly around the origin. By analyzing the trajectory motion in different regions of the phase plane, the characteristics of the trajectory repeated rotation around the origin are shown, and the trajectories may be either divergent or periodic. The necessary conditions for generating stable periodic orbits are presented. It is proved that there are forward invariant sets and that the trajectories from the invariant sets diverge in different quadrants and along a family of invariant lines parallel to the diagonal bisector of the quadrant. We reveal why there are island chains with invariant line segments in a chaotic sea. The method for determining invariant line segments is presented. For the map with two hyperbolic fixed points in both partitions, it is proved that there are forward invariant sets in the phase plane and all trajectories from the invariant sets diverge along a family of hyperbolic invariant sets.

Cartesian coordinates of an intrinsically defined curve

Summarizing some old research on the dynamics of a pointall body along its own trajectory, this paper established the differential relationships between the principal curvatures of a 3D curve, that is the normal curvature and the torsional curvature, and its Cartesian coordinates. The differential system thus derived is actually a dynamical system of a representative point of the curve moving along it. This dynamic system is analyzed to see the possibilities of finding analytical solutions in finite terms, using Frobenius' integrability theorem for the general case and usual integration methods for the particular case consisting of the constant ratio between the two curvatures.

Form-finding for Free-form Reticulated Shell Structure Considering Structural Performance and Regularization of Node Connection Patterns

In this study, we propose a form-finding method for the free-formed reticulated shell structure, considering both structural performance and regularization of node connection patterns. The coordinates of the nodes of the reticulated shells are represented on the non-uniform rational B-spline (NURBS) surface. According to the principal curvature, the clustering classifies the node connection patterns into specified groups. Multi-objective optimization is performed to minimize the strain energy and maximum deviation in clusters. A multi-objective genetic algorithm is used to solve the optimization problem. The mechanical properties of the optimal solution obtained in a numerical example are discussed to confirm the effectiveness of the proposed method.

Comparison of Ring Current Proton Losses Between Contributions From Scattering by Field Line Curvature and EMIC Waves

AbstractAlthough both the field line curvature (FLC) and electromagnetic ion cyclotron (EMIC) waves are recognized to contribute importantly to the loss of energetic protons in the Earth's ring current, their quantitative differences in scattering ring current protons are still not fully understood. In this study, using a realistic, nondipolar magnetic field model, we perform a detailed analysis of the scattering effects of ring current protons and the resultant proton lifetimes due to FLC and EMIC waves at different L‐shells under different levels of geomagnetic activity. We find that compared with EMIC wave‐driven scattering, FLC scattering has a stronger dependence on L‐shell and geomagnetic activity. Pitch angle scattering efficiency due to FLC enhances significantly as proton energy increases. In contrast, scattering by H+ band EMIC waves tends to be weaker with increasing proton energy, while the scattering rates by He+ band EMIC waves increase first and then decrease. The results of proton lifetime against pitch angle scattering show that EMIC wave‐driven scattering dominates the loss of ring current protons at lower L‐shells (L < 6), especially during geomagnetically quiet conditions. During geomagnetically active conditions at L ≥ 6, EMIC wave‐driven scattering dominates at lower proton energies, while FLC scattering dominates at higher proton energies. This study improves the current understanding of the relative contributions of scattering by FLC and EMIC waves to the Earth's ring current proton dynamics.

A kind of rotational surfaces with a light-like axis in conformally flat pseudo-spaces of dimensional three

In this work, we define the rotational surface with a light-like axis in conformally flat pseudo-spaces $\left(\mathbb{E}_3^1\right)_\lambda$, where $\lambda$ is a radial-type conformal factor. We relate the principal curvatures of a non-degenerate surface that belongs to conformally equivalent spaces $\left(\mathbb{E}_3^1\right)_\lambda$ and $\mathbb{R}_1^3$, based on the radial conformal factor. Thus, we establish a relationship between the radial conformal factor and the profile curve of the rotational flat surface in $\left(\mathbb{E}_3^1\right)_\lambda$, but also for that of the rotational surface with zero extrinsic curvature.

Ruled surface family with common special curve

In this paper, we investigate the problem of the construction of a ruled surface family passing through the striction curve (c) of a ruled surface φ. We first define ruled surface family with common striction curve (c). Using the geodesic Frenet frame {e,t,g} of the ruled surface φ, we present sufficient conditions for that striction curve to be geodesic, line of curvature and asymptotic curve. Finally, we give some examples to show ruled surfaces with common geodesic, line of curvature and asymptotic curve.

Free Boundary Minimal Annuli Immersed in the Unit Ball

We construct a family of compact free boundary minimal annuli immersed in the unit ball B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document} of R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document} foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in B3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^3$$\\end{document}. Their existence solves in the negative a problem proposed by Wente in 1995.

Dispersion and spatial structure of coupled Alfvén and slow magnetosonic modes in the dipole magnetosphere

Numerical analysis of the coupling between the Alfvén and slow azimuthally small scale modes in the dipolar model of the magnetosphere is performed. The field line curvature, inhomogeneity both across the magnetic shells and along the field lines, and finite plasma pressure are taken into account. The field line curvature causes coupling Alfvén and slow modes, while the contribution from the fast mode is neglected due to the assumed small scale of the oscillations in the azimuthal direction. The plasma pressure decreases with distance from the Earth, which is a characteristic for the ring current region. The wave's transverse dispersion, that is, dependence of the radial wave vector kr on the frequency ω, was studied. It was found that the wave can exist in two frequency ranges. Both ranges are bounded by resonant frequency on one side and the cutoff frequency on the other. The higher frequency range corresponds to the Alfvén mode. However, its properties are modified due to the coupling with the slow mode. For example, the divergence of the plasma displacement and the magnetic field compressional component appear. In the slow magnetosonic region, in contrast, the cutoff frequency is always smaller than the resonant one. If the pressure gradient is strong and negative, the slow mode cutoff frequency can disappear, that is, the radial wave vector squared even for the zero frequency. It means that the kr value goes to zero at imaginary frequency. Mode structure along the field line for different plasma pressure values and its pressure gradients was calculated.