Principal Curvatures Research Articles

A Classification of Parallel Normalized Biconservative Submanifold in the Minkowski Space in Arbitrary Dimension

IIn this paper, we examine PNMCV-MCGL biconservative submanifold in a Minkowski space $\mathbb{E}_1^{n+2}$ with nondiagonalizable shape operator, where PNMCV-MCGL submanifold denotes a submanifold with parallel normalized mean curvature vector and the mean curvature whose gradient is lightlike ($\langle\nabla H,\nabla H\rangle=0$). We obtain some conditions about connection forms, principal curvatures and some results about them. Then we use them to obtain a classification of such submanifolds. Finally, we showed that there is no biconservative such submanifold in Minkowski space of arbitrary dimension.

Зрительные иллюзии и восприятие классической архитектуры

<p>We carried out psychophysical experiments to study perception mechanisms of images that are elements of building projections in ancient and classical architecture. The curvature of isolated lines and lines superimposed on a fan picture (Wundt-Hering illusion), as well as the orientation of lines in the presence of distractors with varying inclination and separation (tilt illusion) were estimated. We used the methods of constant stimuli and the forced choice procedure. The experiments were carried out on three subjects with a large number of presentations of stimuli, which made it possible to evaluate the statistical significance of the obtained dependences. The values of distortions in the illusion of Wundt-Hering (WH) and in the tilt illusion were determined. It is shown that the curvature of single lines without fan lines is estimated adequately. The analysis of the results led to the conclusion that corrections of the orientation and curvature of elements in classical architecture compensate for the distortions of visual perception. At the same time, the curvature of some elements may be due to the aesthetics of perception.</p>

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.

New Insights into Flow for a Low-Bypass-Ratio Transonic Fan with Optimized Rotor

In this paper, a three-dimensional aerodynamics optimization system is built and applied to optimize a rotor blade to balance the conflicts between stall margin, total pressure ratio, adiabatic efficiency, and mass flow rate for the high-loading and transonic-flow fan. A novel flow diagnostic method based on vorticity dynamics theory is utilized to analyze the reasons for the improvement in aerodynamic performance in the optimized transonic fan. In the established aerodynamic optimization method, use the blade profile camber line curvature and its leading edge metal angle as the optimization variables, which are optimized by modifying the coordinates of their control points and introducing a genetic algorithm. Finally, the vorticity dynamics parameters, such as the boundary vorticity flux (BVF), azimuthal vorticity and skin-friction lines are used to diagnose the key flow features in the optimized and baseline fan passage. The results indicate that, by controlling skillfully the blade camber line curvature in the optimization improves the aerodynamic performance of the fan stage, increasing the total pressure ratio by 1.90% while increasing the mass flow rate and adiabatic efficiency by 5.82% and 4.45%, respectively. The formulas from the vorticity dynamics diagnosis method indicate a close link between the aerodynamic performance and vorticity dynamic parameters for the axial fan/compressor passage flow, and that both azimuthal vorticity and boundary vorticity flux have significant influence on fan stage performance. Moreover, the boundary layer separation flow on the rotor blade surface is accompanied by a spike of entropy and static pressure, and their derivative/gradient also suffer drastic changes under the effect of shock waves. Detailed flow information can be obtained about the on-wall with high accuracy based on the vorticity dynamics diagnosis method, which provides researchers with a novel method for the turbomachinery aerodynamic design and analysis in the aero-engine engineering development field.

SIMILARITIES BETWEEN MORPHOLOGICAL INDICATORS OF THE STOMACH OF HUMANS AND WHITE-BREEDLESS RATS

. Information about the anatomy of the stomach of white-breedless rats is very rare in the scientific literature. The shape, structure, and topography of rat stomachs are significantly different from those of vertebrates, but the same sections as shark stomachs can be found in them. This structure of white rat stomach sections characterizes the sac-like formation of the digestive tract. In the anterior part of the abdominal cavity of rats, on the left side of the midline, there is a hole for the heart between the esophagus and the line of the lesser curvature of the stomach. The liver is located in the small curvature of the stomach and partially covers it. The greater curvature of the stomach touches the caecum and cecum.

3D flame surface measurements in low-turbulence Bunsen flames via scanning and orthogonal cross-planar techniques

Quasi-instantaneous 3D flame surfaces were reconstructed in a turbulent Bunsen flame via a laser scanning technique based on an acoustic-optic-deflector (AOD). Instantaneous flame surface areas were reconstructed from a series of flame contours at different vertical planes via droplet Mie scattering with a Nd:YLF high-frequency laser at 527 nm. Mean total flame surface areas 〈A〉 were estimated and compared with estimated area demands based on total mass flow rates of fuel/air mixture divided by density and unstrained laminar flame speed, m˙/(ρusL). Flame surface density (FSD, both 2D and 3D) and flame surface curvatures were also measured and compared with those acquired using 3D cross-planar methods at particular locations. Differences of up to 67% were measured at a given location in the flame brush, with 3D FSD values generally higher than corresponding 2D measurements. The present measurements are the first comparisons of FSD and 3D curvatures comparing scanning and orthogonal cross-planar techniques allowed the measurement of 2D flame contours on two orthogonal planes, leading to improved accuracy in the determination of principal curvatures in 3D flame surfaces. Novelty and significance statementTurbulent Bunsen flame surfaces were reconstructed by a high frequency scanning technique. Values of the 3D flame surface area were for the first time measured and compared to the required area for burning the total incoming mass flow rate. Values of local 3D FSD were measured via the scanning technique and for the first time compared to corresponding measurements using cross-planar technique. True 3D principal curvatures of flame surfaces were measured by combining scanning and cross-planar techniques to provide a sufficient accuracy.

A classification of triharmonic hypersurfaces in a pseudo-Riemannian space form

The purpose of this article is to investigate triharmonic hypersurfaces Mtn in a pseudo-Riemannian space form Nt1n+1(c). Assuming that the shape operator is diagonalizable, we show that every CMC triharmonic hypersurface either has constant scalar curvature or it is minimal if cε>0 and every CMC triharmonic hypersurface is minimal if cε<0, where ε is the inner product of the unit normal vector of Mtn. Furthermore, we prove that any CMC triharmonic hypersurface with diagonalizable shape operator in a pseudo-Euclidean space Et1n+1 is actually minimal provided that zero is a principal curvature of multiplicity at most one.

Boundary estimates for solutions of the Monge–Ampère equation satisfying Dirichlet–Neumann type conditions in annular domains

We consider smooth solutions of the Monge–Ampère equation on an annular domain, whose boundary consists of two smooth, closed, strictly convex hypersurfaces, subject to mixed boundary conditions. In particular we impose a homogeneous Dirichlet condition on the outer boundary and a Neumann condition on the inner boundary. We demonstrate that in general, global C2 estimates cannot be obtained unless we impose extra restrictions on the principal curvatures of the inner boundary and on the Neumann condition itself. The latter is illustrated by the construction of an explicit counterexample. Under these conditions, we prove a priori C2 estimates and show that our problem admits a smooth solution.

Visual analysis of interface deformation in multiphase flow

In multiphase flows, the evolution of fluid-fluid interfaces is of interest in many applications. In addition to fluid dynamic forces governing the flow in the entire volume, surface tension determines droplet interfaces. Here, the analysis of interface kinematics can help in the investigation of interface deformation and the identification of potential breakups. To this end, we developed a visualization technique using metric and shape tensors to analyze interface stretching and bending. For interface stretching, we employ the eigenpairs of the metric tensor defined for the deformation rate of the fluid surface. For interface bending, we present a technique that locally captures the interface curvature change in terms of a shape tensor, extracting its principal directions and curvatures. We then visualize interface deformation by combining both representations into a novel glyph design. We apply our method to study multiphase flow simulations with particular emphasis on interface effects. These include the interplay between fluid dynamics and surface tension forces leading to breakup processes following droplet collisions, as well as droplet-droplet interactions of different fluids where Marangoni convection along the surface is explicitly taken into account.Graphical abstract

Null hypersurfaces in 4-manifolds endowed with a product structure

AbstractIn a 4-manifold, the composition of a Riemannian Einstein metric with an almost paracomplex structure that is isometric and parallel defines a neutral metric that is conformally flat and scalar flat. In this paper, we study hypersurfaces that are null with respect to this neutral metric, and in particular we study their geometric properties with respect to the Einstein metric. Firstly, we show that all totally geodesic null hypersurfaces are scalar flat and their existence implies that the Einstein metric in the ambient manifold must be Ricci-flat. Then, we find a necessary condition for the existence of null hypersurface with equal nontrivial principal curvatures, and finally, we give a necessary condition on the ambient scalar curvature, for the existence of null (non-minimal) hypersurfaces that are of constant mean curvature.

Umbilics of Surfaces in the Lorentz–Minkowski 3-Space

In this paper, we prove several fundamental properties on umbilics of a space-like or time-like surface in the Lorentz–Minkowski space {{mathbb {L}}}^3. In particular, we show that the local behavior of the curvature line flows of the germ of a space-like surface in {{mathbb {L}}}^3 is essentially the same as that of a surface in Euclidean space. As a consequence, for each positive integer m, there exists a germ of a space-like surface with an isolated C^{infty }-umbilic (resp. C^1-umbilic) of index (3-m)/2 (resp. 1+m/2). We also show that the indices of isolated umbilics of time-like surfaces in {{mathbb {L}}}^3 that are not the accumulation points of quasi-umbilics are always equal to zero. On the other hand, when quasi-umbilics accumulate, there exist countably many germs of time-like surfaces which admit an isolated umbilic with non-zero indices.

Dynamically Characterizing the Structures of Dirac Points via Wave Packets

Topological non-trivial band structures are the core problem in the field of topological materials. We investigate the topological band structure in a system with controllable Dirac points from the perspective of wave packet dynamics. By adding a third-nearest-neighboring coupling to the graphene model, additional pairs of Dirac points emerge. The emergence and annihilation of Dirac points result in hybrid and parabolic points, and we show that these band structures can be revealed by the dynamical behaviors of wave packets. In particular, for the gapped hybrid point, the motion of the wave packet shows a one-dimensional Zitterbewegung motion. Furthermore, we also show that the winding number associated with the Dirac point and parabolic point can be determined via the center of mass and spin texture of wave packets, respectively. The results of this work could motivate new experimental methods to characterize a system’s topological signatures through wave packet dynamics, which may also find applications in systems of other exotic topological materials.

Axial curvatures for corank 1 singular n-manifolds in Rn+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^{n+k}$$\\end{document}

For singular n-manifolds in Rn+k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {R}}}^{n+k}$$\\end{document} with a corank 1 singular point at p∈Msingn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in M^n_{{\ ext {sing}}}$$\\end{document} we define up to l(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l(n-1)$$\\end{document} different axial curvatures at p, where l=min{n,k+1}.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l=\\min \\{n,k+1\\}.$$\\end{document} These curvatures are obtained using the curvature locus (the image by the second fundamental form of the unitary tangent vectors) and are therefore second order invariants. In fact, in the case n=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2$$\\end{document} they generalise all second order curvatures which have been defined for frontal type surfaces. We relate these curvatures with the principal curvatures in certain normal directions of an associated regular (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-manifold contained in Msingn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^n_{{\ ext {sing}}}.$$\\end{document} We obtain many interesting geometrical interpretations in the cases n=2,3.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n=2,3.$$\\end{document} For instance, for frontal type 3-manifolds with 2-dimensional singular set, the Gaussian curvature of the singular set can be expressed in terms of the axial curvatures. Similarly for the curvature of the singular set when it is 1-dimensional. Finally, we show that all the umbilic curvatures which have been defined for singular manifolds up to now can be seen as the absolute value of one of our axial curvatures.

Effect of embedded periodic fiber placement gap defects on the microstructure and bistable behavior of thermoplastic composite laminates

AbstractBy inducing periodic gaps, conventional automated fiber placement avoids overlap defects, which are generally prohibited in composite component manufacturing. However, bistable laminates are highly sensitive to imperfections, and it is critical to explore the effect of periodic gaps on the performance of bistable laminates. The principal curvatures and load–displacement curves of thermoplastic composite laminates with manually embedded periodic gaps was studied using experimental and numerical methods in this paper. SEM microscopy was used to observe the gap areas of the cross section of laminates to assess the microstructure of gaps with width variations after curing. In addition, a numerical method was proposed based on the experimentally determined evolution of gap widths and thicknesses of laminate. Subsequently, the effects of gap width, single‐layer thickness, and total thickness on the principal curvature were investigated. Gaps embedded in the 0° and 90° layers of bistable laminates demonstrated a substantial decrease in the snap load of the laminates. Interestingly, by embedding different gap widths in the 0° layers of the laminates, the snap load discrepancy between the snap‐through and snap‐back processes increased as the gap width widened, and the same trend was also found in the curvatures in the two stable states.Highlights Bistable characteristics of laminates with periodic gap defects. Utilizing gaps embedding in 0° layer to tailor the mechanical properties.

SVD-based maximum principal curvatures for segmentation of blood vessels from diabetic retinopathy fundus images using level sets

SVD-based maximum principal curvatures for segmentation of blood vessels from diabetic retinopathy fundus images using level sets

Dispersion and spatial structure of the coupled Alfvén and slow magnetosonic oscillations in the Solar corona

Abstract Numerical and analytical analysis of the magnetohydrodynamic (MHD) waves in Solar coronal arcades is performed. A semi-cylinder slab model of arcade is used where the field lines are represented by half-circles intersecting the photosphere, the magnetic shells are represented by nested coaxial semi-cylinders. The finite plasma pressure is taken into account. The “corrugational”perturbations are considered, that is, the perturbations with short wavelength in the direction along the arcade. In this limit, there are two oscillation modes, the Alfvén and slow magnetosonic modes, coupled due to the field line curvature. The transverse dispersion of the modes, that is, the dependence of the radial wave vector’s component kr on the wave frequency ω, is considered. It was found that the wave is concentrated in two regions of mode’s existence, where $k_r^2&amp;gt;0$: the Alfvén and magnetosonic transparent regions. On one side, each of them is bounded by the resonance surface, where $k_r^2 \rightarrow \infty$. On the resonance surface, the wave’s frequency is determined by the Alfvén and slow magnetosonic modes dispersion relations, respectively. On the other side, the transparent regions are bounded by cut-off frequencies where $k_{r}^2 =0$. In both transparent regions, the perturbations have both transverse electric field (characteristic for the Alfvén mode) and field aligned velocity (characteristic for the slow mode). The wave structure along the field line for several models of plasma parameters is calculated.

Body weight estimation of beef cattle with 3D deep learning model: PointNet++

It is necessary to measure the live body weight (BW) of beef cattle to assess their health and nutritional degree. Instead of continual manual measuring in holding frames or crates currently used in the breeding industry, this paper has proposed a novel non-contact BW prediction with 3D deep learning model PointNet++. The three-dimensional (3D) point cloud data (PCD) of the whole contour surface of a beef cattle is proximally obtained by a light detection and ranging (LiDAR) sensor. To preserve the quality of the silhouette, the fusion of classical PCD preprocessing is employed. To better the accuracy of manual selection to the position of body conformation, the data extension of PCD and dataset annotations expressed by the feature regions are employed. The PCD part segmentation of deep learning models is validated and redefined with datasets of beef cattle, where the PointNet++ model with the highest accuracy of 95.1% is finally redesigned. With the advantages of being suitable for most beef breeds for the classical Johnson’s live BW prediction, the Gaussian curvature and principal curvature of contour points of feature regions are calculated to extract the body measurement points of body size, and the body length (BL) and chest girth (CG) are calculated and fit-calibrated. With the experimentation of 100 cattle in breeding farms, the results of live BW prediction have shown that the mean absolute percentage error (MAPE) is 3.2%, the root mean square error (RMSE) is 10.2 kg. The effective and practical resolution of BW estimation can improve the weighing process and animal welfare in large-scale breeding farms, and the real silhouette dataset of large animals can help the automated recognition and classification of body conformation.

Surface Family Pair with Bertrand Pair as Mutual Curvature Lines in Three-Dimensional Lie Group

This paper is on deducing the necessary and sufficient conditions of a surface family pair with a Bertrand pair as mutual curvature lines in three-dimensional Lie group G. As a result, the consequence for the ruled surface family pair is also extrapolated. Meanwhile, examples are specified to show the surface family with common Bertrand geodesic curves.

Total torsion of three-dimensional lines of curvature

A curve gamma in a Riemannian manifold M is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when gamma lies on an oriented hypersurface S of M, we say that gamma is well positioned if the curve’s principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that gamma is three-dimensional and closed. We show that if gamma is a well-positioned line of curvature of S, then its total torsion is an integer multiple of 2pi ; and that, conversely, if the total torsion of gamma is an integer multiple of 2pi , then there exists an oriented hypersurface of M in which gamma is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of gamma vanishes when S is convex. This extends the classical total torsion theorem for spherical curves.

Spike formation theory in three-dimensional flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in three-dimensional flows with arbitrary time dependence. Based on the exact evolution of the largest principal curvature on near-wall material surfaces, our theory identifies fixed and moving separation. Our approach is effective over short time intervals and admits an instantaneous limit. As a byproduct, we derive explicit formulas for the evolution of the Weingarten map and the principal curvatures of any surface advected by general three-dimensional flows. The material backbone we identify acts first as a precursor and later as the centrepiece of Lagrangian flow separation. We discover previously undetected spiking points and curves where the separation backbones connect to the boundary and provide wall-based analytical formulas for their locations. We illustrate our results on several steady and unsteady flows.