Principal Curvatures Research Articles

Alignable Lamella Gridshells

Alignable lamella gridshells are 3D grid structures capable of collapsing into a planar strip. This feature significantly simplifies on-site assembly and also ensures compactness for efficient transport and storage. However, designing these structures still remains a challenge. This paper tackles the inverse design problem of alignable lamella gridshells leveraging concepts from differential geometry and Cartan's theory of moving frames. The study unveils that geodesic alignable gridshells, where lamellae are disposed tangentially to the surface, are limited to forming shapes isometric to surfaces of revolution. Furthermore, it demonstrates that alignable gridshells with lamellae arranged orthogonally to a surface can be realized only on a specific class of surfaces that meet a particular curvature condition along their principal curvature lines. Finally, drawing on these theoretical findings, this work introduces novel computational tools tailored for the design of these structures.

Modified Sweeping Surfaces in Euclidean 3-Space

In this study, we explore the sweeping surfaces in Euclidean 3-space, utilizing the modified orthogonal frames with non-zero curvature and torsion, which allows us to consider the spine curves even if their second differentiations vanish. If the curvature of the spine curve of a sweeping surface has discrete zero points, the Frenet frame might undergo a discontinuous change in orientation. Therefore, the conventional parametrization with the Frenet frame of such a surface cannot be given. Thus, we introduce two types of modified sweeping surfaces by considering two types of spine curves; the first one’s curvature is not identically zero and the second one’s torsion is not identically zero. Then, we determine the criteria for classifying the coordinate curves of these two types of modified sweeping surfaces as geodesic, asymptotic, or curvature lines. Additionally, we delve into determining criteria for the modified sweeping surfaces to be minimal, developable, or Weingarten. Through our analysis, we aim to clarify the characteristics defining these surfaces. We present graphical representations of sample modified sweeping surfaces to enhance understanding and provide concrete examples that showcase their properties.

Isotropic ↔ anisotropic surface geometry transitions induced by adsorbed surfactants at water/vapor interfaces.

Adsorbates at a water/vapor interface change the surface geometry through altered surface tension, yet detailed theoretical studies are relatively sparse, and many applications focus on ensemble average characteristics. Here, we demonstrate that different interpretations of surface geometry emerge when considering the distributions of surface curvature and orientation as a function of adsorbed surfactant concentration and sterics. At low surface densities, the tributyl phosphate (TBP) sorbed water/vapor surface has an increased presence of ridges that are defined by principal curvatures κ1 and κ2 of opposite signs yet close in magnitude. As the TBP surface density increases, the difference in principal curvatures slowly increases. There is a distinct transition of the surface geometry, where the ridge-like features become much more pronounced, having sides whose orientation is normal to a flat interfacial plane. Thus, as the TBP surfactant is added to the surface, the surface curvatures become anisotropic in terms of the difference in magnitude of κ1 and κ2. We label this an isotropic → anisotropic geometric transition. Comparing the surface geometry as a function of the carbon tail length of the alkyl phosphate surfactant reveals that smaller surfactants also anisotropically enhance surface curvatures and that adsorbed alkyl tails to the surface stabilize and increase the symmetry of surface waves along the two principal curvature axes. We label this an anisotropic → isotropic geometric transition. These results reflect the opportunity to incorporate more realistic distributions of surface geometry within the collective understanding of statistical theories of surfaces, including capillary wave theory.

On the inverse mean curvature flow by parallel hypersurfaces in space forms

We consider the inverse mean curvature flow by parallel hypersurfaces in space forms. We show that such a flow exists if and only if the initial hypersurface is isoparametric. The flow is characterized by an algebraic equation satisfied by the distance function of the parallel hypersurfaces. The solutions to the flow are obtained explicitly when the distinct principal curvatures have the same multiplicity. This is an additional assumption only for isoparametric hypersurfaces of the hyperbolic space or of the sphere with two or four distinct principal curvatures. The boundaries of the maximal interval of definition, when finite, are determined in terms of the number g g of distinct principal curvatures, their multiplicities m m and the mean curvature H H of the initial hypersurface. We describe the collapsing submanifolds of the flow at the boundaries of the interval. In particular, we show in the Euclidean space the solutions are eternal, while in the hyperbolic space there are eternal and immortal solutions. Starting with a connected isoparametric submanifold of the sphere, we show that the flow is an ancient solution, that collapses into a minimal hypersurface whose square length of its second fundamental form and its scalar curvature are constants given in terms of g g and n n . The minimal hypersurface is totally geodesic when g = 1 g=1 , it is a Clifford minimal hypersurface of the sphere when g = 2 g=2 and it is a Cartan type minimal submanifold when g ∈ { 3 , 4 , 6 } g\in \{3,4,6\} .

Properties of relativistic electron precipitation: a comparative analysis of wave-induced and field line curvature scattering processes

Properties of relativistic electron precipitation: a comparative analysis of wave-induced and field line curvature scattering processes

On quaternionic bisectional curvature

AbstractIn this article we study the concept of quaternionic bisectional curvature introduced by Chow and Yang in (J Differ Geom 29(2):361–372, 1989) for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space $$\mathbb {H}\textrm{P}^n$$ H P n . We also show that all symmetric quaternion-Kähler manifolds different from $$\mathbb {H}\textrm{P}^n$$ H P n admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.

Nightside Electron Precipitation Patterns as Observed by ELFIN and CIRBE CubeSats

AbstractThe rapidly expanding fleet of low‐altitude CubeSats equipped with energetic particle detectors brings new opportunities for monitoring the dynamics of the radiation belt and near‐Earth plasma sheet. Despite their small sizes, CubeSats can carry state‐of‐the‐art instruments that provide electron flux measurements with finer energy resolution and broader energy coverage, compared to conventional missions such as POES satellites. The recently launched CIRBE CubeSat measures 250–6,000 keV electrons with extremely high energy resolution, however, CIRBE typically only measures locally‐trapped electrons and cannot directly measure the precipitating electrons. This work aims to develop a technique for identifying indications of nightside precipitation using the locally‐trapped electron measurements by the CIRBE CubeSat. This study focuses on two main types of drivers for nightside precipitation: electron scattering by the curvature of magnetic field lines in the magnetotail current sheet and electron scattering by resonance with electromagnetic ion cyclotron (EMIC) waves. Using energy and pitch‐angle resolved electron fluxes from the low‐altitude ELFIN CubeSat, we reveal the features that distinguish between these two precipitation mechanisms based solely on locally‐trapped flux measurements. Then we present measurements from four CIRBE orbits and demonstrate the applicability of the proposed technique to the investigation of nightside precipitation using CIRBE observations, enabling separation between precipitation induced by curvature scattering and EMIC waves in nearby regions. Our study underscores the feasibility of employing high‐energy‐resolution CIRBE measurements for detecting nightside precipitation of relativistic electrons. Additionally, we briefly discuss outstanding scientific questions about these precipitation patterns that could be addressed with CIRBE measurements.

Underwater Line Monitoring Using Optimally Placed Inclinometers

Underwater monitoring presents challenges related to maintaining a continuous power supply and communication, necessitating the use of a smaller number of sensors to effectively cover the entire line. An underwater line tracking method is proposed to evaluate global behaviors and stresses in real time. The method employs angles at several points on the line, as well as displacements and curvatures at both ends. In this method, any line displacement, angle, and curvature are expressed as Fourier series, and Fourier coefficients are obtained by utilizing sensor data. Then, the behavior of any line location is assessed. In addition, to reduce the number of sensors and improve accuracy, optimal inclinometer locations are determined by a genetic algorithm. The proposed line tracking algorithm was validated through two numerical examples; one with an inclined tunnel and one with a marine steel catenary riser attached to a Floating Production Storage and Offloading (FPSO) vessel. Through these examples, the proposed algorithm was proven to capture global behaviors accurately when optimally located sensors are used. In the riser monitoring case, the optimized sensor placement with eight intermediate sensors achieved an average mean distance error of 1.91 m, which is lower than the 2.65 m error obtained with ten intermediate sensors without optimization.

On curvature-adapted hypersurfaces of 𝕊2 × 𝕊2 and ℍ2 × ℍ2

In this paper, we first give the characterizations of curvature-adapted hypersurfaces in both of the two Kähler surfaces [Formula: see text] and [Formula: see text]. Then, focusing on two subclasses of the curvature-adapted hypersurfaces in [Formula: see text] and [Formula: see text], we classify the Hopf hypersurfaces with two distinct principal curvatures as well as the Hopf hypersurfaces whose Reeb vector field [Formula: see text] and Reeb function [Formula: see text] satisfying [Formula: see text].

Local rigidity of constant mean curvature hypersurfaces in space forms

In this paper, we study the local rigidity of constant mean curvature (CMC) hypersurfaces. Let x:Mn→Mn+1(c),n≥4, be a piece of immersed constant mean curvature hypersurface in the (n+1)-dimensional space form Mn+1(c). We prove that if the scalar curvature R is constant and the number g of the distinct principal curvatures satisfies g≤3, then Mn is an isoparametric hypersurface. Further, if Mn is a minimal hypersurface, then Mn is a totally geodesic hypersurface for c≤0, and Mn is either a Cartan minimal hypersurface, a Clifford minimal hypersurface, or a totally geodesic hypersurface for c>0, which solves the high dimensional version of Bryant Conjecture.

A New Fractional Curvature of Curves and Surfaces in Euclidean Space Using the Caputo’s Fractional Derivative

In this paper, the authors generalize the fractional curvature of plane curves introduced by Rubio et al. in 2023, to regular curves in the Euclidean space R3, and study the geometric properties of the curve using Caputo’s fractional derivative. Furthermore, we introduce a new definition of fractional curvature and fractional mean curvature of a regular surface, using fractional principal curvatures; and prove that such concepts are invariant under isometries; i.e., they belong to the intrinsic geometry of the regular surface. Also, a geometric interpretation is given to Caputo’s fractional derivative of algebraic polynomials.

The Impact of the 8–10 March 2012 Geomagnetic Storm on Inner Zone Protons as Measured by Van Allen Probes

AbstractThe Relativistic Electron Proton Telescope (REPT) instrument on the Van Allen Probes observed a double‐peaked inner zone proton population throughout the 7 year lifetime of the mission. M. Hudson et al. (2023) showed that a strong SEP event accompanied by a CME‐shock in early March 2012 provided the Solar Energetic Proton (SEP) source for the higher L trapped proton population, which then diffused radially inward to be observed by REPT at . The study followed trajectories of SEP protons launched isotropically from a sphere at 7 Re for 2.5 hr in fields calculated by the LFM‐RCM global MHD model, which includes electric fields needed to model the transport and trapping of the protons by the shock, and then a radial diffusion simulation was run for 2 years using the result from the test‐particle simulation as the initial condition. The simulation result was compared with REPT measurement in November 2013 and showed reasonable agreement. However, the simulation overestimated the Phase Space Density by a factor of four due to lack of field line curvature scattering during the storm in the model. In this study, a test‐particle simulation is performed for 2 days following the injection and trapping of protons in March 2012 using TS05 fields to simulate the field line curvature scattering of the trapped SEP due to the buildup of the ring current during the geomagnetic storm. The resulting sample distribution was then weighted using the flux at the end of the two‐hour MHD‐test particle simulation. A radial diffusion simulation is then run using the initial profile that included the loss effect, with improved comparison with REPT measurements after 2 years.

A two-layered cross-ply laminate model for a single-ply satin woven fabric composite under thermal loads

The dominant fiber direction of a single-ply satin woven fabric is different depending on whether it is the upper or lower side. This affects the thermal distortion of laminates and principal curvature directions. Therefore, we propose a two-layered cross-ply laminate model for a single-ply satin woven fabric composite to consider the dominant fiber directions. To verify our model, we derive analytical solutions of antisymmetric angle-ply laminates under thermal loads for the conventional and proposed models. By comparing the analytical solutions with experimental results of unidirectional and satin woven fabric composites, we confirm that the proposed model accurately predicts the principal curvature direction.

Flux atop an advancing slip face and the brink line curvature of barchan dunes

Flux atop an advancing slip face and the brink line curvature of barchan dunes

Möbius geometry of locally conformal flat hypersurfaces

In this paper, we consider the Möbius geometry of locally conformal flat hypersurfaces. First, we prove that any locally conformal flat hypersurface in the sphere Sn+1(n≥4) is a canal hypersurface, which is the envelope of a one-parameter family of principal curvature hyperspheres. Conversely, canal hypersurfaces in Sn+1(n≥3) without umbilical points are locally conformal flat. Finally, by applying the maximum principle, we show that a compact umbilical points hypersurface with isotropic Blaschke tensor in Sn+1(n≥3) is Möbius equivalent to a compact minimal hypersurface with constant scalar curvature in Sn+1.

A novel approach to construct timelike developable surface with a common line of curvature

A novel approach to construct timelike developable surface with a common line of curvature

Characterization of Trabecular Bone Microarchitecture and Mechanical Properties Using Bone Surface Curvature Distributions.

Understanding bone surface curvatures is crucial for the advancement of bone material design, as these curvatures play a significant role in the mechanical behavior and functionality of bone structures. Previous studies have demonstrated that bone surface curvature distributions could be used to characterize bone geometry and have been proposed as key parameters for biomimetic microstructure design and optimization. However, understanding of how bone surface curvature distributions correlate with bone microstructure and mechanical properties remains limited. This study hypothesized that bone surface curvature distributions could be used to predict the microstructure as well as mechanical properties of trabecular bone. To test the hypothesis, a convolutional neural network (CNN) model was trained and validated to predict the histomorphometric parameters (e.g., BV/TV, BS, Tb.Th, DA, Conn.D, and SMI), geometric parameters (e.g., plate area PA, plate thickness PT, rod length RL, rod diameter RD, plate-to-plate nearest neighbor distance NNDPP, rod-to-rod nearest neighbor distance NNDRR, plate number PN, and rod number RN), as well as the apparent stiffness tensor of trabecular bone using various bone surface curvature distributions, including maximum principal curvature distribution, minimum principal curvature distribution, Gaussian curvature distribution, and mean curvature distribution. The results showed that the surface curvature distribution-based deep learning model achieved high fidelity in predicting the major histomorphometric parameters and geometric parameters as well as the stiffness tenor of trabecular bone, thus supporting the hypothesis of this study. The findings of this study underscore the importance of incorporating bone surface curvature analysis in the design of synthetic bone materials and implants.

Rayleigh-type wave in non-planar pre-stressed nearly incompressible elastic structure

The primary objective of the current work is to establish the impacts of planar boundary and non-planar boundary on the displacement components (DCs) of Rayleigh-type wave propagation in an elastic structure. The elastic structure, precisely, incorporates a pre-stressed nearly incompressible elastic material in the half-space along with the non-planar boundary at the free surface. The methodology applied features variable separable technique and Taylor’s series expansion along with substitution method to establish the expressions for DCs and frequency relation. The frequency relation of Rayleigh-type wave is obtained for the situation considering planar free surface, which is further reduced as a special case of the study and validated with the classical result present in the literature. The DCs of Rayleigh-type wave have been numerically computed, and their reliance on wave number have been exhibited graphically. In addition to this, the influences of affecting parameters viz., hydrostatic stress, principal Cauchy stress and curvature parameter on DCs have been traced out. The computational result also manifests the domain of existence of Rayleigh-type wave propagating in the considered model in the presence of planar free surface.

Multi-segmented fifth-order polynomial–shaped shells under hydrostatic pressure

The design and construction of submerged complex shells for new applications in offshore structures are increasingly popular. Therefore, the purpose of this study is to present the analytical model and Lagrange multipliers associated with the constraint equation for large displacement analysis of a fifth-order polynomial–shaped shell under hydrostatic pressure for the first time. The shell geometry can be computed using differential geometry with a fifth-order polynomial. The energy functional of the fifth-order polynomial–shaped shell is derived based on the principle of virtual work and written in the appropriate form. The nonlinear static responses of the fifth-order polynomial–shaped shell under hydrostatic pressure can be calculated using the nonlinear finite element method via the fifth-order polynomial shape function. This study develops the model using one-dimensional beam elements divided along the shell radius. To avoid the slope of a meridian curve at the equatorial plane approaching infinity, the shell is divided into two regions defined by different surface parameters. At the junction of two adjacent regions, the continuity requirements are established as the constraint conditions using Lagrange multipliers. The numerical results from the proposed methods are demonstrated and discussed, along with the effects of varied seawater depth, thickness, and elastic modulus on the deformed configuration and principal curvature at the deformed state. The results show that the nonlinear displacement is higher than the linear one in the case of the hydrostatic pressure, whereas the case of the internal pressure has an opposite result. For principal curvatures at the apex, the principal curvatures increase as the seawater depth increases, whereas the principal curvatures decrease when the thickness and elastic modulus increase.

Posterior Eye Shape in Myopia

PurposeTo explore prevalence and associated factors of abnormalities of the posterior eye shape in dependence of axial length. DesignPopulation-based study. ParticipantsOut of the participants (n=3468) of the Beijing Eye Study, we included all eyes with an axial length of ≥25 mm, and a randomized sample of eyes with an axial length of <25 mm. MethodsUsing 30°-wide, serial horizontal, and fovea-centered radial, optical coherence tomographic (OCT) images, we examined location and depth of the most posterior point of the retinal pigment epithelium/Bruch´s membrane line (PP-RPE/BML). Main Outcome ParametersPrevalence and depth of an extrafoveal PP-RPE/BML. ResultsThe study included 366 eyes (314 individuals;age:63.7±9.7 years). On the radial OCT scans, the PP-RPE/BML was located in the foveola in 190 (51.9%;95%CI:46.9,56.9) eyes, in 121 (33.1%;95%CI:28.1,38.1) eyes in the 6 o´clock part of the vertical meridian at a mean distance of 1.73±0.70mm from the foveola, and in 54 (14.8%;95%CI:11.1,18.4) eyes in the 12 o´clock part of the vertical meridian at a distance of 2.01±0.66mm from the foveola. Deeper PP-RPE/BML depth increased with longer axial length (beta:0.28;B:12.3;95%CI:7.5,17.1;P<0.001), female sex (beta:0.17;B:30.2;95%CI:10.7,49.6;P=0.002), and higher corneal astigmatism (beta:0.12;B:13.9;95%CI:1.50,26.3;P=0.03). On the horizontal OCT scans, the PP-RPE/BML was located in the foveola in 304 (83.1%;95%CI:79.2,86.9) eyes, between the foveola and the optic disc in 36 (9.8%;95%CI:6.8,12.9) eyes (mean depth: 119±142μm; mean distance:1.59±0.76mm from the foveola), and temporal to the foveola in 26 (7.1%;95%CI:4.5,9.8) eyes (depth:71±46μm; distance:1.20±0.60mm). Higher prevalence of an extrafoveal PP-RPE/BML correlated with longer axial length (OR:1.55;95%CI:1.28,1.89;P<0.001), higher corneal astigmatism (OR:1.78;95%CI:1.14,2.79;P=0.01) and female sex (OR:2.74;95%CI:1.30,5.77;P=0.008). The curvature of the RPE/BM line at the posterior pole was similar to the RPE/BM line curvature outside of the posterior pole in 309 (84.4%;95%CI:80.7,88.2) eyes, and it was steeper (i.e., smaller curvature radius) in 57 (15.6%;95%CI:12.1,19.1) eyes. In these eyes, axial length was longer (24.41±1.78mm versus 27.74±1.88mm;P<0.001). ConclusionsWith longer axial length, the foveola is more often located outside of the geometrical posterior pole. It may be of importance for biometric axial length measurements. An extrafoveal location of the PP-RPE/BML may be due to an axial elongation-associated, meridionally asymmetric enlargement of Bruch´s membrane in the fundus midperiphery.