Curve Research Articles

A safety-guaranteed game-theoretical velocity planning for autonomous vehicles on sharp curve roads

In this paper, a safety-guaranteed game-theoretical velocity planning framework in a hierarchical manner is proposed to generate safe, ride comfort, and travel efficiency-balanced velocity for autonomous vehicles (AVs). In the upper layer, a bang-bang decision-making method is utilized to determine which planning mode to be implemented based on acceleration and jerk constraints, including a comfort mode, an efficiency mode, and a game mode. In the lower layer, asymmetric jerk limits based on comfort characteristics sensibility analysis and safe velocity simultaneously considering longitudinal and lateral stability are firstly developed to maintain ride comfort and driving safety, respectively on curve roads, especially sharp curves where vehicle stability may be not fully considered in most researches. Based on these, a non-cooperative game-theoretical velocity planning method is presented to solve the conflict between comfort mode and efficiency mode by optimizing his own objective based on the other’s action. Finally, for the sake of solving efficiency and accuracy, a chaos optimization-based algorithm (COA) is designed to solve for the Stackelberg equilibrium solution of the bilevel game optimization problem. Three experimental tests are carried out to comprehensively demonstrate the effectiveness, robustness, and real time of the proposed framework. The results show that the proposed method can provide the great performance of ride comfort, travel efficiency, and longitudinal-lateral stability in real time in the velocity planning process.

Sobolev metrics on spaces of manifold valued curves

We study completeness properties of reparametrization invariant Sobolev metrics of order $n\ge 2$ on the space of manifold valued open and closed immersed curves. In particular, for several important cases of metrics, we show that Sobolev immersions are metrically and geodesically complete (thus the geodesic equation is globally well-posed). These results were previously known only for closed curves with values in Euclidean space. For the class constant coefficient Sobolev metrics on open curves, we show that they are metrically incomplete, and that this incompleteness only arises from curves that vanish completely (unlike "local" failures that occur in lower order metrics).

Kink soliton solution of integrable Kairat-X equation via two integration algorithms

<p>In order to establish and assess the dynamics of kink solitons in the integrable Kairat-X equation, which explains the differential geometry of curves and equivalence aspects, the present investigation put forward two variants of a unique transformation-based analytical technique. These modifications were referred to as the generalized ($ r+\frac{G'}{G} $)-expansion method and the simple ($ \frac{G'}{G} $)-expansion approach. The proposed methods spilled over the aimed Kairat-X equation into a nonlinear ordinary differential equation by means of a variable transformation. Immediately following that, it was presumed that the resultant nonlinear ordinary differential equation had a closed form solution, which turned it into a system of algebraic equations. The resultant set of algebraic equations was solved to find new families of soliton solutions which took the forms of hyperbolic, rational and periodic functions. An assortment of contour, 2D and 3D graphs were used to visually show the dynamics of certain generated soliton solutions. This indicated that these soliton solutions likely took the structures of kink solitons prominently. Moreover, our proposed methods demonstrated their use by constructing a multiplicity of soliton solutions, offering significant understanding into the evolution of the focused model, and suggesting possible applications in dealing with related nonlinear phenomena.</p>

Influence of Curve Geometry Factors on Driver’s Speed Decision Making When Passing through a Horizontal Curve

Influence of Curve Geometry Factors on Driver’s Speed Decision Making When Passing through a Horizontal Curve

Impact of smoothing interpolations on the non-conventional machining of pockets

Smoothing curves have been used in computer aided design software for years. This is the reason why machinists very early wondered about the possibilities of creating toolpaths directly generated from this type of mathematical curves. In addition, in recent research work, we have shown that the non-uniform rational Bspline interpolation is the most favorable towards high-speed machining of complex shapes. As a result, we are thinking of simulating by this type of interpolation the pocket recess machining strategies proposed by a computer aided manufacturing software, currently developed in linear and circular interpolations only. In this article we propose a study of the influence of the parameters of Bspline curves on the hollowing out of pockets by smoothing initial cutting methods. To do this, we chose the smoothing of the zigzag method by Bspline and NURBS while varying the order and the discretization parameters to know the impact on the dynamics of the high-speed machining mechanism and on the non-conventional machining precision. This work has shown that parametric curves are favored for this type of machining provided that this tool is mastered.

On the convergence of the Willmore flow with Dirichlet boundary conditions

Very little is yet known regarding the Willmore flow of surfaces with Dirichlet boundary conditions. We consider surfaces with a rotational symmetry as initial data and prove a global existence and convergence result for solutions of the Willmore flow with initial data below an explicit, sharp energy threshold. Strikingly, this threshold depends on the prescribed boundary conditions — it can even be made to be 0. We show sharpness for some critical boundary data by constructing surfaces above this energy threshold so that the corresponding Willmore flow develops a singularity. Finally, a Li–Yau inequality for open curves in H2 is proved.

Assessing driving behavior influence on fuel efficiency using machine-learning and drive-cycle simulations

Consumption of fossil fuel-based energy for vehicle propulsion and associated emissions are a global concern.One pathway to energy reduction is to examine situations where high-energy consumption occurs on roadways, e.g., speed volatility at work zones, or on sharp curves, which has been understudied. Harnessing second-by-second data from the naturalistic driving study and using the concept of driving volatility, this paper explores driving styles in work-zones and curves using machine learning approaches (k-medoids, hierarchical clustering) and drive-cycle simulations from Autonomie®. Results show that aggressive driving account for 12.2 % and 15.4 % of events that occurred in work zones and on curves and leads to 23 % increase in fuel consumption as opposed to normal driving.These results have implications for transportation agencies to improve work-zone configurations and provide signage or technology on curves to reduce fuel consumption and emissions. Moreover, automated and connected vehicles can smooth out traffic flow with advanced advisories and warnings.

Rail Wear Evolution on Small-Radius Curves under Mixed Traffic Conditions, In-Field Investigations

This paper’s aim is to assess the rail wear evolution of railway tracks under mixed traffic conditions for sharp curves. Field investigations were conducted over a period of five years, at 800 points on a rail track, for two curves that underwent maintenance interventions: sleeper and rail replacements during measurements. Lateral and vertical wear measurements were monitored with mechanical instruments, and Prokon and Sap simulations were performed in order to determine the bearing capacity of the rail at different stages of wear. The results obtained after five years of investigation show wear speed propagation on the outer rail of small-radius curves as well as the bending moments and admissible stress variation. At more than 18% of the studied points, lateral wear exceeded the maximum admissible limit, showing a faster propagation speed on the outer rail compared to the vertical wear rate.

The boundary value problem for the two-dimensional Laplace equation with transmission conditions given on open Lipschitz curve

We consider boundary value problem for Laplace equation in bounded two-dimensional Lipschitz domain with transmission boundary condition given upon open curve. This conditions includes itself the jump of solution of boundary value problem and the meaning of boundary value of its normal derivative. We prove the equivalence of considered boundary value problem and obtained variational problem. As a result we prove existence and uniqueness of solution of the posed problems in appropriate functional spaces. Based on the integral representation of the solution the considered boundary value problem is reduced to the system of boundary integral equation which has unique solution.

Cause–effect relationship between model parameters and damping performance of hydraulic shock absorbers

Despite long-term research and development of modern shock absorbers, the effect of variations of several crucial material and model parameters still remains dubious. The goal of this work is therefore a study of the changes of shock absorber dynamics with respect to typical parameter ranges in a realistic model. We study the impact of shim properties, as well as geometric features such as discharge coefficients and bleed orifice cross section. We derive cause–effect relationships by nonlinear parameter fitting of the differential equations of the model and show digressive and progressive quadratic damping curves for shim number and thickness, sharp exponential curves for discharge coefficients, and leakage width, as well as a linear decrease of damping properties with bleed orifice area. Temperature increase affecting material properties, such as density and viscosity of the mineral oil, is found to have a mostly linear relationship with damping and pressure losses. Our results are not only significant for the general understanding of shock absorber dynamics, but also serve as a guidance for the development of specific models by following the proposed methodology.

The Differential Geometry of Curves in B1×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^1\ imes \\mathbb {R}$$\\end{document}

The three-dimensional pseudohermitian manifolds with positive, zero, and negative constant pseudohermitian sectional curvature, respectively, with vanishing pseudohermitian torsion are the Heisenberg group H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_1$$\\end{document}, the CR sphere S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^3$$\\end{document}, and the circle bundle B1×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^1\ imes \\mathbb {R}$$\\end{document}. While the differential geometry of curves in H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_1$$\\end{document} and S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S^3$$\\end{document} has been studied before, there has not been any study on the differential geometry of curves in B1×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^1\ imes \\mathbb {R}$$\\end{document}. In this paper, we take the first step in this direction. Among other things, we find the explicit formulas of the p-curvature and T-variation for curves in B1×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^1\ imes \\mathbb {R}$$\\end{document}. We also prove the fundamental theorem of curves in B1×R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B^1\ imes \\mathbb {R}$$\\end{document}.

A nonlinear contraction channel design inspired by typical mathematical curves: Boosting net power and water discharge of PEM fuel cells

The next-generation fuel cell needs to fit high current density operating conditions, necessitating improved flow fields with superior capabilities in mass transfer and drainage. This study introduces four innovative nonlinear contraction channels aimed at enhancing net power and mitigating waterflooding under high current density. This design draws inspiration from classical mathematical curves, which can provide targeted mass transfer enhancement contingent upon species distribution. Multiple quantitative indicators are employed to evaluate the efficacy of nonlinear structures in cross-scale mass transfer, drainage, temperature uniformity, and power consumption. Numerical results indicate that the exponential-like curve channel exhibits notably superior overall performance. Its nonlinear contraction region induces purge airflow at the tail of channel, facilitating drainage and boosting the local mass transfer flux in oxygen-starved region by 48.75%. The utilization of nonlinear contraction structures alters the diffusion-convection distribution mechanism and increases the convective mass transfer proportion. Under conditions of high current density, there is a noticeable rise in the Peclet number. Considering the pumping power loss, the hyperbolic tangent, quarter-ellipse, parabolic, and exponential-like curve flow channels can enhance the net power by 3.10%, 6.82%, 8.57%, and 10.04% over straight channels. Additionally, the exponential-like curve channel enhances temperature uniformity in the catalyst layer under rated working conditions, whereas the hyperbolic tangent flow channel demonstrates superior uniformity in higher current density.

Mathematical Models for Predicting the Elastic and Thermal Behavior of Heterogeneous Materials through Curve Fitting

The aim of this study is to establish a new method for predicting the effective elastic and thermal behavior of heterogeneous materials through mathematical curve fitting. The research initiates with the collection of 3D microstructures comprising 200 spherical inclusions embedded within a linear elastic matrix. This database is acquired by varying the volume fraction of the inclusions (10%, 15%, 20%, or 25%) and the contrast between the two phases. The contrast is quantified in terms of the ratio of the Young’s modulus and thermal conductivity modulus, EiEm and λiλm, with values ranging from 10 to 200. These microstructures are then used to estimate the elastic and thermal properties by calculating the effective bulk, shear, and thermal conductivity moduli via the finite element method (FEM). The compiled database is a crucial asset for the development of mathematical equations that precisely fit the curves of various moduli based on changes to input parameters such as the volume fraction and the contrast. The process includes analyzing the data, identifying patterns, and establishing mathematical relationships that effectively reflect the moduli’s observed behavior. By integrating these individual equations and taking into account their interdependencies, the resultant comprehensive 3D model provides an extensive representation of the material’s behavior and takes into consideration the impacts of varying the volume fraction and contrast on the different moduli. This approach enables a better understanding of the material’s response under diverse conditions. Results exhibit the accuracy and reliability of the chosen mapping functions and parameters. Thereby, these proposed functions, with respect to boundary conditions and analytical limits, confirm the relevance of the proposed model to capture such information with a suitable level of precision.

Twisted curve geometry underlying topological invariants

Topological invariants such as winding numbers and linking numbers appear as charges of topological solitons in diverse nonlinear physical systems described by a unit vector field defined on two and three dimensional manifolds. While the Gauss-Bonnet theorem shows that the Euler characteristic (a topological invariant) can be written as the integral of the Gaussian curvature (an intrinsic geometric quantity), the intriguing question of whether winding and linking numbers can also be expressed as integrals of some other kinds of intrinsic geometric quantities has not been addressed in the literature. In this paper we provide the answer by showing that for the winding number in two dimensions, these geometric quantities involve torsions of the two evolving space curves describing the manifold. On the other hand, in three dimensions we find that in addition to torsions, intrinsic twists of the space curves are necessary to obtain a nontrivial winding number and linking number. They arise from the hitherto unknown connections that we establish between these topological invariants and the corresponding appropriately normalized global space curve anholonomies (i.e., geometric phases) that can be associated with the unit vector fields on the respective manifolds. An application of our results to a 3D Heisenberg ferromagnetic model supporting a topological soliton is also presented.

Assessment of the Effect of Garcinia kola (Clusiaceae) on the Carbon Sequestration Capacity and Productivity of Cocoa (Theobroma cacao L) in Cocoa-Based Agroforestry Systems

Aims: Ivorian cocoa production reaching 2 million ton annually at the expense of the forest reserve, and an annual deforestation rate around 2.78%. The integration of agroforestry into cocoa plantations appears to be a suitable approach to combating deforestation of the remaining forest reserves and restoring the degraded cocoa landscape. The choice of species is the key to the success of this program. Our objective was to evaluate the interaction between Garcinia kola and cocoa in traditional agroforestry systems. Study Design: Around each Garcinia Kola tree identified in the orchard, two curves lines with radii of 10 and 20 m respectively were delimited in order to assess the interaction of Garcinia Kola on the cocoa tree according to their proximity to Garcinia Kola. Place and Duration of Study: The study was conducted at Gueyo and Buyo in cocoa-based agroforestry systems in western Côte d'Ivoire. from June 2020 to November 2021. Methodology: The influence of Garcinia Kola on cocoa trees was assessed by measuring the biomass and carbon sequestration rate of Garcinia kola and cocoa trees. The productivity of cocoa trees and Garcinia kola was also determined. Results: Our results showed that Garcinia kola trees have an influence on the density and yield of cocoa trees. Cocoa trees close to Garcinia Kola trees within a 10 m radius showed low density and low pod production compared to those within a 20 m radius. Meanwhile, cocoa biomass and carbon sequestration levels were higher in the 10 m radius than the 20 m radius. This study has shown that associated cocoa and Garcinia kola stands make different contributions depending on their proximity. Conclusion: This complementarity in terms of carbon sequestration capacity and productivity between Garcinia Kola and cocoa is an asset that can be exploited for Cocoa agroforestry system to improve the sustainable production system and restore the cocoa landscape.

Deep Open-Curve Snake for Discriminative 3D Neuron Tracking.

Open-Curve Snake (OCS) has been successfully used in three-dimensional tracking of neurites. However, it is limited when dealing with noise-contaminated weak filament signals in real-world applications. In addition, its tracking results are highly sensitive to initial seeds and depend only on image gradient-derived forces. To address these issues and boost the canonical OCS tracker to a new level of learnable deep learning algorithms, we present Deep Open-Curve Snake (DOCS), a novel discriminative 3D neuron tracking framework that simultaneously learns a 3D distance-regression discriminator and a 3D deeply-learned tracker under the energy minimization, which can promote each other. In particular, the open curve tracking process in DOCS is formed as convolutional neural network prediction procedures of new deformation fields, stretching directions, and local radii and iteratively updated by minimizing a tractable energy function containing fitting forces and curve length. By sharing the same deep learning architectures in an end-to-end trainable framework, DOCS is able to fully grasp the information available in the volumetric neuronal data to address segmentation, tracing, and reconstruction of complete neuron structures in the wild. We demonstrated the superiority of DOCS by evaluating it on both the BigNeuron and Diadem datasets where consistently state-of-the-art performances were achieved for comparison against current neuron tracing and tracking approaches. Our method improves the average overlap score and distance score about 1.7% and 17% in the BigNeuron challenge data set, respectively, and the average overlap score about 4.1% in the Diadem dataset.

Geometry of relativistic nuclei collision and hurst correlation curves

Geometry of relativistic nuclei collision and hurst correlation curves

Applications of Differential Geometry of Curves in Roads Design

Abstract An important role in the roads design is played by plane curves, if the road lies on a plane surface and by space curves, if the road can not be embeded in a plane. This article presents some work done in the two and three dimensional roads design, by using the differential geometry methods. A study of a general case for plane roads is given.

Flanging, squeal and corrugation-induced noise: Insights from recent measurement campaigns at interconnected tight curves

High noise events at tight curves often comprise of squeal, flanging and corrugation-induced noise. All noise types have the potential to cause annoyance at sensitive receivers in close proximity to the rail line. The presence of squeal, flanging and other high noise events at three interconnected sharp passenger curves on the metropolitan network in Sydney, New South Wales is investigated in the context of measured parameters. Rail and noise parameters were measured at ten locations in 2021, with a repeat of the measurements undertaken at three locations in 2022. Differences in high noise events were analysed and quantified to understand the underlying causes of curving noise in this area. The role of the wheel-rail interface contact conditions, train type, rail roughness, friction and speed is investigated to provide insights into the mechanisms leading to curving noise events at this site.

Technical note: The Graetz problem at small Peclet number and its application to sizing condensation particle counters

We extend to small Peclet number Pe the familiar high-Pe solution to the Graetz problem of heat/mass transfer in a tube with parabolic flow with a wall temperature discontinuity at an axial position z = 0. Retaining diffusion in the axial direction results only in slight shifts in the eigenfunctions describing the problem. This enables the characterization of the saturation field S by separation of variables, which we implement for Pe = 0, 0.1, 0.25. Calculations based on discretization of the transport partial differential equations are also used to analyze the problem at Pe up to 5. As Pe tends to 0, the heat and mass transfer problems are both described by the same dimensionless function θ(z,r) of the axial and radial coordinates z and r. S then depends on the single quantity θ, and its maximum value Smax is exactly the same along all streamlines. This would result in sharp activation curves in a hypothetical low-Pe condensation particle counter, similarly as in the special circumstance when the ratio Le between heat and mass diffusivities is unity. Calculations are run to determine how the radial uniformity of Smax is degraded at increasing Pe. When accounting for axial diffusion, Smax at the wall is independent of Pe or Le. The commonly considered high Pe limit fails to satisfy this requirement, and therefore predicts an incorrect Smax at distances from the wall discontinuity small compared to Pe−1/2.