Vertical Curvature Research Articles

Ways of increasing the efficiency of physical exercises of teenage school students with impaired health

The work is devoted to the definition of anatomical and physiological features of the formation of the movement culture of schoolchildren. The relevance of the issue of increasing the effectiveness of physical exercise classes for primary school girls with postural disorders is substantiated and is the formation of a culture of movements. It has been established that postural disorders are a prerequisite for the appearance of accompanying diseases and abnormalities in physical development and negatively affect the culture of movements, distorting the physique, plasticity of movements, gait, impair motor achievements during physical exercises, lead to a decrease in motivation for motor activity, negatively affect well-being children, lead to the emergence of psychological complexes and complicate socialization in the student body. According to the index method, which included the determination of the shoulder index, which characterizes the level of stooping (kyphotic posture), and the index of vertical curvature of the spine, which indicates the presence of scoliotic posture, it was found that 26.3% of schoolgirls have various posture disorders, while 82 of them .8% of girls have slouching (kyphotic posture), the other 17.2% of schoolgirls have scoliotic posture in various variants of vertical curvature of the spine. Since the level of some motor indicators indirectly determines the culture of girls' movements and affects the formation of a posture stereotype, pedagogical testing of various types of strength, coordination abilities and flexibility of elementary school students was conducted. At the same time, a significant difference was recorded between the motor development of schoolgirls with various postural disorders in the indicators of strength endurance of the muscles of the abdomen, back, flexibility, plasticity and static balance of healthy girls and schoolgirls with postural disorders. The conducted linear correlation analysis using the Pearson method between posture indicators and movement indicators showed a reliable relationship of moderate strength between the posture indicator and some movement indicators, which justifies the need to develop a methodology for teaching the movement culture of elementary school students with posture disorders in the process of physical education.

Distribution and correlation of refractive parameters in children with different corneal curvatures in southeast China.

To analyze the distribution of refractive status in school-age children with different corneal curvatures (CC) and the correlation between CC and refractive status. A total of 2214 school-aged children of grade 4 in Hangzhou who were screened for school myopia were included. Uncorrected distance visual acuity (UCDVA), non-cycloplegic refraction, axial length (AL), horizontal and vertical corneal curvature (K1, K2) were measured and spherical equivalent (SE), corneal curvature radius (CCR) and axial length/corneal radius of curvature ratio (AL/CR) were calculated. UCDVA<5.0 and SE≤-0.50 D were classified as school-screening myopia. According to the different CCRs, the patients were divided into the lower corneal curvature (LCC) group (CCR≥7.92) and the higher corneal curvature (HCC) group (CCR<7.92). Each group was further divided into the normal AL subgroup and the long AL subgroup. The refractive parameters were compared to identify any differences between the two groups. Both SE and AL were greater in the LCC group (P=0.013, P<0.001). The prevalence of myopia was 38% in the LCC group and 44% in the HCC group (P<0.001). The proportion of children without screening myopia was higher in the LCC group (62%) than in the HCC group (56%). Among these children without screening myopia, the proportion of long AL in the LCC group (24%) was significantly higher than that in the HCC group (0.012%; P<0.001). The change of SE in the LCC group was less affected by the increase of AL than that in the HCC group. School-aged children in the LCC group have a lower incidence of screening myopia and longer AL. Low CC can mask SE reduction and AL growth to some extent, and the change of AL growth change more in children with low CC than high CC. Before the onset of myopia, its growth rate is even faster than that after the onset of myopia.

Corneal Rigid Gas Permeable Contact Lens in Corneal Scarring: a Case Report

Introduction : Corneal scarring is part of corneal wound healing response to injury, surgery, or infection to the cornea. Corneal scarring causes significant visual reduction. This case report highlights visual acuity (VA) improvement in patient with corneal scarring following rigid glass permeable contact lens (RGP-CL)&#x0D; Case Illustration : A 25 year-old male presented blurry vision in right eye (RE) due to trauma. His VA was 3/60 on RE and 6/6 on left eye (LE). The best corrected visual acuity (BCVA) of RE was 6/45 (S-1.25 C-5.00 X 180°). Slit lamp examination of RE showed corneal scarring in central area and deep neovascularization. Keratometry readings obtained 39.50D (8.48mm) at 170 degree and vertical curvature of 44.00D (7.68mm) at 80 degree, with markedly irregular mires of RE. A trial fitting was performed for RE by using tisilfocon A lens with base curve of 8.30 mm, power S-4.00D and diameter 9.20 mm. The over-refraction with trial lens obtained a total refractive error of S-6.00 with cylindrical correction neutralize with RGP. The achieved BCVA by over-refraction was 6/45 with comfortable fit.&#x0D; Discussion : Rigid permeable lens (RGP) is an option in dealing with irregular astigmatism due to corneal scarring. The RGP fitting works best especially when the corneal scarring is not in the center of visual pathway. Rigid lens works by providing tear-lens power and creating smooth surface to neutralize the irregular cornea. Good RGP-CL fitting could help patient with corneal carring gain better VA.&#x0D; Conclusion : Good rigid permeable contact lens fitting could help patients with corneal scarring gain better VA.

MaxEnt machine learning model predicts high groundwater potential areas in a fractured volcanic aquifer system

Groundwater constitutes a vital resource for public water supply, and thus, it is imperative to recognize the areas of highest potential for increasing availability. The present study employs the MaxEnt model to discern the most favorable areas for locating high-yield wells in Caxias do Sul, Rio Grande do Sul, southern Brazil, where the Serra Geral Aquifer System, a fractured volcanic aquifer, emerges. This aquifer system is characterized by its heterogeneous, discontinuous, and highly anisotropic nature. A dataset comprising 83 wells with high flow rates (≥10 m³/h) was selected from the municipal registry of deep tubular wells, along with 14 factors that influence groundwater occurrence (specific capacity, transmissivity, altitude, slope, horizontal curvature, vertical curvature, relief dissection index, drainage density, distance to drainage, topographic wetness index, distance to lineament, lineament density, precipitation, and soil hydrological group). The model output was a Groundwater Potential Map, which stochastically expresses the probability of obtaining flow rates ≥10 m³/h. The map was validated through cross-validation, resulting in an average accuracy of 65.14%, and by the Receiver Operating Characteristic analysis, resulting in an Area Under the Curve value of 0.911, indicating satisfactory validation. While the MaxEnt model is widely used in ecology to model species distribution, its application in groundwater prediction remains limited, particularly in fractured aquifers associated with volcanic rocks. Apart from optimizing the use of groundwater resources, this study also enhances the understanding of natural phenomena in this type of aquifer.

Larsemann Hills: Geomorphometric modeling and mapping

Geomorphometric modeling is widely used in geosciences. However, geomorphometric modeling and mapping of Antarctic oases has not been performed so far. This article presents the first results of our work on geomorphometric modeling and mapping of the Larsemann Hills obtained in the frameworks of the 68th Russian Antarctic Expedition in January–April 2023. As input data, we used a fragment of the Reference Elevation Model of Antarctica (REMA). From the extracted and edited digital elevation model, we derived digital models and maps of the following 17 morphometric variables: slope, aspect, horizontal curvature, vertical curvature, mean curvature, Gaussian curvature, minimal curvature, maximal curvature, unsphericity curvature, difference curvature, vertical excess curvature, horizontal excess curvature, ring curvature, accumulation curvature, catchment area, topographic index, and stream power index. We also conducted a field geomorphometric interpretation work to provide correct physical geographic, geological, and geomorphological interpretations of morphometric maps. In the fieldwork, we carried out 54 foot routes with the total length of about 422 km. During the routes, we collected 150 rock samples for further petrological and mineralogical analyses as well as three-dimensional modeling of the samples. Derived morphometric maps can be useful for structural geological and process-oriented hydrological studies. The ultimate goal of the ongoing work is to create a large-scale geomorphometric atlas of Antarctic oases and other ice-free Antarctic territories.

Effects of geometric attributes of horizontal and sag vertical curve combinations on freeway crash frequency

The geometric design of the combinations of horizontal and sag vertical curves (sag combinations or sag combined curves) is vital to road safety. However, there is little research that investigates the safety effects of their geometric attributes based on the analysis of real-world crash data. To this end, the crash, traffic, geometric design, and roadway configuration data are collected from 157 sag combinations on six freeways in Washington State, during 2011–2017. Poisson, negative binomial (NB), hierarchical Poisson, and hierarchical NB models are developed for analyzing the crash frequency of sag combinations. The models are estimated and compared in the context of Bayesian inference. The results indicate that significant over-dispersion and cross-group heterogeneity exist in the crash data and that the hierarchical NB model yields the best overall performance. The parameter estimates show that: five geometric attributes, including horizontal curvature, vertical curvature, departure grade, the ratio of horizontal curvature to vertical curvature, and the layout of front dislocation, have significant effects on the crash frequency of sag combinations. Freeway section length, annual average daily traffic, and speed limits are also important predictors of crash frequency. The analysis results and the proposed model are useful for evaluating the safety performance of freeway sag combinations and optimizing their geometric design based on substantive safety evaluation.

Inequalities on Riemannian Warped Product Submersions for Vertical Casorati Curvatures

Inequalities on Riemannian Warped Product Submersions for Vertical Casorati Curvatures

Геоморфометрическое моделирование и картографирование антарктического оазиса Холмы Ларсеманн

Geomorphometric modeling is widely used in geosciences. However, geomorphometric modeling and mapping of Antarctic oases has not been performed so far. This article presents the first results of geomorphometric modeling and mapping of the Larsemann Hills oasis obtained in the 68th Russian Antarctic Expedition in January–April 2023. As input data, we used a fragment of the Reference Elevation Model of Antarctica (REMA). From the extracted and edited digital elevation model, we derived models and maps of the following 17 morphometric variables: slope, aspect, horizontal curvature, vertical curvature, mean curvature, Gaussian curvature, minimal curvature, maximal curvature, unsphericity curvature, difference curvature, vertical excess curvature, horizontal excess curvature, ring curvature, accumulation curvature, catchment area, topographic index, and stream power index. We conducted a field geomorphometric interpretation to provide with correct physical geographic, geological, and geomorphological interpretations of morphometric maps. In this fieldwork, we carried out 54 foot routes with the total length of about 422 km. During the routes, we collected 150 rock samples for further petrological and mineralogical analyses and three-dimensional modeling of the samples. Morphometric maps can be useful for geological, geomorphological, soil, biological, and hydrological studies. The ultimate goal of the author’s multiyear project is to create a digital geomorphometric atlas of Antarctic oases and other ice-free Antarctic territories.

Posture correction in children 11-12 years old by means of swimming with elements of aquafitness

The article is devoted to the justification of the use of a physical therapy program using swimming equipment to improve the physical condition of children aged 11-12 years with postural disorders. The relevance of the study is due to the critical sharp increase in the incidence of disease among children all over the world, in particular, in Ukraine in recent years. The evaluation method was the measurement of Moshkov's rhombus, acromial, digital, iliac crest and lower angles of the scapula of the point. According to the results of the study, it was found that among boys aged 11-12 years under the influence of classes under the rehabilitation program using swimming equipment with elements of aquafitness, the number of people with an index of vertical curvature of the spine of 100.0-90.0% increased by 30.77%, among girls - by 38.46%. At the same time, the number of persons with a spine curvature index of 79.9-70.0% in the main groups of boys and girls decreased significantly compared to the control groups. The feasibility of using the proposed program of complex rehabilitation with the use of swimming equipment with the use of elements of aquafitness was proven due to a significant improvement in the indicators of posture assessment in the studied children 11-12 years old.

The position and morphology characteristics of multiple impacted anterior teeth in the unilateral maxillary area: A retrospective study based on cone-beam computed tomography

This study investigated the position and morphologic characteristics of multiple impacted anterior teeth in the unilateral maxillary area. Cone-beam computed tomography images of 21 patients (11 males and 10 females; median age 9.42 years [9.08-11.29]) with multiple teeth impacted were collected and imported into Dolphin Imaging software (Dolphin Imaging and Management Solutions, Chatsworth, Calif). The vertical height, crown orientation, twist direction, and root curvature of each impacted tooth were described. The crown length, root length, and root canal width of impacted and homonym teeth were measured. The positions of the impacted lateral incisors are lower than that of the other 2 anterior teeth. Most crowns of impacted central incisors are positioned distally, labial surfaces in mesial torsion, with most roots, bent toward the distal and labial. Crowns of impacted lateral incisors are positioned mesiolabially, with labial surfaces mostly in distal torsion, and most roots bent toward the mesial and labial. Crowns of impacted canines are mostly positioned mesiolabially, with labial surfaces in mesial torsion. The crowns and roots of the impacted central and lateral incisors were shorter than those of the homonym (P<0.05); however, the difference in crown length is clinically negligible, and there was no difference in root canal widths. There was no difference in the comparison of parameters for the canine group. There are certain rules in the vertical height, crown orientation, twist direction, and root curvature of multiple impacted anterior teeth in the unilateral maxillary area. Root development of impacted central and lateral incisors was restricted.

An integrated remote sensing and biomarker maturity parameter-based approach to trace petroleum migration in a complex fault zone lineament of the Brazilian Equatorial Margin

The present paper investigates the oil migration and lineament distribution in fault zones within a sedimentary basin, showcasing the results from an integrated use of geochemical, and remote sensing data along the Fazenda Belém Fault Zone (FBFZ) in the Potiguar Basin, Brazilian Equatorial Margin. Lineaments were detected from a LiDAR digital elevation model data set using the quantitative method of morphology recognition based on horizontal and vertical curvature calculation and mapping. The polynomial regression-based spatial distribution of 20S/(20S + 20R) C29 steranes is successfully used in oils from offshore and onshore reservoirs of the Potiguar Basin, tracing field-scale oil migration direction. These multidisciplinary analyses revealed that the oil initially migrated from an offshore area along the NE-oriented fault zone, and then parts moved along the branches of this zone to finally reach the areas of the onshore reservoirs. In addition, NE-oriented lineaments enhanced the near-surface fluid circulation, while they commonly correspond to the topographic features and fault zones. Anisotropy tensors and slip- and dilation tendency calculated from the lineament map also indicate that the geometric complexity of the lineament pattern is influenced by present-day and partitioned stress fields in the Potiguar Basin. The integration of remote-sensing and geochemistry analyses at the FBFZ area proved to be effective in approaching the tectonic aspects of a fault zone in a petroleum basin, as well as in investigating the field-scale migration of oil along the fracture corridor in a fault zone.

New formulation of the two-dimensional steep-slope shallow water equations. Part I: Theory and analysis

Two-dimensional (2D) depth-averaged shallow water equations (SWE) are widely used to model unsteady free surface flows, such as flooding processes, including those due to dam-break or levee breach. However, the basic hypothesis of small bottom slopes may be far from satisfied in certain practical circumstances, both locally at geometric singularities and even in wide portions of the floodable area, such as in mountain regions. In these cases, the classic 2D SWE might provide inaccurate results, and the steep-slope shallow water equations (SSSWE), in which the restriction of small bottom slopes is relaxed, are a valid alternative modeling option. However, different 2D formulations of this set of equations can be found in the geophysical flow literature, in both global horizontally-oriented and local bottom-oriented coordinate systems. In this paper, a new SSSWE model is presented in which water depth is defined along the vertical direction and flow velocity is assumed parallel to the bottom surface. This choice of the dependent variables combines the advantages of considering the flow velocity parallel to the bottom, as can be expected in gradually varied shallow flow, and handling vertical water depths consistent with elevation data, usually available as digital terrain models. The pressure distribution is assumed linear along the vertical direction and flow curvature effects are neglected. A new formulation of the 2D depth-averaged SSSWE is derived, in which the two dynamic equations represent momentum balances along two spatial directions parallel to the bottom, whose horizontal projections are parallel to two fixed orthogonal coordinate directions. The analysis of the mathematical properties of the new SSSWE equations shows that they are strictly hyperbolic for wet bed conditions and reduce to the conventional 2D SWE when bottom slopes are small. Finally, it is shown that the SSSWE predict a slower flow compared with the conventional SWE in the theoretical case of a 1D dam-break on a frictionless channel with fixed slope. The capabilities of the proposed model are demonstrated in a companion paper on the basis of numerical and experimental tests.

Numerical Optimization of Gravitational Water Vortex Turbine using Computational Flow Analysis

Gravitational Water Vortex Power (GWVP) plants are considered one of the most favourable micro-hydro extraction methods suitable for rural communities in mountain areas. Many researchers have focused on enhancing the performance of GWVP plants since it is less efficient than other commercial hydro turbines. This study mainly focuses on optimization of vortex turbine situated in conical vortex basin using the computational fluid dynamic (CFD) analysis. The basic turbine blade design was modified with inclination, height, vertical twist, and horizontal curvature. CFD results were analysed at each step concerning air-water interface, pressure, and velocity while identifying the relationship of flow behaviour with performance. A similar blade inclination with the conical basin enhanced performance, and the increasing of blade height also showed good performance. Vertical twist and horizontal curvature were investigated separately, and the horizontal curvature blade showed slightly better performance than the vertical twisted blade recording the highest efficiency as 40.4%.

Continuous Extraction and Continuous Backfill Mining Method Using Carbon Dioxide Mineralized Filling Body to Preserve Shallow Water in Northwest China

The exploitation and utilization of coal resources are not only prone to causing water table lowering, but also produce a large amount of CO2 and coal-based solid waste. A scientific concept that employs the CO2 and solid wastes to develop filling bodies and inject them into the mined-out area, to sequestrate CO2 and mitigate the overburden migration and thus preserve the overlying aquifer, is proposed. Continuous extraction and continuous backfill (CECB) mining was selected as the mining method to meet the aforementioned objectives. Additionally, carbon dioxide mineralized filling body (CMFB) under ambient temperature and pressure was developed, with fly ash as aggregate, and CO2 gas, silicate additives and cement as accessories. The uniaxial compressive strength (UCS) and tensile strength of CMFB with various curing times and fly ash contents were tested indoors. A physical analogue simulation and FLAC3D numerical calculation were then successively implemented on the premise of determining a similar material ratio of CMFB in analogue simulation and calibrating the parameters of the CMFB in numerical simulation. The deformation of aquifuge and water level lowering while using CECB and CMFB with various proportion of fly ash were obtained. When using the CMFB with 75% fly ash content and 28 d curing time, the maximum values of vertical displacement, horizontal displacement, inclination, horizontal deformation and curvature of aquiclude were 26 mm, 6.5 mm, 0.12 mm/m, 0.08 mm/m and 0.0015 mm/m2, respectively, and the water table decreased 0.47 m. The results show that the CMFB with 75% fly ash is the most appropriate ratio to realize water preservation mining, CO2 sequestration and harmless treatment of solid wastes, contributing to the green and sustainable development of coal areas.

Vertical Clearance Assessment of Overpassing Structures on Sag Vertical Curves

The paper investigates areas on sag vertical alignments where overpassing structures impose restrictions on truck drivers’ sight distance. The assessment is performed for the least desired case where the driver’s sightline is equivalent to the defined stopping sight distance (SSD). The overall objective is to define the boundaries where such overpassing structures limit the available sight distances for truck drivers below the required SSD and thus to provide valuable guidance to practitioners. Initially, the authors addressed the SSD calculation on variable grades during the braking process through an earlier approach based on the point mass model and the laws of mechanics. As a second step, truck deceleration rates were determined from the literature. The effect of various design speeds was examined, corresponding to specific ranges of grade values and sag vertical curvature rates. The methodology was based on calculating the difference between the truck driver’s line of sight and the road alignment at fixed locations and associating the defined value with the desired vertical clearance rate (VC). The analysis revealed that critical cases arise for design speed values between 90 km/h and 110 km/h. The examined cases for Vdesign = 100 km/h and Vdesign = 110 km/h revealed more critical concerns since the assessment showed that even a VC value of 4.90 m is not adequate to address SSD provision. Such cases require further attention, especially when the entering downgrade to the sag vertical curve is long and steep.

Exploring and exploiting ant colony optimization algorithm for vertical highway alignment development

AbstractThe vertical alignment optimization is about developing a minimum cost curvilinear vertical profile of constrained grade sections and appropriate non‐overlapping vertical curves passing through fixed control points with elevation constraints. Variations in ground profile and discreteness in unit cutting and filling costs make it a non‐convex, noisy, constrained optimization problem with many local minima. Further, the gradient related constraints and vertical curvature are non‐linear. This paper presents an innovative exploring and exploiting ant colony optimization (E&amp;E‐ACO) algorithm with an appropriate point sampling, vertical curve fitting strategies, and an intuitive feasible region identification approach for solving the vertical alignment optimization problem. The E&amp;E‐ACO algorithm extensively explores the feasible search space to generate a set of potential solutions and effectively exploit the space around the potential solutions for developing the optimized vertical alignment. The efficacy of the proposed method is demonstrated using two case studies. In one case study, the optimized solution by the proposed method had a marginally better objective function value and about three times lesser computational time than the solution by the mesh adaptive direct search method. The optimized alignment satisfied the elevation constraints of fixed control points and imitated the manually designed real‐world vertical alignment. The linearly varying exploration and exploitation parameters had better convergence rate than the other tested variations. Further, the proposed method at the end of 1000 iterations yielded about six times better result than the traditional ACO algorithm.

Accuracy and predictability of novel MS Excel-based surgically induced astigmatism calculator

Study of steepening, flattening, clockwise, and counter-clockwise torque effect is indispensable to understand and design surgical induced astigmatism calculator. Hence, in this study by constructing a novel Microsoft Office Excel 2007 based astigmatic calculator following cataract surgery, analysis on the accuracy and predictability evaluated for the performance. Post-cataract surgery patients from May 2019 to January 2020 at a tertiary medical institution recruited for this present study. Based on Pythagoras principle, MS Excel calculator designed and the law of cousins for calculating the vector magnitude and axis respectively. Manual keratometry measurements for pre and postoperative horizontal (Kh) and vertical (Kv) curvatures established, and statistical analysis for the resultant SIA magnitude and axis deduced with Medcalc software comparing with the existing SIA 2.1 version calculator. A total of 29 eyes of 25 patients studied with a mean age of 62.55 (±8.08) years, males contributing to 14 (56%), and right laterally in 17 (58%) eyes. MS Excel and SIA 2.1 versions calculated a mean SIA magnitude of 0.66 (±0.47) D and 0.64 (±0.55) D respectively. Pearson coefficient correlation (r=-0.16, p=0.40), paired-two sample test (t value= 0.11, p= 0.91) and ROC curve analysis (AUC = 0.75, p= 0.34, 95% CI= 0.25 to 0.99) calculated. Regression equation (y = 0.75 + -0.14 x) and limits of agreements (95% CI -0.29 to 0.31) analyzed, and, 95% of data points distributed within ±1.96 SD of the line of equality on Bland-Altman difference plots.The present calculator proclaimed an acceptable accuracy and agreement with a prediction of 0.61 Diopter for every unit change in the magnitude of SIA 2.1 software in addition to consideration of interchangeability.

The relationship between choroidal thickness and axis length and corneal curvature in high myopia.

Objectives:To measure choroidal thickness (CT) in highly myopia eyes using enhanced depth imaging spectral-domain optical tomography (EDI SD-OCT) and explore the relationship between the CT and axial length (AL) and corneal curvature.Methods:This study assessed 66 eyes of 33 patients with both eyes of high myopia (equivalent sphericity ≥-6D or AL≥26mm) who underwent treatment at the department of ophthalmology, fourth hospital of Hebei Medical University from August 2020 to August 2021(High-myopia group). The control group included 54 eyes from 27 healthy volunteers. The patients in the two groups were 18~40 years old. EDI SD-OCT was used to measure the CT of subfoveal choroidal thickness (SFCT) and CT from fovea to nasal and temporal sides at an interval of 500~2500μm. The intraocular pressure, diopter, ocular axis, corneal curvature, and CT were compared between the two groups. The correlation between CT, diopter and AL was measured by Pearson’s r.Results:The diopter, AL, vertical corneal curvature, and CT in the high-myopia group were significantly higher than those in the control group (P<0.05). There was significant correlation between SFCT and AL in both groups (P<0.05). There was a negative correlation between AL and CT in high-myopia group (r=-0.395, P<0.05), and a positive correlation between AL and CT in control group (r=0.29, P<0.05). There was a weak correlation between AL and gender, intraocular pressure, and horizontal corneal curvature(P<0.05), and a negative correlation between AL and diopter (r=-0.861, P<0.001).Conclusion:EDI SD-OCT can quantitatively measure the CT of high myopia. The CT of patients with high myopia was significantly thinner than that of the control group of the same age. There was a significant correlation between diopter, AL and CT, suggesting that AL is a parameter indicating the degree of myopia, and the change of CT may play a role in the occurrence and development of high myopia.

Lower Bounds on Statistical Submersions with vertical Casorati curvatures

In this paper, we obtain lower bounds for the normalized scalar curvature on statistical submersion with the normalized [Formula: see text]-vertical Casorati curvatures. Also, we discuss the conditions for which the equality cases hold. Beside this, we determine the statistical solitons on statistical submersion from statistical manifolds and illustrate an example of statistical submersions from statistical manifolds.

Torsion and vertical curvature of motion-trajectory curves

The paper discusses difference between super-elevation of recorded motion-trajectory (MT) curves and constant super-elevation of curve plane and demonstrates that vertical curvature used in railroad literature is not necessarily associated with curve out-of-plane bending or twist. Vertical curvature can be highly nonlinear while the curve remains planar and untwisted. A computational procedure is proposed for identifying MT planar curves using computer-simulation or experimentally recorded data. Data-driven science (DDS) approach allows measuring Frenet vertical-development and super-elevation that define centrifugal inertia force. Difference between motion-independent curve-plane and motion-dependent osculating plane (OP) which contains velocity, acceleration, and inertia centrifugal forces is explained. Three Frenet angles: horizontal-curvature, vertical-development, and bank angles; are used to define planar MT geometry. Zero-torsion and zero-vertical-curvature conditions are derived for identifying planar curves and demonstrating that vertical curvature is not always associated with curve torsion. An orthogonal transformation is used to define OP sweeping angle whose derivative defines curve curvature regardless of curve-plane orientation. Difference between Frenet horizontal-curvature angle and OP sweeping angle is discussed. It is shown that super-elevation of planar-curve surface is equal to Frenet super-elevation if Frenet vertical-development angle is zero. An analytical 3D, yet planar and untwisted, curve is used to demonstrate that planar curves can have non-vanishing and nonlinear curvature, vertical-elevation, vertical curvature, and super-elevation.