Main Boundary Research Articles

Morphotectonic assessment of the Gaula river basin, Kumaun lesser Himalaya, Uttarakhand

Morphotectonics is a discipline of geomorphology that explores how landforms are formed or modified by tectonic action. In the present study we uthe sed application of geospatial technology to evaluate the active deformation owing to Main Boundary Thrust (MBT) and Ramgarh Thrust (RT) in Outer part of Kumaun Lesser Himalaya. Active deformation inside river valleys is ascribed downcuttinging and incision of bedrock, offset of river channel, lateral migration of the river, and development of Knickpoints. Therefore, we used quantitative geo-mathematical analysis of The Gaula river basin using ASTER DEM along with survey of India Toposheet (1:50,000) supplemented by field observations. Four morphotectonic metrics, including the basin elongation ratio, the asymmetry factor, the channel sinuosity, and the hypsometric integral, were derived to examine the relative index of active tectonics (RIAT). We applied Stream Power Incision Model (SPIM) by analysing the steepness index (Ksn) using the stream power law. Additionally, a comprehensive evaluation of tributary streams of the Gaula watershed has been carried out to understand relative tectonic activity. Further we used Chi integral (χ) to measure transience dynamics and geometric disequilibrium of Gaula basin. We conclude that most sub-basins are tectonically enhanced, structurally organized and elongated in nature. The landform in the study area is largely controlled by presence of MBT, RT and subsidiary thrusts spread over in the vicinity. Based on our activity classes we concluded that approximately 31% of the area in Gaula river basin falls in very active whereas ∼34% is potentially active. Furthermore, we suggest that the results will be very useful for estimation of hazard potential zones in sub-Himalayan region of foothill zone of Kumaun Himalaya.

Quaternary stratigraphy of Poland – current status

A critical verification of the previous stratigraphic Quaternary subdivisions has updated the setting of the stratigraphic units in Poland. Inconsequently applied classification and terminology in the Polish Quaternary stratigraphy has been accompanied by arbitrary correlation with marine isotope stages. This has resulted in the creation of several stratigraphic units, occasionally with ambiguous stratigraphic setting and chronology, and usually devoid of the type sections. A record of most of the Early and Middle Pleistocene is full of sedimentary hiatuses. The detailed stratigraphic setting of 5 glaciations (Nidanian, Sanian 1, Sanian 2, Odranian and Vistulian) and 4 interglacials (Podlasian, Ferdynandovian, Mazovian and Eemian) has been established in the Pleistocene of Poland. The palaeomagnetic Brunhes/Matuyama boundary was determined within the Podlasian Interglacial and therefore, the oldest glaciation (Nidanian) has no equivalent anywhere in Europe. The stratigraphic units distinguished are correlated with those in Western Europe and with the marine isotope stages. The Quaternary stratigraphy in Poland is faced with the necessity of how to define regional stratotypes of the main stratigraphic units and boundaries. A crucial issue is to change the approach from a qualitative description of the stratigraphic units to one based also on selected quantitative criteria.

Features of Phase Formation in the CsOH–H2SO4–H3PO4–H2O System and the Growth of the Cs6(SO4)3(H3PO4)4 Crystals

In this paper, the conditions of phase formation in the system CsOH–H2SO4–H3PO4–H2O are considered for the first time. The phase formation of Cs6(SO4)3(H3PO4)4 at t = 50 °C has been studied extensively. The main concentration boundary conditions for this compound are considered for the first time. The solubility congruence of Cs6(SO4)3(H3PO4)4 is shown. Conditions and approaches for obtaining crystals by isothermal evaporation and saturated solution temperature reduction methods are considered. The results of obtaining Cs6(SO4)3(H3PO4)4 crystals with maximum dimensions of ~20 mm are presented.

New Probabilistic Seismic Hazard Model for Nepal Himalayas by Integrating Distributed Seismicity and Major Thrust Faults

Nepal is one of the most seismically active regions in the world, as highlighted by the recent devastating 2015, Mw~7.8 Gorkha earthquake, and a robust assessment of seismic hazard is paramount for the design of earthquake-resistant structures. In this study, we present a new probabilistic seismic hazard assessment (PSHA) for Nepal. We considered data and findings from recent scientific publications, which allowed us to develop a unified magnitude homogenized seismicity catalog and propose alternative seismic source characterization (SSC) models including up-to-date parameters of major thrust faults like main frontal thrust (MFT) and main boundary thrust (MBT), while also considering existing SSC models and various seismic hazard modeling strategies within a logic tree framework. The sensitivity analyses show the seismic hazard levels are generally higher for SSC models integrating the major thrust faults, followed by homogenous volume sources and smoothed seismicity approach. The seismic hazard maps covering the entirety of Nepal are presented as well as the uniform hazard spectra (UHS) for five selected locations (Kathmandu, Pokhara, Biratnagar, Nepalganj, and Dipayal) at return periods of 475- and 2475-years considering Vs,30 = 760 m/s. The results obtained are generally consistent with most recent studies. However, a notable variability in seismic hazard levels and several discrepancies with respect to the Nepal Building Building Code NBC105: 2020 and global hazard model, GEM are noted, and possible causes are discussed.

Mio‐Pliocene exhumation of the hinterland and sediments provenance of the Neogene strata from Kumaun Himalaya of northwest India: Insights from detrital fission‐track thermochronology

Fission‐track (FT) thermochronological analysis of detrital apatite and zircon derived from the lower and middle Siwalik Group of the sub‐Himalaya allows the reconstruction of the long‐term exhumation history of the Higher Himalayan Crystalline (HHC) and Lesser Himalayan Sequence (LHS), and also the provenance of the Siwalik sediments. Here, we report 14 detrital apatite and zircon fission‐track (AFT/ZFT) ages of lower and middle Siwalik subgroups of the Kumaun region, northwest Himalaya. Detrital FT ages suggest that all sandstone samples were never buried below the temperature range of 90° to 100°C and remained as unreset to retain a signal of source‐area exhumation. A detrital AFT age from sandstone sample at close proximity to the Main Boundary Thrust (MBT) is 4.4 ± 0.5 Ma (Pliocene time) which is completely reset and younger than the depositional age, while the detrital ZFT ages of the same sample are unrest. The AFT age represents the time of reactivation of the MBT during which ZFT ages were not affected. This study revealed that the provenances of Siwalik sediments are located majorly in the MBT hinterland areas and the nearest sources are Amritpur granite, the outer Lesser Himalayan meta‐sedimentary (LHMS) zone, Almora klippe, Ramgarh thrust sheet, and the overlying Tethyan Himalayan Sequence. These source areas can be attributed to the development of local drainage networks in the Ramganga‐Kosi drainage basin during late Miocene to Pliocene (i.e., ~9 to 4 Ma). Based on detrital FT datasets, it has been envisaged that different tectonic activities in the MBT hinterland area between Miocene and Pliocene Periods cause the tectonic upliftment and exhumation of source areas and supply the sediment to the Siwalik foreland basin.

Study on the environmental effect on human being detection using the UWB bio-radar

Objective: Non-contact penetrating detection and sensing of human beings (vital signs) through nonmetallic obstacles (ruins, wall, and smog) using the ultra-wideband (UWB) bio-radar plays a significant role in various post-disaster rescue operations in national public security events, like earthquake, building collapse, and factory explosion. In practical application scenarios, the narrowband radio frequency interference (RFI) and surrounding movement interference (SMI) are the two most common and major types of environmental interference (EI), which would cause serious effects on the penetrating detection performance of the UWB bio-radar.Methods: Therefore, through establishing a quantitative and controllable experiment system and combining a proposed quantitative evaluation method, this paper quantitatively investigates and evaluates the influencing characteristics and laws of these two interferences on human vital sign detection using the UWB bio-radar. In the quantitative experiments, two key environmental spatial parameters (interference distance and angle) of interference sources and two kinds of system parameters (main lobe width and time window) of the UWB bio-radar are considered.Results: Numerous experiments at different interference positions and corresponding statistical results demonstrated that both the RFI and SMI have a high negative correlation with the interference distance. Meanwhile, the SMI, which is highly related to the interference angle, could be screened or isolated by the detecting time window and main lobe boundary of UWB bio-radar, but the RFI could not be detected.Discussion: This study shows potential in providing transcendental knowledge and guidance for special EI suppression study, and even for the future design, manufacture, and rational use of the UWB bio-radar, facilitating improvement in the practical performance and effectiveness of the UWB bio-radar search and rescue system.

Elastic-plastic torsion of a two-layer rod

We study the elastic-plastic torsion of a two-layer rod under the action of torque in this article. It is assumed that the rod consists of two layers. Each layer has its own elastic properties, but the plastic properties of both layers are the same. The contact boundary of the layers is located along the oh axis. The lateral boundary of the rod is stress-free, displacements and stresses are continuous at the interface. The components of the stress tensor at a point are calculated using contour integrals derived from conservation laws calculated along the lateral boundary. Next, the second invariant of the stress tensor is compared with the yield strength. At those points where the yield point is reached, the plastic state is realized, in the rest -elastic. This allows you to build a boundary between the plastic and elastic regions. This technique provides a way to calculate elastic-plastic boundaries for the main rolling profiles of rods. This is supposed to be done in subsequent works. We remind you that earlier, with the help of conservation laws, the main boundary value problems for a plastic two-dimensional medium, elastic-plastic torsion of isotropic rods and elastic media for bodies of finite dimensions were solved.solutions.

Research on time series prediction of the flow field in supersonic combustor based on deep learning

The detection of supersonic combustion flow field provides valuable reference information for the efficient combustion of hypersonic vehicles. From the perspectives of scramjet operation during flight as well as experiment where the measured information is limited, flow field prediction using deep learning technology is a promising method that can provide future flow field evolution in scramjet combustors. This study proposes a convolutional neural network model of multi-temporal pressure information fusion to predict combustion flow field based on wall pressure. In addition, it analyzes the effect of various wall pressures on the flow field prediction. An experimental dataset of supersonic ground-pulse wind tunnel tests with different injection pressures is built through the combustion chamber wall under self-ignition conditions of a hydrogen-fueled scramjet. The results showed that the predicted flow field of the combustor agrees with the ground test, average correlation coefficient reached more than 90% and when the wall pressure number is 32, the average peak signal to noise ratio reaches about 23. and the main wave structure and turbulent boundary layer region are restored even if there is a shock boundary layer interaction.

Бикубическая поверхность на фиксированном каркасе: расчет и визуализация

The paper proposes an algorithm for calculating a composite bicubic surface with a fixed frame formed by longitudinal (along the x axis) and transverse (along the y axis) cubic splines. Frame line equations are taken as the main boundary conditions. According to the proposed algorithm, the problem is divided into two stages: first, the frame line equations are found and then the coefficients included in the equations of the bicubic portions forming the bicubic surface are calculated. This approach reduces the size of the characteristic matrix of the linear equation system by reducing the number of coefficients in the surface equation. The matrix size is reduced from 16mn to 9mn, where m and n are the number of bicubic portions along the x and y axes. Surface visualization is reduced to building a grid of longitudinal and transverse generators, the equations of which are formed from the surface equation by substituting y=const (longitudinal generators) or x=const (transverse generators). In this paper we calculate and visualize bicubic surfaces with a frame formed by a mixed set of cubic splines and straight lines. The clarity of the examples is ensured by indicating the numerical values of all calculated magnitudes with an accuracy of up to nine significant figures.

Effect of nano-Cu addition on the diffusion behaviors of Dy in hot-deformed Nd-Fe-B magnets

It is a major challenge for Nd-Fe-B based permanent magnets to improve the utilization of heavy rare-earth resources. For enhancing the diffusion efficiency of heavy rare-earth elements in the hot-deformed Nd-Fe-B magnets, an intergranular diffusion of DyF3 aided by a trace of nano-Cu was implemented by dual alloy process. Through the trace addition (0.2 wt%) of nano-Cu, the coercivity increment of the Cu-containing magnet is higher than that of the Cu-free magnet after the diffusion of DyF3, and the remanence was maintained at a high level. SEM results revealed that a considerable amount of bulky elongated Dy-rich phase was blocked by (Nd,Pr)-rich phase and hardly infiltrated into powder flakes in the Cu-free magnet. But most of the bulky Dy-rich phase dissolved and infiltrated into adjacent powder flakes with the aid of nano-Cu. TEM results revealed that the Dy concentrations of main phase and grain boundary phase in the Cu-containing magnet are both higher than those in the Cu-free magnet. SEM and TEM results confirmed that the trace addition of nano-Cu contributes to the diffusion of Dy into the main phase and grain boundary phase at the surface of powder flakes, which is the main reason for the further enhancement of coercivity. Further discussions on the enhancement mechanism of coercivity were made by using micromagnetic simulations.

3D Geophysical and Geological Modeling of the South Orkney Microcontinent (Antarctica): Tectonic Implications for the Scotia Arc Development

AbstractThe opening of the Scotia Arc resulted in the final breakup of the land bridge between South America and the Antarctic Peninsula. The South Orkney Microcontinent (SOM) constituted part of this former connection and it is now the largest continental block in the Southern Scotia Arc. We present the first 3D model of the SOM that, given its strategic position and characteristics, allows us to advance the knowledge of the tectonic processes involved in the development of the Scotia Arc. Due to the scarcity of reliable geological data, the initial approximation of the deep structure of the SOM was supported by the calculation of three main geological boundaries from geophysical data: the acoustic basement, the boundary of the magnetic anomaly source and the Moho depth. The 3D model was built, refined and validated by forward modeling and joint inversion of gravity and magnetic data. We have accurately defined the geometry of the sedimentary cover, determined the geometry of the intrusive igneous body causing the Pacific Margin Anomaly (PMA) and mapped the heterogeneity of the crustal thickness. These structural features show a clear relationship to each other and are consistent with an important E‐W extension to the east of the SOM during early stages of the Scotia Arc formation, prior to the opening of the Powell Basin.

Lithofacies and Depositional Environments of the Garubathan Alluvial Fan, North Bengal, India

Abstract The sedimentology of the Garubathan alluvial fan of north Bengal foothills is a product of Himalayan neotectonics and monsoon climate. The active neo-tectonics of the Main Central Thrust (MCT), Ramgarh Thrust (RT) and Ghis Transverse Fault (GTF) produce a huge amount of angular clasts in the upper reaches of the river Chel and Mal. During the wet season, either sediment flow by gravity or turbulent stream flow or both, move the clasts to the downhill. The tectonic zone of Main Boundary Thrust (MBT) and Main Frontal Thrust (MFT) are relative passive thrusts than the MCT, RT and GTF in the study area. A very low angle slope is observed in the area of MBT and MFT of the study area. So, the downhill deposition occurred in MBT and MFT zones of the study area. The river Chel and Mal are perinial rivers. During dry periods (pre-monsoon and post-monsoon), in the absence of sediment flow by gravity or turbulent stream flow, the low stream competency deposit the fine sediments. To identify the depositional environments three, seven and six lithofacies varieties were identified in the upper, middle and lower fan surfaces, respectively. The non-cohesive sediment flows by gravity deposited large angular boulders, cobbles and pebbles in the upper fan area. These deposits are either chaotic, unsorted, un-imbricated or graded and imbricated. Deposition of sand is scanty. Both, sediment flow by gravity and fluvial processes deposited angular to round clasts with matrix in the middle fan area. In this area the deposits are reverse graded, reverse to normally graded, laminated and cross-bedded. The high facies diversity of this area indicates highly variable nature of the depositional environments. In the lower fan area, presence of both clast-supported chaotic and matrix-supported laminated or cross-bedded layers suggests a complex nature of the depositional environment of sediment flow by gravity and fluvial processes.

Reconciling Organic-Medical and Psychological Paradigms in the Study of Counselors’ Cognitive Flexibility: a Critical Literature Review

Cognitive flexibility of a counselor or therapist refers to their ability to adapt their cognitive approach to the unique needs and circumstances of their clients. This requires the ability to recognize and adapt to the unique cognitive processes and thought patterns of each individual. In the contextual development of cognitive flexibility in counselors, the organic-medical paradigm and the psychological paradigm have led to a revolution. The existence of these two paradigm perspectives necessitates a unique study that bridges the concept of understanding the two paradigms holistically in order to enhance the cognitive flexibility of the counselor. The purpose of this study is to investigate where the organic-medical paradigm and the psychological paradigm intersect when examining the cognitive flexibility of the counselor. The research design was quantitative and based on a critical literature review. The research focuses on three major components: (1) context issues regarding counselor cognitive flexibility from the perspective of the medical organic paradigm and psychological paradigm; (2) issues regarding the main implicit and boundary problems of counselor cognitive flexibility on professional work performance based on literature research data; and (3) the most promising questions for future research development. In this article, the research findings are discussed further.

Fracture Pattern Analysis of the Upper Cretaceous-Eocene Carbonates along with the Ghumawan Dome, Hazara Basin

Deformational history of the Hazara basin indicates a primitive collision of the two landmasses that undergoes an episodic deformation with NE-SW structural trend. Panjal Thrust (PT) and Main Boundary Thrust (MBT) demarcate the northern and southern extremities of the basin, respectively. The area bounded between these two thrusts is the core consideration of the present research. Different stratigraphic units juxtapose along the Hazara Kashmir Syntaxes (HKS), while the strike-slip component is indicated by imbrication due to thrusts. The study is amied to analyze the paleo-stresses along with developed fracture patterns. Field data were collected via Circle Inventory Method from various localities of the Ghumawan dome, Hazara basin. The zones of upper Cretaceous to Eocene carbonates were mainly targeted during the data collection. Win-Tensor was the key software that helps to analyze the paleo-stresses and fracture pattern of the study area. NW-trending fracture pattern was observed with a highly non-symmetric to dense fracture pattern. The local thrust system lead to severely de-shape the study area. N-S oriented σ1 indicated the compressional tectonic condition that prevailed during deformation of this area. Some segments also show extensional features i.e. normal faulting.

Testing erosional and kinematic drivers of exhumation in the central Himalaya

This study investigates the influence of fault geometry, kinematics, and displacement on the exhumation history of the central Himalaya using geologic mapping constraints, a new interpreted cross-section, and a suite of 176 thermochronometer ages through the Marsyangdi valley in central Nepal. Guided by the cross-section, we integrate a forward model of fold-thrust belt evolution with a 2D thermokinematic model. Model-predicted and measured thermochronometer ages were compared to evaluate the sensitivity of thermochronometer ages to the geometry and location of structures, and their rates of deformation. Results indicate 84% of the measured data can be reproduced with a largely in-sequence system of faulting where displacement occurs on the Main Central thrust (MCT) from 23-16 Ma, the Ramgarh-Munsiari thrust (RMT) from 16-7.5 Ma, the Trishuli thrust (TT) from 7.5-6 Ma and the Main Boundary thrust (MBT) from 6-3 Ma. Our cross-section solution shows the development of a duplex that initiates at 4 Ma with the TT and MBT as the roof thrust. The duplex is translated over the Main Himalayan thrust (MHT) ramp, concurrent with forward propagation of faulting in the synorogenic Siwaliks from 3 Ma to present. Notably, the 2-0.5 Ma apatite fission-track ages between the MCT and Tethyan strata 45 km north of the MCT are not reproduced by the wide range of in-sequence deformation scenarios we explored. Instead, these data are consistent with simulations that include significant displacement on two out-of-sequence (OOS) faults in the last ∼1 Ma: 10 km of OOS faulting near the MCT followed by 5 km of displacement on a fault 15 km south of the MCT. Modeled OOS thrusting both builds significant topography in the hinterland and flexurally suppresses topography in the foreland. Consequently, our preferred kinematic solution has ∼10 km of final fault motion in the Siwaliks to rebuild topography at the MBT and the active MFT from ∼0.5 Ma to the present.Modeled exhumation rates are highly variable through time, and are highest during translation of rocks over high (∼10 km) ramps and during significant changes in architecture, such as duplex formation and OOS faulting. Notably high exhumation rates include 10.5 mm/yr during the first 4 Myr of MCT motion, 4-6 mm/yr at ∼7 Ma, due to the translation over an ∼10 km high TT ramp, and 7-12 mm/yr from 4 Ma to present driven by the development of the duplex and its translation over the MHT ramp, followed by OOS thrusting. The high (7-12 mm/yr) exhumation rates documented here from 4 Ma, correlate with a time of distinct climate change, including strengthening of the monsoon, and northern hemisphere glaciation, lending support to potential climate-tectonic feedbacks.

Target Selection and Validation of DESI Emission Line Galaxies

The Dark Energy Spectroscopic Instrument (DESI) will precisely constrain cosmic expansion and the growth of structure by collecting ∼40 million extragalactic redshifts across ∼80% of cosmic history and one-third of the sky. The Emission Line galaxy (ELG) sample, which will comprise about one-third of all DESI tracers, will be used to probe the universe over the 0.6 < z < 1.6 range, including the 1.1 < z < 1.6 range, which is expected to provide the tightest constraints. We present the target selection for the DESI Survey Validation (SV) and Main Survey ELG samples, which relies on the imaging of the Legacy Surveys. The Main ELG selection consists of a g-band magnitude cut and a (g − r) versus (r − z) color box, while the SV selection explores extensions of the Main selection boundaries. The Main ELG sample is composed of two disjoint subsamples, which have target densities of about 1940 deg−2 and 460 deg−2, respectively. We first characterize their photometric properties and density variations across the footprint. We then analyze the DESI spectroscopic data that have been obtained from 2020 December to 2021 December in the SV and Main Survey. We establish a preliminary criterion for selecting reliable redshifts, based on the [O ii] flux measurement, and assess its performance. Using this criterion, we are able to present the spectroscopic efficiency of the Main ELG selection, along with its redshift distribution. We thus demonstrate that the Main selection 1940 deg−2 subsample alone should provide 400 deg−2 and 460 deg−2 reliable redshifts in the 0.6 < z < 1.1 and the 1.1 < z < 1.6 ranges, respectively.

Deep velocity structure of the eastern margin of the Sarmatian protocraton based on the «Aleksandrovka» seismic station data from the receiver function technique

A velocity section was obtained to a depth of about 250 km on the eastern margin of the Sarmatia protocraton (East-European Craton) based on P receiver functions (PRF). Seismograms of the new broadband station «Aleksandrovka» were used as initial data. The section reveals the main seismic boundaries, and also shows the presence of mid-lithospheric discontinuity in the upper mantle (MLD).

Isogeometric indirect BEM solution based on virtual continuous sources placed directly on the boundary of 2D Helmholtz acoustic problems

An indirect boundary element method (BEM) based on isogeometric analysis (IGA) is proposed for 2D Helmholtz acoustic problems using virtual continuous sources placed directly on the problem boundary. The virtual sources form a virtual boundary identical to the main problem boundary. The proposed solution couples the CAD model with the analysis model by approximating both the CAD model and the virtual continuous sources with the same non-uniform rational B-spline functions (NURBS). Moreover, neither domain discretization nor truncation boundaries at the far-field are required. This solution creates only one coefficient matrix by directly arranging the linear system of equations.The solution follows a collocation scheme on the boundaries based on Greville abscissae with offsets wherever C0 continuity is encountered to permit an easy prediction for the normal directions and the free-terms at the collocation points required in the cases of Neumann and Robin boundary conditions. It allows us also to treat all integrals with standard Gauss quadrature points.Several numerical examples for exterior and interior acoustic problems are discussed to verify the proposed solution with comparisons to analytical solutions and previously published numerical results. These examples prove the robustness of the proposed solution even for high wavenumbers, in contrast to the previous attempts which implemented extensive investigations to find out the optimum place of the sources outside the domain producing minimum errors. Furthermore, no fictitious eigenfrequency problem is observed for exterior acoustic problems.

Geochemical Studies of Rare Earth Elements (REE) in Ion Adsorption Clays (IAC) in Gua Musang, Kelantan

Rare earth element (REE) become the ‘critical metals’ for green technology development that have been rapidly expanded worldwide in these days. REE is mainly originated from granitic rocks. REE in ion adsorption clay (IAC) is the product from weathering of granite. IAC are believed to store high concentration of heavy rare earth element (HREE) and light rare earth element (LREE). Gua Musang is selected for this study because it is located on the three longitudinal belts that composed of acid volcanic igneous rocks from Main Range, Senting and Boundary Range Granites. In this study, the characteristics of ion adsorption clays and REE distribution in Gua Musang have been studied by mineralogy and geochemical analyses. Rocks and soil samples were collected closed to the granite bodies and its surrounding to represent its weathering products. Polarised optical microscopy was used for petrography and mineralogy studies. From fieldwork observation, Gua Musang lithologies composed of carbonate facies, argillaceous facies and pyroclastic facies. X-Ray Fluorescence (XRF), X-Ray Diffraction (XRD) and Inductive Coupled Plasma Microspectrometry (ICP-MS) were used accordingly to characterise the composition of major and trace elements in IAC samples. REE value in Pulai are the highest concentration as iron nodule have been found in the sampling area. Sample from Boundary Range granite also reported store high concentration of REEs in this study.

Бикубическая ленточная поверхность

A bicubic ribbon is a surface of constant width extended along the Ox-axis and formed by a set of rectangular bicubic portions connected to each other with smoothness C1 (continuity of gradient between portions) or C2 (continuity of curvature). Each portion is limited by cubic parabolas lying in vertical planes x=const, y=const. The article presents algorithms for calculating a bicubic band based on the use of boundary curve equations as the main boundary conditions. The «flat corners» conditions are accepted as additional boundary conditions. The proposed approach makes it possible to reduce the size of the characteristic matrix of a system of linear equations with respect to the coefficients included in the equations of bicubic portions. For example, the calculation of 16 coefficients of the equation of a bicubic portion passing through fixed boundary curves reduces to solving a system of four linear equations. Criteria for smooth joining of bicubic portions are formulated (in the form of theorems). Theorem 1 formulates and proves the continuity conditions for the gradient. Theorem 2 contains conditions for the continuity of curvature. Examples of calculation and visualization of C1 and C2- smooth ribbon surfaces, consisting of two or three bicubic portions, are presented.