Stereographic Projection Research Articles

What if the Universe Expands Linearly? A Local General Relativity to Solve the “Zero Active Mass” Problem

Modern cosmology presents important challenges such as the Hubble Tension, El Gordo’s collision, or the impossible galaxies (z > 10). Slight modifications to the standard model propose new parameters (e.g., the early and dynamical dark energy). On the other hand, alternatives such as the coasting universes (e.g., the hyperconical model and the spatially flat R h = ct universe) are statistically compatible with most of the observational tests, but still present theoretical problems in matching the observed matter contents since they predict a “zero active gravitational mass.” To solve these open issues, we suggest that general relativity might be not valid at cosmic scales, but it would be valid at local scales. This proposal is addressed from two main features of the embedding hyperconical model: (1) the background metric would be independent of the matter content, and (2) the observed cosmic acceleration would be fictitious and because of a distorted stereographic projection of coordinates that produce an apparent radial inhomogeneity from homogeneous manifolds. Finally, to support the discussion, standard observational tests were updated here, showing that the hyperconical model is adequately fitted to Type Ia supernovae, quasars, galaxy clusters, baryon acoustic oscillations, and cosmic chronometer data sets.

QCD Anderson transition with overlap valence quarks on a twisted-mass sea

In this work we probe the QCD Anderson transition by studying spectral distributions of the massless overlap operator on gauge configurations created by the (tmfT) with 2+1+1 flavors of dynamical quarks and the Iwasaki gauge action. We assess finite-size and discretization effects by considering two different lattice spacings and several physical volumes, and mimic the approach to the continuum limit through stereographic projection. Fitting the inflection points of the participation ratios of the overlap Dirac eigenmodes, we obtain estimates of the temperature dependence of the mobility edge, below which quark modes are localized. We observe that it is well described by a quadratic polynomial and systematically vanishes at temperatures below the pseudocritical one of the chiral transition. In fact, our best estimates within errors overlap with that of the chiral phase transition temperature of QCD in the chiral limit. Published by the American Physical Society 2024

The impact of fractures and planar structures on the quality of the Upper Jurassic Mozduran reservoir, Kopet Dagh basin (Northeast Iran)

In this study, based on stereographic projections, two sets of planar structures were identified in the Mozduran reservoir, a prolific reservoir in Northeast Iran. The first set includes shallow structures, consisting of stylolites, solution seams, dense veins, and bedding planes. These structures were created by both sedimentary and diagenetic processes and do not have much effect on reservoir quality. The second set includes structures are related to deformation, and these have a substantial impact on increasing the reservoir quality. To investigate the relationships between the types of fractures and their respective influences on reservoir quality, the production logging tool (PLT) gas production rate data was utilized. This analysis revealed that in fractured zones 2 and 4 of Mozduran reservoir, where fractures and flat structures have increased the quality of the reservoir, the PLT gas production rate also increases. Conversely, in areas where filled fractures and diagenetic processes have negatively affected reservoir quality, the PLT gas production rate decreases. This pattern indicates the contrasting impact of planar features on reservoir quality. Moreover, a higher frequency of fractures corresponds to a higher permeability in the Mozduran reservoir of Well #B, whereas a lower frequency of fractures corresponds to lower permeability in that reservoir in Well #A. These relationships confirm the positive influence of fractures on the Mozduran reservoir quality. The study results reveal that deformation variables in the examined reservoir have greater impacts on its quality than the sedimentary and diagenetic porosity/permeability components, but those impacts can have both positive and negative consequences.

Non-inertial interpretation of the Dirac oscillator

Non-inertial physics is seldom considered in quantum mechanics and this contrasts with the ubiquity of non-inertial reference frames. Here, we show an application to the Dirac oscillator which provides a fundamental model of relativistic quantum mechanics. The model emerges from a term linearly dependent on spatial coordinates added to the momentum of the free-particle Dirac Hamiltonian. The definition generates peculiar features (mutating vacuum energy, non-Hermitian momentum, accidental degeneracies of the spectrum, etc). We interpret these anomalies in terms of inertial effects. The demonstration is based on the decoupling of the Dirac equation from the stereographic projection that maps the 3D geometry of the dynamical problem to the complex plane. The decoupling shows that the fundamental mechanical model underpinning the Dirac oscillator reduces to the representation of the oscillator in the rotating reference frame attached to the orbital angular momentum. The resulting Coriolis-like contribution to the Hamiltonian accounts for the peculiarities of the model (mutating vacuum energy, form of the non-minimal correction to the momentum, classical intrinsic spin and gain of its quantum value, accidental degeneracies of the energy spectrum, supersymmetric potential). The suggested interpretation has an interdisciplinary character where stereographic geometry, classical physics of the Coriolis effect and quantum physics of Dirac particles contribute to the definition of one of the few exactly soluble models of relativistic quantum mechanics.

ANALYSIS OF ROCK QUALITY AND LANDSLIDE POTENTIAL IN NORTH JERING HILLS, SLEMAN, YOGYAKARTA

The study was to identify the direction and type of potential landslides, as well as assess the rock quality on the slopes of North Jering Hills. The scanline method, consisting of stretching the meter, was employed to collect data on the plane of failure. The information gathered encompassed the direction, slope level, and condition of the unsuccessful plane, aligned with the parameters in the RMR table of Bieniawski from 1989. Following the collation of data, kinematic and rock mass quality analysis was undertaken using stereographic projection and RMR parameter weighting respectively. The kinematic analysis revealed a planar avalanche type on the slope. The RMR value of the research location's slope was 80, classifying it as good rock.

Текстура рекристалізації та анізотропії пружних властивостей листів сталі DC04

We studied steel sheets DC04 (0.06% C, up to 0.35% Mn, up to 0.40% Si, ~ 0.025% S and P) with a thickness of 0.95 mm as delivered. Sheets of A4 size were annealed in a laboratory oven (6000C in an argon atmosphere, hold for 1 hour). The structure of DC04 steel sheets after recrystallization annealing was studied. The microstructure of the steel sheets under study is presented from the side of the rolling plane and in the section of the sheet perpendicular to the direction. In the plane of the sheets, the grains are elongated; in the cross section, the grains are approximately equiaxed. Pole figures (PF) were constructed based on the results of electron backscatter diffraction (EBSD) on a LEO 1455 VP electron microscope at an accelerating voltage of 20 kV from the plane of the sheets and from the section of the sheet perpendicular to the rolling direction. To improve statistics, PF were constructed by averaging reflex stereographic projections from 20 different representative volumes of material relative to the rolling direction and transverse direction. The texture and anisotropy of Young's modulus in the plane and cross section of steel sheets DC04 after recrystallization annealing was studied using EBSD method. A connection has been obtained between ideal orientations that describe the texture in two mutually perpendicular planes and the corresponding integral characteristics of texture (ICT). Rectangular samples with a length of 100 and a width of 10 mm at different angles to the rolling direction every 150 to measure Young's modulus. Samples were processed in a bag to ensure uniform dimensions. Young's modulus was determined by the dynamic method from the frequency of natural transverse vibrations. Three batches of samples were used to construct Young's modulus anisotropy curves. The anisotropy of the Young's modulus in the plane of steel sheets, calculated from the ICT based on the results of EBSD data, is in good agreement with the results of direct measurements. The value of Young's modulus in the direction normal to the plane of the sheet and in the section plane in the direction normal to the plane of the sheet, calculated from the ICT and the values of the compliance constants of iron, coincide. Keywords: texture, pole figure, anisotropy, integral characteristics of texture, Young's modulus.

Dynamic Projection Method of Electronic Navigational Charts for Polar Navigation

Electronic navigational charts (ENCs) are geospatial databases compiled in strict accordance with the technical specifications of the International Hydrographic Organization (IHO). Electronic Chart Display and Information System (ECDIS) is a Geographic Information System (GIS) operated by ENCs for real-time navigation at sea, which is one of the key technologies for intelligent ships to realize autonomous navigation, intelligent decision-making, and other functions. Facing the urgent demand for high-precision and real-time nautical chart products for polar navigation under the new situation, the projection of ENCs for polar navigation is systematically analyzed in this paper. Based on the theory of complex functions, we derive direct transformations of Mercator projection, polar Gauss-Krüger projection, and polar stereographic projection. A rational set of dynamic projection options oriented towards polar navigation is proposed with reference to existing specifications for the compilation of the ENCs. From the perspective of nautical users, rather than the GIS expert or professional cartographer, an ENCs visualization idea based on multithread-double buffering is integrated into Polar Region Electronic Navigational Charts software, which effectively solves the problem of large projection distortion in polar navigation applications. Taking the CGCS2000 reference ellipsoid as an example, the numerical analysis shows that the length distortion of the Mercator projection is less than 10% in the region up to 74°, but it is more than 80% at very high latitudes. The maximum distortion of the polar Gauss-Krüger projection does not exceed 10%. The degree of distortion of the polar stereographic projection is less than 1% above 79°. In addition, the computational errors of the direct conversion formulas do not exceed 10−9 m throughout the Arctic range. From the point of view of the computational efficiency of the direct conversion model, it takes no more than 0.1 s to compute nearly 8 million points at 1′×1′ resolution, which fully meets the demand for real-time nautical chart products under information technology conditions.

Quick Extraction of Joint Surface Attitudes and Slope Preliminary Stability Analysis: A New Method Using Unmanned Aerial Vehicle 3D Photogrammetry and GIS Development

The present study proposes a preliminary analysis method for rock mass joint acquisition, analysis, and slope stability assessment based on unmanned aerial vehicle (UAV) photogrammetry to extract the joint surface attitude in Geographic Information Systems (GIS). The method effectively solves the difficulties associated with the above issues. By combining terrain-following photogrammetry (TFP) and perpendicular and slope surface photogrammetry (PSSP), the three-dimensional (3D) information can be efficiently obtained along the slope characteristics’ surface, which avoids the information loss involved in traditional single-lens aerial photography and the information redundancy of the five-eye aerial photography. Then, a semi-automatic geoprocessing tool was developed within the ArcGIS Pro 3.0 environment, using Python for the extraction of joint surfaces. Multi-point fitting was used to calculate the joint surface attitude. The corresponding attitude symbols are generated at the same time. Finally, the joint surface attitude information is used to perform stereographic projection and kinematic analysis. The former can determine the dominant joint group, and the latter can obtain the probability of four types of failure, including planar sliding, wedge sliding, flexural toppling, and direct toppling. The integrated stability evaluation method studied in this paper, which combines a 3D interpretation of UAV and GIS stereographic projection statistical analysis, has the advantages of being efficient and user-friendly, and requires minimal prior knowledge. The results can aid in the geological surveys of slopes and guide engineering practices.

Estimating accurate and precise strain in low-magnitude penetratively deformed rocks: Integrating forward strain modeling and natural data

Estimating grain-scale strain using commonly used strain methods from a low-magnitude penetratively deformed rock is typically fraught with uncertainties as the measured strain is influenced by the pre-deformed grain shapes and grain-scale strain partitioning. Therefore, developing kinematic and mechanical models of orogenic wedge evolution based on such strain data can be misleading without knowing how accurate and precise those data are. To determine accuracy we created five 2-D synthetic rock images and deformed them with known strain under pure shear, simple shear, and general shear. We generated 4160 strain ratio (Rs) and long-axis orientation (φ) data. To determine precision we calculated 2σ (95%) uncertainties in Rs and φ from 102 quartz-rich rocks in Sikkim and generated 977 uncertainty data. Results show that the shape matrix eigenvector and stereographic projection methods record the lowest root mean square error, mean absolute error, and 2σ uncertainty values suggesting they are the most accurate and precise, regardless of strain ratio and deformation types. However, the accuracy and precision of φ deteriorate at Rs < 1.5. For most accurate and precise results, using the above mentioned methods, we recommend a minimum of 100 and 75 markers for Rs < 1.5 and Rs > 1.5, respectively.

Parametric generation of optimal structures through discrete exponential functions: unveiling connections between structural optimality and discrete isothermicity

This study discusses an equilibrium state of structures endowed with integrability and relates the structural optimality for Michell’s classic problem and the isothermicity in discrete differential geometry. This discussion leads to a new approach for the parametric generation of quasi-optimal layouts of bar members. The layout of bar members is determined by taking the diagonals of a quadrilateral mesh constructed from a discrete exponential function. The configuration of the planar layout can be changed by adjusting the parameters of a discrete exponential function. In addition, the inverse stereographic projection allows for obtaining spherical shapes from the planar layouts, and the Möbius transformations enable the generation of eccentric near-optimal shapes. It is also demonstrated that the structural layouts generated in this study are the exact optimal or near-optimal solution to Michell’s optimization problem.

Litho-Lineament Mapping of Rocks in Ila Orangun Area Southwestern Nigeria Using Remote Sensing and Aeromagnetic Data: Implications for Mineral Exploration

Ila Orangun is located north of Okemesi and falls within latitudes 7054l and 8000l N and longitudes 40 53l and 50 00l E respectively. Field studies revealed six lithological units in the study area namely quartzites, granites, granite gneiss, porphyritic granites, amphibolite schist and pegmatites. The aim of the research is to elucidate the geology and structure as well as evaluate the metallic mineral potentials of the study area. An integrated multi-technique study approach was adopted for reconnaissance survey and structural interpretation which involved the use of acquired remotely sensed satellite imageries such as Landsat-8 OLI, Shuttle Radar Topographic Mission, Radar Sentinel-2A and a geophysical method involving the use of aeromagnetic data. Results obtained from interpretation of remote sensing data was used to produce different lineament maps which displayed fractures of varying lengths and trend dominantly in the NE-SW directions and subsidiary fractures also orientate in the NW-SE, E-W and N-S directions. Results from the stereographic projection plots showed that the dominant orientation of foliation planes is in the NE-SW direction with the NE and SE sections of the study area recording the highest cluster of the foliation planes. Hence, the eastern section spotted as the mineralization zone.

Determination of Black Sea Coastline Length with Oblique Stereographic Projection Using Affine Transformation

The Black Sea basin has rich oil and natural gas resources. As a result, determining the continental shelf in international relations and maritime law is a critical issue for countries with a shoreline on the Black Sea, which is a semi-enclosed sea. Global projections are generally used for the projection of satellites used to image the earth. The use of global projections causes increased deformation in applications in local areas. For this reason, the deformation of satellite images used as a base can be reduced by converting them to the appropriate projection. In this study, the coastline lengths of the countries neighboring the Black Sea were calculated by using remote sensing images and selecting the appropriate cartographic projection due to the aim of minimum map deformation. By using different numbers of control points to determine the six parameters of the 2D Affine transformation used for coordinate transformation, the change in the accuracy of the transformation depending on the number of control points was examined. The Black Sea coastline, which was digitized in the local coordinate system via the MODIS satellite image, was transformed into coordinates in the oblique stereographic projection system with 2D affine transformation. 11 test points were used in the affine transformation parameters calculated using different control points, and root mean square error (RMSE) of approximately 6 km on the X axis and approximately 10 km on the Y axis was achieved using 25 control points. As a result of the transformation, the coastline lengths of each country bordering the Black Sea were determined in oblique stereographic projection.

Stereographic Projection of Theoretical Orientation Relationships Between Crystals of Any Types in Phase Transformation and Precipitation

AbstractStereographic projection is a classic technology to represent the angular relationships of lattice planes and directions. In some cases, it is necessary to project arbitrary lattice planes or directions of a crystal onto an arbitrary lattice plane of another crystal, e.g., when representing orientation relationships (ORs) in phase transformation or precipitation. Commercial EBSD (electron backscatter diffraction) software cannot illustrate theoretical orientation relationships based purely on the parallelism conditions. This work presents a generalized and unified formulation for stereographic projection of any lattice planes or directions onto any lattice planes of any crystals in the 32 point groups, which is utilized to represent the theoretical orientation relationships between any crystal types. Examples are given to illustrate the correlations of common orientation relationships in both cubic and hexagonal crystals. A procedure is also established to color code and simultaneously project all low‐index Miller/Bravais planes or directions of a crystal onto an arbitrary lattice plane of the same crystal, which can be utilized to visualize the crystal symmetry and determine the standard stereographic triangle (SST). All these illustrations are realized in a computer program written in C++ using the OpenGL graphic libraries.

Magnon Topological Transition in Skyrmion Crystal

We study the magnon spectrum in skyrmion crystal formed in thin ferromagnetic films with Dzyaloshinskii–Moriya interaction in presence of magnetic field. Focusing on two low-lying observable magnon modes and employing stereographic projection method, we develop a theory demonstrating a topological transition in the spectrum. Upon the increase in magnetic field, the gap between two magnon bands closes, with the ensuing change in the topological character of both bands. This phenomenon of gap closing, if confirmed in magnetic resonance experiments, may deserve further investigation by thermal Hall conductivity experiments.

Your journey to Prada Marfa, McDonald Observatory and Santa Elena Canyon: Probes for a hot planet, dark skies and cool gaze

This article takes readers on a brief journey through the remote regions of far west Texas, site of the Greater Big Bend International Dark Sky Reserve, the largest dark sky reserve in the world. Spanning 15,000 square miles, the reserve is home to The University of Texas at Austin McDonald Observatory and Big Bend National Park, along with many other natural areas and art installations. This article uses Prada Marfa, McDonald Observatory and Santa Elena Canyon as probes for a hot planet under dark skies, while also deploying a cool gaze into the Milky Way. In so doing, cool media takes on a profound new meaning for our shared destiny on a tiny planet, getting ever hotter as it hurtles through the vast universe.

Spinorial higher-spin gauge theory from IKKT model in Euclidean and Minkowski signatures

We explore the semi-classical relation between the fuzzy 4-hyperboloid HN4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {H}_N^4 $$\\end{document} and non-compact quantized twistor space ℙN1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{P}}_N^{1,2} $$\\end{document} at large N. This provides two backgrounds of the IKKT matrix model via two natural stereographic projections, leading to higher-spin gauge theories with Euclidean and Minkowski signature denoted by HS-IKKT. The resulting higher-spin gauge theory can be understood as an uplift of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills to twistor space. The action of HS-IKKT is written using a spinor formalism in both in Euclidean and Minkowski signature. We then compute the tree-level amplitudes of the massless sector within the Yang-Mills part of the HS-IKKT model in the flat limit in Euclidean signature. All n-point tree-level scattering amplitudes for n ≥ 4 of this sector are found to vanish in the flat limit.

Relational Reprojection Platform: Non-linear distance transformations of spatial data in R

When mapping relationships across multiple spatial scales, prevailing visualization techniques treat every mile of distance equally, which may not be appropriate for studying phenomena with long-tail distributions of distances from a common point of reference (e.g., retail customer locations, remittance flows, and migration data). While quantitative geography has long acknowledged that non-Cartesian spaces and distances are often more appropriate for analyzing and visualizing real-world data and complex spatial phenomena, commonly available GIS software solutions make working with non-linear distances extremely difficult. Our Relational Reprojection Platform (RRP) fills this gap with a simple stereographic projection engine centering any given data point to the rest of the set, and transforming great circle distances from this point to the other locations using a set of broadly applicable non-linear functions as options. This method of reprojecting data allows users to quickly and easily explore how non-linear distance transformations (including square root and logarithmic reprojections) reveal more complex spatial patterns within datasets than standard projections allow. Our initial release allows users to upload comma separated value (CSV) files with geographic coordinates and data columns and minimal cleaning and explore a variety of spatial transformations of their data. We hope this heuristic tool will enhance the exploratory stages of social research using spatial data.

Rock mass joint set identification at Draa Sfar mine in Morocco through stereographic projection and K-means clustering

Rock mass joint set identification at Draa Sfar mine in Morocco through stereographic projection and K-means clustering

Large constellations for fast acquisition and delivery of information: Design and analysis by stereographic projection onto the equatorial plane

SummaryRecent space projects are designed by satellite constellations with a large number of spacecraft, a global character (i.e., from equatorial to high inclination orbits), and the possibility to transfer information to different satellites of the constellation (inter satellite link) in order to deliver the information to the ground as soon as possible. To cope with a large number of parameters, a fast tool for constellation design, performance evaluation and networking strategies is needed. The aim of this paper is to obtain the performance of any constellation in less than 1 s, even when the number of satellites in the constellation and the duration of the analysis is large (e.g., more than 200 satellites in a period of some days). The proposed algorithm is based on analytical formulae obtained by using the stereographic projection on the equatorial plane of the satellite orbits and the projection of the target and ground station orbits. It is believed that the two‐dimensional projection proposed here can offer some advantages with respect to the spatial analysis of satellite orbits and their ground tracks, such as the reduction in time required to calculate station/satellite and satellite/satellite encounter conditions, and the clear and simple representation of the motion of the satellite of the constellation and of the ground stations of interest.

Texture, electronic and magnetic states of Mn atoms in polycrystalline MnSi thin films facing a c-sapphire substrate studied by soft X-ray magnetic circular dichroism

Polycrystalline MnSi is a highly efficient magnetic skyrmionic material and information carriers for future spintronic devices, deposited on a c-cut sapphire substrate by magnetron sputtering with a low base pressure. Due to their fascinating crystal structures and spin textures these Mn-based silicides are most extensively studied now a days. In order to study the appearance of any impure phases like MnxSiy, hysteresis behavior and the inception of ferromagnetic/superparamagnetic domains present in the MnSi film, we have performed X-ray diffraction (XRD), XRD pole figures in stereographic projection, vibrating sample magnetometry and powerful element specific X-ray absorption spectroscopy (XAS) and X-ray magnetic circular dichroism techniques. The room temperature hysteresis curve points towards the phenomenon of long-range ferromagnetic ordering. The Mn L2,3 edge XAS spectra did not show a Mn2+-like multiplet structure. The overall magnetization increases with the increase in applied magnetic field from 0.1 T to 2 T on MnSi thin film. The observed spin and orbital magnetic moments ∼ 0.03 ̶ 0.1 μB and 0.01 to 0.04 μB, respectively.