Stereographic Projection Method Research Articles

A fuzzy K-Means algorithm based on Fisher distribution for the identification of rock discontinuity sets

Rock discontinuities significantly impact the mechanical and hydraulic behavior of rock masses. A crucial aspect of rock engineering involves classifying discontinuities with similar orientations into groups. For this purpose, clustering algorithms, such as K-Means and Fuzzy K-Means (FKM), have been employed. However, the outcomes of these algorithms are influenced by the selection of initial cluster centers. This paper proposes an improved FKM algorithm to automatically identify rock discontinuity sets based on the Fisher distribution (FFKM). The algorithm uses the Fisher distribution to generate and select appropriate initial cluster centers. The performance of FFKM was initially validated using a published data set, and its results were compared with other clustering methods commonly used for grouping discontinuities. Results demonstrated the superior performance of FFKM over the FKM algorithm, comparable to other methods. Subsequently, the proposed algorithm was employed to analyze a fracture data set sampled at an open-pit iron mine. The FFKM facilitated identifying the correct number of sets and produced results consistent with the fracture sets observed in the field. Finally, the algorithm was verified using an artificial discontinuity data set, and the results demonstrated that the method correctly identified the number of sets and provided discontinuity sets similar to the original data set. The FFKM algorithm offers significant advantages: it maintains the essential characteristics of the FKM algorithm, effectively addresses the challenge of selecting suitable initial cluster centers, requires only the expected number of discontinuity sets as an input parameter, processes data within an acceptable computation time, serves as a tool for defining the number of discontinuity sets, and mitigates the drawbacks of the stereographic projection method.

Controlling effect of palaeo-tectonic stress field on gas occurrence

The paleo-tectonic stress fieldwas hereby inverted by using the stereographic projection method through field and underground observations of conjugate shear joints. On the basis of analyzing and studying the characteristics of gas occurrence in mining areas, the control effect of paleo-tectonic stress field on gas occurrence was discussed from three aspects of gas generation, preservation environment and gas migration. The results show that: (1) During the Indosinian and early-middle Yanshan period, the coal seam was buried deep, and the temperature and pressure conditions were suitable for massive gas generation, especially during the Indosinian period featuring massive gas generation and weak gas migration; (2) During the late Yanshan period, the metamorphic evolution rate of coal seams accelerated, secondary hydrocarbon generation occurred in the coal seams, and a large amount of gas was generated. Meanwhile, the gas migration was enhanced. The gas generation amount was much larger than the emission amount, therefore, making it still a period of massive gas generation in general; (3) During the Himalayan period, the coal measure stratum was in the uplift stage, and a large number of geological structures were developed in the stratum. The tectonic stress field in this period caused the escape of massive coal seam gas. Multi-stage tectonic stress field acted on coal measures strata in turn, resulting in gas generation in coal seam and gas migration at the same time. Besides, gas occurrence is the superposition effect of gas generation, preservation conditions, and gas migration in coal seam.

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.

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.

Application of the deterministic block theory to the slope stability design of an open-pit mine in Morocco

Purpose. Discontinuities in rock masses are natural fractures that delimit various block shapes and sizes, which can fall, slide or topple from the excavation and collapse under their own weight inducing probably severe damage. Thus, it is essential to carry out a block analysis before beginning any surface or underground excavation project. This paper proposes a methodology based on key block theory analysis to select the suitable slope of different discontinuous rock masses of an open-pit mine in Morocco. Methods. At first, the main discontinuities of each bench are determined and projected onto a stereonet with a maximum dip angle of the excavation plane. Then, it is possible to identify the removable blocks by using the theorem of removability according to block theory. After that, a limit equilibrium analysis is performed to determine the failure mode and the friction angle required to stabilize the blocks. When the selected dip angle of the slope plane is found to be unsuitable, it is changed and reduced by one degree, and the same approach is repeated until the maximum safe slope dip angle is obtained. Findings. The results of the proposed methodology based on key block theory analysis have shown that the maximum safe slope angles of the studied benches are in the range of 63-73°. When compared to the slope angles used in the mine, which are between 58-78°, the results of this study are close to in-situ conditions. Originality. In this research, the maximum safe slope angle of fractured rock masses was optimized by eliminating slope angles inducing unstable blocks (key blocks) and by using the stereographic projection method of key block theory. Practical implications. Using this methodology, stability of rock slopes in civil or mining-engineering projects can be designed or assessed when geotechnical data are very limited.

Estimating Effective Leaf Area Index of Winter Wheat Based on UAV Point Cloud Data

Leaf area index (LAI) is a widely used plant biophysical parameter required for modelling plant photosynthesis and crop yield estimation. UAV remote sensing plays an increasingly important role in providing the data source needed for LAI extraction. This study proposed a UAV-derived 3-D point cloud-based method to automatically calculate crop-effective LAI (LAIe). In this method, the 3-D winter wheat point cloud data filtered out of bare ground points was projected onto a hemisphere, and then the gap fraction was calculated through the hemispherical image obtained by projecting the sphere onto a plane. A single-angle inversion method and a multi-angle inversion method were used, respectively, to calculate the LAIe through the gap fraction. The results show a good linear correlation between the calculated LAIe and the field LAIe measured by the digital hemispherical photography method. In particular, the multi-angle inversion method of stereographic projection achieved the highest accuracy, with an R2 of 0.63. The method presented in this paper performs well in LAIe estimation of the main leaf development stages of the winter wheat growth cycle. It offers an effective means for mapping crop LAIe without the need for reference data, which saves time and cost.

Stability analysis of rock slope under Sujiaba overpass in Chongqing City based on kinematic and numeric methods

Rock slopes have the characteristics of complex geological conditions, weak structural surface development, steep slope, and great damage. In this study, the Sujiaba overpass slope in Chongqing was selected as the evaluation object, and the main stability evaluation methods of rock slope were analyzed. Combined with the special geological conditions and geographical location of the rock slope, through a geological survey, the slope was qualitatively analyzed based on the stereographic projection method, and the slope stability safety factor was calculated by using the finite element strength reduction method. FLAC3D was used to simulate the initial stress state of the unstable rock mass, the limit state stability of the unstable rock mass before bolt support, and the stability after bolt support. The simulation results show that the stability coefficients of selected unstable rock masses W1, W2, W6, and W7 under the limit state before bolt support are, respectively, 1.22, 1.80, 5.90, and 2.10. Unstable rock masses separate from the parent rock, causing a large displacement due to their instability and downward sliding. After bolt support, stability coefficients for those four unstable rock masses are 1.60, 2.40, 8.60, and 3.20, respectively. Under the same reduction coefficient, rock masses are stable and the displacement is small. The results show that the calculation results of the initial stress state of the rock slope are consistent with the theoretical understanding and field investigation. After the implementation of bolt support, the anti-sliding stability of unstable rock is obviously improved. The research results have important scientific guiding significance and practical value for revealing the failure mechanism of rock slope and analyzing the stability of unstable rock mass.

RETINAL VASCULAR BED AREA IN EYES WITH RETINAL VEIN OCCLUSION ON ULTRA-WIDEFIELD FLUORESCEIN ANGIOGRAPHY: WAVE Study.

To compute retinal vascular bed area (RVBA) in square millimeters on distortion corrected ultra-widefield fluorescein angiography images in eyes with retinal vein occlusion (RVO). Prospective observational study. The peripheral distortion of baseline ultra-widefield fluorescein angiography (Optos 200Tx) images of 30 patients with RVO from the WAVE study (NCT01710839) and 13 control eyes of normal subjects was corrected using the stereographic projection method to compute RVBA in square millimeters. In comparison with age- and sex-matched normal control eyes, eyes with RVO had a decreased global RVBA for the entire retina (50.5 ± 20.4 mm 2 vs. 62.6 ± 12.2 mm 2 , P = 0.023). Eyes with RVO and the unaffected fellow eye had a similar RVBA globally (50.5 ± 20.4 mm 2 vs. 46.2 ± 18.9 mm 2 , P = 0.523). The RVBA was observed to negatively correlate with nonperfusion area (R = -0.47, P = 0.009). However, RVBA was not related to the severity of macular edema ( P > 0.05). Eyes with RVO have a similar RVBA to the unaffected fellow eyes but with a reduction when compared with normal control eyes. Retinal vascular bed area appears to be a surrogate biomarker of retinal ischemia on ultra-widefield fluorescein angiography but not the extent of macular edema.

Context-Dependent Parameter Sensitivities in Rock Slope Stability

Context-dependent sensitivities of parameters and reliability-based design (RBD) of rock slopes are the subjects of this paper. The similarities and differences between the design points in RBD and those of partial factor design methods are discussed. It is demonstrated that partial factors provided by the design point of the first-order reliability method (FORM) can provide case-specific insights and guidance to partial factor design methods like Eurocode 7 (EC7) and the load and resistance factor design (LRFD). It is suggested that conducting RBD-via-FORM in tandem with partial factor designs is more illuminating and meaningful than calibration of partial factors of parameters which can be sensitive in one case but insensitive in another case. Three cases are analysed probabilistically with respect to plane sliding in rock slopes with one or more discontinuities. In the first two cases, different deterministic solution procedures are used for the single block and two-block mechanisms, for comparison with stereographic projection method and closed form equation, respectively, prior to extending the cases into RBD. The third case involves a failed slope in a limestone quarry, analysed using FORM in this paper, for comparison with Monte Carlo simulation.

Soliton based director deformation in a twist grain boundary liquid crystal

We investigate the director dynamics of a twist grain boundary liquid crystal under the one constant approximation for the different elastic constants representing the various deformation present in the liquid crystal medium. The free energy density is deduced to a higher-order vector nonlinear partial differential equation by balancing the torque experienced by the nematic molecules under a viscous field and the molecular field arises due to the presence of elastic constants. Upon employing the stereographic projection method we further reduced the vector nonlinear differential equation into a complex scalar nonlinear partial differential equation. We obtain a series of localized solutions for the complex scalar nonlinear partial differential equation through the standard tanh method.

Stability prediction and optimal angle of high slope in open-pit mine based on two-dimension limit equilibrium method and three-dimension numerical simulation

The optimization design of the open pit slope can ensure safety in production at the same time bring considerable economic benefits, so it is necessary to evaluate the stability of the initial excavation scheme of the high slope in an open-pit mine and optimize the initial slope angle according to the evaluation results. In this paper, an exhaustive study of Anyi open-pit mine in mouding County Yunnan province China has been made, and the following results have been achieved: (1)Confirmed the geological factors which affected the slope stability of open-pit mine; (2)Quantified the stability of the open-pit mine slope; (3) Under the premise of safety, the angle of the open pit high slope has been optimized on the base of the calculations of the open-pit mine high slope stability so as to ensure the maximum economic benefit. In this paper, the study area was divided into six zones. Firstly, qualitative analysis of slope stability was made by analogy method of engineering geology and stereographic projection method according to the orebody location, the properties of rock and soil in slope, the height of slope, and so on. The analysis shows that the preliminary design of slope angle is so large to ensure safety. Secondly, the stability of different slopes of a high slope is the quantitative analysis of different zones in high slope was made by two-dimensional limit equilibrium method and the finite difference method. In this paper, the mechanical parameters of rock are obtained by a laboratory test, and the stability of the high slope is quantitatively analyzed by two-dimensional limit equilibrium and three-dimensional numerical simulation.

A Study on Evaluation of Slope Stability and Range of Rockfall Hazard of Daljeon-ri Columnar Joint in Pohang, Korea

In this study, we evaluated the slope stability of the Pohang Daljeon-ri columnar joint (Natural Monuments # 415) and calculated the maximum energy, jumping height and moving distance of rockfalls using a simulation. Based on the results, we established the range of rockfall risk. The slopes of the Pohang Daljeon-ri columnar joint have dip directions of 93.79°, 131.99°, 165.54° and 259.84° from left (SW) to right (NE). Furthermore, they have a fan-like shape. The Pohang Daljeon-ri columnar joints are divided into four sections depending on the dip direction. The measurement results of the discontinuous face show that zone 1 is 125, zone 2 is 261, zone 3 is 262, zone 4 is 43. The results of slope stability analyses for each section using a stereographic projection method correspond to the range of planar and toppling failure. Although it is difficult to diagnose the type of failure, risk evaluation of currently falling rocks requires further focus. The maximum movement distance of a rockfall in the simulation was approximately 66 m and the rockfall risk range was the entire area under slope. In addition, it is difficult to forecast where a rock will fall as it rolls in various directions due to topographic factors. Thus, the installation of measures to prevent falling is suggested to secure the stability based on the results of the rockfall simulations and its probabilistic analysis.

Determination of particular singular configurations of Stewart platform type of fixator by the stereographic projection method

Hexapod-type external fixators based on a general 6-6 Stewart platform structure are extensively used to manage orthopaedic disorders. While implementing these robotic devices, a practical visual aid is needed to quickly identify their uncontrollable states referred to as singularities. Thus, a visible correlation between the singularity of hexapod-type external fixators and their particular configurations has been explored geometrically in this work. A novel method called stereographic projection is utilized for that purpose. A mathematical procedure has been established to determine the characteristic values of the singular states. It is found that in case four- out of six-rod directions intersect each other at a common point, two different singular robot configurations result. Besides, if four joint angles at the top and bottom rings of the hexapod are equal, rings being parallel, then five-rod directions are intersected by two lines each passing through the end joints of the fifth rod.

Triple helix versus skyrmion lattice in two-dimensional noncentrosymmetric magnets

It is commonly assumed that a lattice of skyrmions, emerging in two-dimensional non-centrosymmetric magnets in external magnetic fields, can be represented as a sum of three magnetic helices. In order to test this assumption we compare two approaches to a description of regular skyrmion structure. We construct (i) a lattice of Belavin-Polyakov-like skyrmions within the stereographic projection method, and (ii) a deformed triple helix defined with the use of elliptic functions. The estimates for the energy density and magnetic profiles show that these two ansatzes are nearly identical at zero temperature for intermediate magnetic fields. However at higher magnetic fields, near the transition to topologically trivial uniform phase, the stereographic projection method is preferable, particularly, for the description of disordered skyrmion liquid phase. We suggest to explore the intensities of the secondary Bragg peaks to obtain the additional information about the magnetic profile of individual skyrmions. We estimate these intensities to be several percents of the main Bragg peak at high magnetic fields.

Ultrawide Field of View by Curvilinear Projection Methods

The rectilinear Perspective projection produces natural-looking results on the condition that the degree of field of view (FoV) is narrow, as raising it causes an exponential increase in visual distortion. Curvilinear perspective projection methods that counter this issue exist in photography, but unlike the rectilinear Perspective projection, these are neither used nor technically documented in computer graphics. This paper contributes by presenting results from a perceptual experiment comparing the rectilinear Perspective projection method to the curvilinear Panini and Stereographic projection methods. These are commonly used with wide-angle lenses in photography and have been digitally recreated for use in computer graphics. The experiment shows a clear preference for the two curvilinear projection methods at high degrees of FoV, as not a single participant prefers the rectilinear Perspective projection at degrees of FoV approaching the human breadth of vision.

Analysis of Highway Slope Failure by Using Stereographic Projection Method

This study was conducted at the Bhasu Bhir of Kailali district in the Far-western Province, along the Khanidanda-Sahajpur road section of Bhimdatta pant highway. The main objective of this study is to identify the potential slope failure area and assess the major control factors along the road section. To accomplish the objective, the dip amount and dip direction of major discontinuity was measured in the field along with hill slope and hill slope direction. The data of 40 locations were analyzed by using stereographic projection method. The result was obtained by using Dips 6.0 software. The major slope failures obtained on the area were wedge failure and plane failure. The hill slope geometry, joint status and rock geometry are found major controlling factors for the slope failure in the study area.

Paleo-tectonic Stress Field and Tectonic Evolution since the Mesozoic in the Eastern Mining Area of Pingdingshan

By observing a large number of conjugate shear joint data in the field and underground of the study area, the conjugate shear joint data is analyzed by the stereographic projection method, and the paleo-tectonic stress field experienced in the mining area since the Mesozoic is inferred. Based on the above research, the tectonic evolution process of the eastern mining area of Pingdingshan is discussed from the aspect of dynamic mechanism. The results show that: (1) The eastern mining area of Pingdingshan experienced the Indosinian, Yanshan early-middle, Yanshanian, and Xishanian tectonic stress fields, with the directions of NS, NW, NE, and NNE, respectively. (2) The tectonic evolution of the mining area can be divided into four stages: (a) The Indosinian tectonic stress field has the weakest effect on the study area, the coal-bearing strata are gradually uplifted by the stress field in this period, and some small structures are formed in the coal-bearing strata; (b) The early-middle Yanshanian tectonic stress field has a certain transformation effect on the coal-bearing strata in the eastern part of Pingdingshan. It mainly forms the secondary structure of the mining area such as Huoyan normal fault and small fault-fracture structure; (c) The late Yanshanian tectonic stress field has the strongest effect, and the control complex fold structure with the axis near NW direction is formed under the tectonic stress field of this period, such as Likou syncline, Baishishan anticline, Lingwushan syncline, Guozhuang anticline, Niuzhuang syncline, etc. The near-NW direction of the control fault structure is also formed under the action of the late Yanshanian tectonic stress field, such as the Niuzhuang reverse fault, the original No.11 mine reverse fault, etc.; (d) The Himalayan tectonic stress field mainly forms some secondary and small structures in the mining area and has a certain transformation effect on the structure formed by the early tectonic stress field. The tectonic pattern of the eastern mining area of Pingdingshan is the result of the above-mentioned tectonic stress field acting in sequence with the coal-bearing strata, causing structural deformation of the coal seam.

Towards an Effective Theory of Skyrmion Crystals

We consider multiskyrmion configurations in 2D ferromagnets with Dzyaloshinskii-Moriya (DM) interaction and the magnetic field, using the stereographic projection method. In the absence of DM interaction, $D$, and the field, $B$, the skyrmions do not interact and the exact multiskyrmion solution is a sum of individual projections. In certain range of $B, D\neq0$, skyrmions become stable and form a hexagonal lattice. The shape of one skyrmion on the plane is fully determined by $D$ and $B$. We describe multiskyrmion configurations by simple sums of individual skyrmion projections, of the same shape and adjusted scale. This procedure reveals pairwise and triple interactions between skyrmions, and the energy of proposed hexagonal structure is found in a good agreement with previous studies. It allows an effective theory of skyrmion structures in terms of variables, referring to individual skyrmions, i.e., their position, size and phase, elliptic distortions etc.

Finite strain of the Tertiary rocks and their relation to tectonic deformation at Al-Tib Anticline in Missan governorate, Southeastern part of Iraq

This paper presents the finite strain results from seven oriented samples data on Tertiary sandstone of Muqdadiya Formation and (400) samples of pebbles and conglomerate of Bai –Hassan Formation at the southwestern limb of Al-Tib Anticline in the Southeastern part of Iraq. Measurement and analysis of finite strain are carried out including these rocks at fluvio- lacustrine environment. The present study followed Fry method. The computed strain was, in the form of ellipses, within three prepared perpendicular planes in a single sample and Center to Center method was used to determine the strain ratio of the these samples. The strain in the studied area is low, this is mainly due to the sampled rocks underwent brittle deformation during folding. The calculated axial ratios (Rs), (three dimensional orientations of the strain axes) of strained rocks are shown which is equivalent to the log Flinn diagram normally used for plotting strain ellipsoids and according to the stereographic projection method, the long axes of this ellipsoid are sub parallel to measured Al-Tib Anticline axis and the short axes are approximately perpendicular to the axis of Al- Tib Anticline in the study area. This orientation of these axes is related to the movement of the Arabian plate.

精确绘制立方晶系极射赤面投影图的新方法：平面几何法

精确绘制立方晶系极射赤面投影图的新方法：平面几何法