Geometric Elements Research Articles

Geometric Error-Based Multi-Source Error Identification and Compensation Strategy for Five-Axis Side Milling

Based on a multi-source error model, this paper discusses the principle of error element identification and uses the mirror bias method to compensate the geometric errors of a process system. Firstly, a nine-line measurement method to determine the geometric error of the linear feed axes of machine tools is introduced, and the geometric error identification model based on the “nine-line method” is established. Then, using a ballbar mounted in the axial, tangential, and radial directions of the machine, the geometric error elements of the rotation axis are identified by three simple measurements in each direction. Subsequently, for the more common flat vise clamping workpiece in actual production, the workpiece position error is identified by using the traditional process of dimensional chain, and the workpiece attitude error is identified by fitting the angle between the positioning plane and the horizontal plane by the least squares method. Finally, based on the tool position points and tool axis vectors obtained from the multi-source error model, the error compensation value is solved using inverse machine tool kinematics to offset the machining error by mirroring the error value of the same size, and based on the “S-shaped specimen” to compensate the processing experiments, after compensation, the processing error is reduced by 30~45%, verifying the effectiveness of the compensation method.

Aviation equipment measurement and assembly analysis method based on robotic system

In the manufacturing industry of aviation equipment (e.g., ballistic missile, rocket, unmanned aerial vehicle (UAV), and airplane), the traditional assembly quality inspection methods mainly rely on manual operation of single- or multi-stations laser trackers, but they are powerless to measure and evaluate complex components on the aviation equipment. This paper introduces a method for automatically detecting and evaluating assembly accuracy: first, the global accurate calibration is performed automatically by deriving the optimized positioning of leapfrog balls; after selecting the aviation equipment type to determine the measurement tasks, each side of the aviation equipment is equipped with a robot and an accompanying FARO scanning arms that can move together on the U-shaped guide rail to the measurement station where the measurement task is to be performed; at each measurement station, the robot drives the movement of the FARO scanning arm through a linkage mechanism to perform scanning; after the whole measurement processes are completed, data from multiple measurement stations is integrated into the same coordinate system for calculation and assembly quality analysis. The algorithm module consists of path planning function for scanning various parts, feature extraction function for various geometric elements, and assembly quality evaluation function. Compared with laser tracker method, the proposed method can effectively and accurately measure and analyze basic measurement tasks: the maximum differences of length, concentricity, and perpendicularity are all less than 0.3%; while the time consumption can be significantly reduced to less than 5% of that of the laser tracker method. Moreover, the proposed method can also achieve the automatic measurement and analysis of complex positioning components that cannot be achieved by laser trackers, e.g., center of positioning hole, position deviation of the bump, and azimuth angle between the positioning surfaces.

Investigating Non-Newtonian Fluid Behavior in Hydrocyclones Via Computational Fluid Dynamics

Expert researchers examine complex patterns of pressure, viscosity, and velocity in a CFD study of viscoelastic food inside hydrocyclones to obtain a detailed grasp of particle behavior and fluid dynamics. Velocity profiles show how fluids and particles flow through the hydrocyclone in complex ways, while pressure distributions show where high and low pressure is found, regions that are critical for maximizing separation efficiency. Furthermore, the analysis of viscosity fluctuations clarifies the intricate relationship between fluid rheology and flow dynamics, providing information on how food's viscoelastic characteristics affect particle trajectories and separation efficiency. Utilizing this comprehensive examination, scientists hope to optimize the design and functioning parameters of the hydrocyclones, which will in turn improve the efficacy and efficiency of particle separation procedures in viscoelastic food solutions. This will ultimately lead to improvements in food processing technology and product quality. Researchers look into the impact of geometric elements on flow patterns and separation efficiency in addition to these characteristics, such as hydrocyclone size and inlet configurations. Additionally, they investigate how different operating parameters, such rotational speed and flow rate, affect how well the hydrocyclone handles viscoelastic food items. Through the integration of these complex analyses, researchers hope to create all encompassing models that can precisely forecast and optimize the behavior of viscoelastic food flows inside hydrocyclones, opening the door to improved process control and food sector product quality.

Mathematical Model and Analysis of Novel Bevel Gear with High Load-Capacity Based on the Geometric Elements

Load-capacity has always been one of the performances that is paid much attention to in the development of bevel gear transmission applications. Consequently, the mathematical model of novel bevel gear with high load-capacity based on geometric elements is proposed in this paper, which could be applied to the aviation, aerospace and other fields. In parallel, the design principle and design method of the novel bevel gear are introduced in detail. Subsequently, the conditions for tooth surface continuity and non-interference are derived. Furthermore, the model of novel bevel gear is established. Finally, the load-bearing characteristics are analyzed, revealing that an increase in the number of contact points could significantly enhance the load capacity of the bevel gear pairs. When the load torque applied to bevel gear II is 100 Nm, the contact pressure endured by the bevel gear pair with five-point contact is decreased by 41.37% compared to the bevel gear pair with single-point contact. When the number of contact points is the same, increasing the distance between the contact points could also reduce the contact stress. This provides strong theoretical support for the application of the bevel gear based on the geometric elements.

Advanced Transparent Glass‐Ceramics via Laser Anisotropic Nanocrystallization

AbstractTransparent glass‐ceramics, merging the attributes of both glass and ceramics, have diverse applications in ballistic protection, armor materials, dentistry, light‐emitting diode, and optical domains. A limitation of conventional methods for glass‐ceramic fabrication is their inadequacy in precisely controlling the morphology of crystallization, consequently constraining their prospective advancements in diverse areas. Here, a method that facilitates precise control over the formation and growth of polarization‐dependent nonperiodic anisotropic nanocrystals, with a short axis ranging from 15 to 280 nm, in Li2O‐Al2O3‐SiO2 (LAS) glass is proposed. The method utilizes the principles of near‐field anisotropy between light and matter, building upon the classical nucleation‐growth model of glass crystallization. Notably, the glass‐ceramics possessing laser‐induced nanocrystals within the Rayleigh size regime showcase an impressive 99.7% transmittance in the visible and near‐infrared wavelengths. Additionally, the glass‐ceramics with non‐periodic anisotropic nanocrystals macroscopically exhibit form‐birefringence properties, thus enabling transparent optical applications, such as polarization elements, geometric phase elements, and multi‐dimensional optical data storage. This work opens up new avenues for morphological manipulation of nanocrystallization, with potential applications in optics, nanophotonics, functional glass‐ceramics, and high mechanical performance devices.

Hyperspectral optical orbital angular momentum modulation from tunable structured waveplates.

Shaping the orbital angular momentum of optical pulses in the spectral domain is a means of managing light in space and time that offers many possible applications. However, these are limited by the small number of techniques available, whose flexibility does not yet rival that of the continuous regime. We propose here to implement a tunable hyperspectral management of the orbital angular momentum of a polychromatic light field. The main idea is to exploit the dispersive nature of geometric phase optical elements by intentionally choosing to work in a regime of high anisotropic optical retardance. An experimental proof of principle is demonstrated in the visible range using a supercontinuum laser and an optically thick, electrically controllable, liquid crystal structured wave plate.

Conceptual RF design of 750 MHz IH cavities for [formula omitted] = 0.10–0.15 ion beams in medical accelerators

In light of the potential interest of Interdigital H-mode (IH) cavities for accelerating carbon ion beams beyond 5 MeV/u, we are reviewing the key geometric elements of the regular cells and end cells to optimize performance in terms of power efficiency, achievable voltage, and dipole field correction.

GEOMETRIC ASPECTS OF THE GEDHE KAUMAN MOSQUE IN YOGYAKARTA

This research aims to analyze ethnomathematics in a mosque named Gedhe Kauman in Yogyakarta. The ethnomathematics studied is related to the mathematical concepts of area and perimeter of two dimensional shapes. This research is qualitative research with an ethnographic approach. The data source in this research is the result of direct observation and documentation by the researcher. Data collection techniques were carried out by observing and documenting mosque buildings. Testing the validity of the data was carried out using technical triangulation. The subject of this research is the historic building of the Gedhe Kauman Mosque. Based on the research results, it was found that the Mosque building is quite closely related to ethnomathematics learning resource. This is proven by the presence of geometric elements in the structure and contents of the building that have easyly reognized patterns. The geometric shapes found are squares, rectangles, circles, triangles and trapezoids. These plane shapes can be integrated into the mathematical concepts of perimeter and area for learning elementary mathematics. This instructional approach might evoke student's disposition towards mathematics.

DeepBranchTracer: A Generally-Applicable Approach to Curvilinear Structure Reconstruction Using Multi-Feature Learning

Curvilinear structures, which include line-like continuous objects, are fundamental geometrical elements in image-based applications. Reconstructing these structures from images constitutes a pivotal research area in computer vision. However, the complex topology and ambiguous image evidence render this process a challenging task. In this paper, we introduce DeepBranchTracer, a novel method that learns both external image features and internal geometric characteristics to reconstruct curvilinear structures. Firstly, we formulate the curvilinear structures extraction as a geometric attribute estimation problem. Then, a curvilinear structure feature learning network is designed to extract essential branch attributes, including the image features of centerline and boundary, and the geometric features of direction and radius. Finally, utilizing a multi-feature fusion tracing strategy, our model iteratively traces the entire branch by integrating the extracted image and geometric features. We extensively evaluated our model on both 2D and 3D datasets, demonstrating its superior performance over existing segmentation and reconstruction methods in terms of accuracy and continuity.

Auxetic dihedral Escher tessellations

The auxetic structure demonstrates an unconventional deployable mechanism, expanding in transverse directions while being stretched longitudinally (exhibiting a negative Poisson’s ratio). This characteristic offers advantages in diverse fields such as structural engineering, flexible electronics, and medicine. The rotating (semi-)rigid structure, as a typical auxetic structure, has been introduced into the field of computer-aided design because of its well-defined motion patterns. These structures find application as deployable structures in various endeavors aiming to approximate and rapidly fabricate doubly-curved surfaces, thereby mitigating the challenges associated with their production and transportation. Nevertheless, prior designs relying on basic geometric elements primarily concentrate on exploring the inherent nature of the structure and often lack aesthetic appeal. To address this limitation, we propose a novel design and generation method inspired by dihedral Escher tessellations. By introducing a new metric function, we achieve efficient evaluation of shape deployability as well as filtering of tessellations, followed by a two-step deformation and edge-deployability optimization process to ensure compliance with deployability constraints while preserving semantic meanings. Furthermore, we optimize the shape through physical simulation to guarantee deployability in actual manufacturing and control Poisson’s ratio to a certain extent. Our method yields structures that are both semantically meaningful and aesthetically pleasing, showcasing promising potential for auxetic applications.

Curve-Surface Pairs on Embedded Surfaces and Involute D-Scroll of the Curve-Surface Pair in E3

Willmore defined embedded surfaces on f:S→E3, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of E3, inducing a Riemannian structure on f(S). The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature H2 on f(S). Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the D∼-scroll of the curve–surface pair and involute D∼-scroll of the curve–surface pair with some differential geometric elements and found D∼(α,M)(s) and D∗∼(α,M)(s)-scrolls of the curve–surface pair (α,M).

Bayesian analysis of uncertainties in circle, straight-line and ellipse fitting considering a-priori knowledge − comparative analysis with total-least-squares approaches

Fitting standard geometric elements into measurement data using Least-Squares techniques is a common task in signal processing across various technical applications. However, the application of these well-established but purely data-based methods does not consider potentially available prior knowledge about the measurand of interest or the measuring device. Thus, up to this day, additional information is usually left unused beyond a few academic applications. By applying a Bayesian approach, this prior knowledge can be incorporated into the fitting task, potentially leading to a reduction in overall uncertainty and fragility of the evaluation result. In this study, Bayesian models are proposed for incorporating prior knowledge into circular, linear, and ellipse fitting tasks. The general approaches as well as specific results are compared to the established Total-Least-Squares method within the example of the application of the F-operator in surface texture measurement illustrating the practical benefits of the approach.

Modified Reliability Theory for Speed-Based Evaluation of Successive Geometric Elements

Modified Reliability Theory for Speed-Based Evaluation of Successive Geometric Elements

A Numerical Study of Blade Geometry Effects in a Vertical-Axes Wind Turbines

Several geometrical elements influence the aerodynamic properties of the Darrieus vertical axis wind turbines (VAWTs). Many extant studies have examined properties, such as solidity, pitching axis position (x/c), length of chord (c), blade quantity (N), diameter (d) of the rotor, and aspect ratio. However, not many have examined the shape of the airfoil (AF), which is a vital property that remains to be thoroughly investigated. Therefore, this present study used computational fluid dynamics (CFD) to investigate many airfoils blade characteristics, such as blade thickness (BT), maximum camber ratio (MCR), MCR location (MCRL), and air speed (AS), to determine their impact on VAWT performance. The results demonstrate a blade thickness BT of 10 to 12%, MCR of 0 to 22%, and MCRL of 24 to 23% yield a comparatively high coefficient of power, adequate optimal blade rotation to airspeed ratio (TSR), broader operational area, and high band efficiency while air velocities of 15 to 10% yield a comparatively higher power coefficient.

The mathematics on classical ballet: connections between the beautiful and the exact

The article addresses the intrinsic relationship between classical ballet and mathematics, exploring the presence of geometric elements and symmetry in the techniques and movements of this form of dance. The research aims to explore the potential use of the golden ratio and mathematical modelling to analyze patterns, proportions, and understand the underlying physical principles of the dancers' leaps and turns. This qualitative and applied research, based on bibliographical analysis and a case study, provides a profound insight into the interaction between art and science, demonstrating how mathematics translates into beauty and harmony in classical ballet. These conclusions enrich the understanding of the interconnection between seemingly distinct disciplines, highlighting mathematics as a form of artistic expression and classical ballet as a means of mathematical expression.

Geometric Analysis of the Proposed Al-Baghdadi Dam Reservoir, Anbar Governorate, Iraq

This study included conducting a geometric analysis of the reservoir of the proposed Al-Baghdadi Dam in the downstream area of the Haditha City northwestern part of Anbar Governorate. The digital elevation model with an accuracy of 10 × 10 m per pixel area used to extract the geometric elements by the ArcGIS, Global Mapper, and Surfer software, which are included: negative volume (storage volume), positive volume (size of islands), negative surface area (uneven area of the bottom of the reservoir), and area Positive surface (uneven area of islands), negative flat area (water surface area), positive flat area (area of the projection of islands), average depth of reservoir, and average thickness of islands at each hypothetical level from 77.5 to 106 ms and with a contour interval of 0.5 m above sea level. The relationship between the geometric elements and the water level was used to clarify the extent of variation of the values of these elements with the water level fluctuation, and to delineate the area that will be subjected to immersion in the event of the construction of the dam and to determine the optimal and safest operating level. In contrast, it produces less submerged areas and less contact with the rock exposures of the formation in the study area.

WURRANDERRA’S SYMBOLS: AN EXPLORATION AND CONTEXTUALISATION OF THE THURK PETROGLYPH SITE (KINGSTON-ON-MURRAY) ON THE MURRAY RIVER, SOUTH AUSTRALIA

This article describes and contextualises the rock art at the Thurk Petroglyph Site on the Murray River in South Australia using multiple methods. The Thurk Aboriginal engravings comprise at least 524 motifs made up predominantly of geometric line elements as well as a small number of other ‘simple’ geometric motifs, two ‘bird tracks’, one figurative design (a ‘fish’) and a possible anthropomorphous figure. This paper provides the first synthesis of rock art sites/complexes and motifs from other sites on the Murray River as well as visual symbols recorded from senior Aboriginal ‘knowledge carriers’. These syntheses allow us to consider the relationship of Thurk to other cultural places and to highlight and honour the traditional knowledges and beliefs which underpin the rock art. Thurk’s placement within the riverscape, its unique geological canvas, lack of observable ‘domestic’ archaeological evidence combined with it being the likely upstream extent of Murray River rock art in South Australia contribute additional dimensions to its cultural significance. That Thurk’s rock art, and the broader site, have been desecrated by gratuitous graffiti, vandalism and infrastructure brings into sharp focus Australia’s poor record of heritage protection and provides a challenge to current and future generations of non-Aboriginal people to remedy this past.

An optical needle with elongated transversal profile created using Airy beams for laser processing of glasses

In recent decades, significant progress has been made in the field of laser beam shaping, driven by the growing demands in both scientific research and industrial applications. Notably, a class of beams known as nondiffracting beams, which exhibit such characteristics as self-healing and resistance to diffraction, has found practical utility in communication, imaging, and laser microprocessing. Although Bessel beams have become commonplace for various optical applications, further advances in beam engineering remain a need. One of the key requirements is the ability to control the elongation of the transverse profile of an optical needle, which is highly sought after for specific applications. In this study, we propose to achieve this control by utilizing nondiffracting Airy beams. The elongation of the optical needle’s transverse profile is achieved through the implementation of a binary phase mask. We evaluate the performance of this method through both numerical simulations and experimental assessments, employing various metrics, such as the major-to-minor axis ratio, the length of the optical needle, and its stability. We present a specific case where a flat optical element is created using geometrical phase induced by nanogratings inscribed within the element’s volume by means of a femtosecond laser. We validate the performance of this geometrical phase element through laser microprocessing of transparent glasses and report the results obtained from these experiments.

Exploring The Intersection of Geometry and Ballet

Ballet originated in the courts of the Italian Renaissance in the 15th century is still a popular dance art today. It is characterized by diversity, and the expression of emotion varies from situation to situation. Geometry plays a crucial role in ballet formation, human form, and movement proportions. This paper focuses on the relationship between ballet and geometry, deeply discusses how geometric principles such as line, triangle, circle, symmetry and asymmetry are integrated to ballet choreography and performance. By analyzing ballets such as Swan Lake and Giselle, this paper highlights how dancers use geometric concepts to create visually compelling formations that emphasize harmony and unity. These geometric elements are shown to enhance the precision, beauty, and storytelling in ballet performances.

The Use of Transition Curves in Airport Runway Rapid Exit Taxiways (RETs)

This paper describes a study result focused on the transition curves used in airport rapid exit taxiways (RETs). Such geometric elements are not currently made explicit by standards, and major international government agencies, such as the International Civil Aviation Organization and the Federal Aviation Administration, are limited to providing some layouts consisting of only constant curvature elements, although the aircrafts speed in RETs is variable. In addition to the clothoid that has unitary shape parameter “n” and the multiparameter clothoids (generalized Cornu spirals), the study analyzes the Bloss curve employment. Regarding the multiparameter clothoids, four schemes with shape parameter equal to 1.25, 1.50, 1.75, and 2.00 were considered, respectively. The proposed curves’ effectiveness was evaluated by analyzing the main geometric and kinematic variables trend, that is, curvature, transversal acceleration, lateral jerk, speed, travel time, longitudinal deceleration, and centrifugal force. To ensure the analysis effectiveness, the study layouts were designed from those used for runways codes 1‐2 and 3‐4 maintaining the main geometric elements dimensions unchanged (length of straight lines, radii of circular curves, exit angles, etc.). The kinematic variables values along the design RETs were calculated “point by point” by MATLAB software to take into account nonuniform motion regimen. The obtained results show that both the Bloss curve and the multiparameter clothoid having a shape factor equal to 2 are capable of better contribution in terms of safety and comfort. Implications in the airport construction field are related to the possibility of using lower deceleration values compared to current standards, that is, allowing higher exit speed for the same deceleration. This allows aircrafts to vacate the runway more quickly and, therefore, to increase runway capacity without taking expensive RETs repositioning.