Principles Of Geometry Research Articles

Autonomous collision avoidance system for unmanned vessels: algorithms and software.

Ship collision is one of the most substantial concerns in the global maritime transportation industry. Hence, navigation safety is considerably cited topic in maritime literature. Recently, Unmanned Navigation (UN) technology is gradually becoming more widely used across in the field of marine robotics. The paper investigates the problem of navigation safety in the movement control of Unmanned Vessels (UVs). The object of the study is the process of avoiding collisions of UVs. The subject of the research is the algorithms of the autonomous decision-making system and software for preventing vessel collisions during UN. The intent of this article is to improve the safety of UN by developing new Decision-Making algorithms for autonomous collision prevention of UVs in real time, taking into account the International Rules for the Prevention of Collisions at Sea, 1972 (COLREGs-72) and the recommendations of the Federal agency for sea and inland water transport of the Russian Federation (Rosmorrechflot).In this article, the fundamental concept and the key functions set of the Autonomous Collision Avoidance System (ACAS) are carried out for UVs which are marine transport vehicles capable of sensing its environment and operating without human involvement. Along this line of research, this work focuses on the development of a software algorithm for determining the most dangerous obstacle located within a radius of 12 miles (recommendations of Rosmorrechflot) around an UV based on the principle of vessels collision avoidance geometry, collision risk assessment and the characteristics of obstacles. Moreover, the proposed algorithms can prevent the collision and address the issues of real-time collision avoidance for UVs. The simulation results also demonstrate the promising application of the proposed algorithms in studying the UN safety. Nonetheless, this study provides a way forward to conduct a new information decision-making system design for UVs collision avoidance. This is currently under development, and will be proposed later.

Fractal Geometry in High-Frequency Trading: Modeling Market Microstructure and Price Dynamics

This theoretical article delves into the intricate world of high-frequency trading (HFT) without empirical testing of real-world data, focusing on the incorporation of fractal geometry principles to enhance our understanding of market microstructure and price dynamics. In the introduction, we outline the significance of this research in the context of modern financial markets and lay out the objectives of our theoretical analysis. The article then takes an in-depth dive into fractal geometry fundamentals, illuminating its core concepts and its relevance within financial markets. Subsequently, the article explores the landscape of high-frequency trading, offering an overview of this dynamic domain and how fractal geometry can be incorporated into trading models. The section on modeling market microstructure presents theoretical approaches to understanding order flow dynamics, including novel derivations and equations. It then transitions into fractal-based approaches for analyzing the complexities of market microstructure, providing both an original perspective and numbered equations. Moreover, this article investigates the theoretical modeling of price dynamics, underscoring the pivotal role of fractal geometry in enriching these models. The discussion revolves around the fundamental autoregressive models and multifractal models, and it elucidates how fractal geometry principles, such as the Hurst exponent, come into play. We explore the self- similarity of price dynamics, fractal dimensions, and how these aspects can be integrated into high-frequency trading strategies. Overall, this article offers a comprehensive theoretical exploration of fractal geometry's implications in the realm of high-frequency trading, providing valuable insights for both researchers and practitioners seeking to fathom the complexities of market microstructure and price dynamics. The incorporation of fractal principles into financial models fosters a deeper understanding of self-similarity and complexity within financial markets, even in the absence of empirical data.

Pole Estimation and Optical Navigation Using Circle of Latitude Projections

Images of both rotating celestial bodies (e.g., asteroids) and spheroidal planets with banded atmospheres (e.g., Jupiter) can contain features that are well-modeled as a circle of latitude (CoL). At large distances, the projections of these CoLs appear as ellipses in images collected by cameras or telescopes onboard exploration spacecraft. This work shows how CoL projections may be used to determine the pole orientation and covariance for a spinning asteroid. In the case of a known planet modeled as an oblate spheroid, it is shown how similar CoL projections may be used for spacecraft localization. These methods are developed using the principles of projective geometry. Numerical results are provided for simulated images of asteroid Bennu (for pole orientation) and of Jupiter (for spacecraft localization).

How the Game of Soccer can Foster Creative Research in Natural Sciences

The systems science approach based on exploration of analogies between distinct fields of inquiry can lead to the creation of new interdisciplinary branches of knowledge. In this essay, a series of such analogies between soccer and natural sciences is elaborated with the objective of recognizing the scientific elements intrinsic to the game of soccer and transcribing them to the creative practice of science. The sources of analogies are multifold, involving the tactical setups of the teamwork, the geometry and cybernetic principles governing the coordination of the players, the kinesthetic intelligence thereof and various psychological factors. The essay also intends to inspire the crafters of academic curricula to consider building courses around the intersections of a sport such as soccer and natural sciences so as to promote sportier and more spirited scientific communities. Along with the humanization of sciences via their cross-fertilization with soccer, the enlightenment of the latter profession may also result from this interdisciplinary marriage, so that the animalistic competitiveness in this sport gets suppressed and its artistic elements accentuated. Keywords: Academic culture, analogical reasoning, art, association football, science.

NUMERICAL SIMULATION OF PASSIVE MICROMIXER WITH NOVEL OBSTACLE STRUCTURE

The micromixer is an important prerequisite for realizing the function of the microfluidic chip system. Micromixers are mainly used in protein crystallization, biological reactions, drug delivery and other fields. Efficient micromixers enable fast bioreaction processes. In this paper, according to the principle of plane geometry, micromixers with three different structural obstacles, circular, equilateral triangle and square, are designed to study how to improve the mixing efficiency of the micromixer. In this paper, the influence of the obstacle structure on the mixing efficiency and pressure drop of the micromixer is analyzed. Through numerical simulation, the results show that when Re = 100, the mixing efficiency of the circular obstacle micromixer is up to 97.4%, and the pressure drop is 15,789.597 Pa. In addition, this paper analyzes that the use of cylindrical obstacles can make the fluid generate horizontal and vertical vortex flow, break the usual laminar flow, and significantly improve the mixing efficiency.

Investigation of loaded contact characteristics of planetary roller screw mechanism based on influence coefficient method and machine learning

The performance of the planetary roller screw mechanism (PRSM) is directly determined by the mechanical behavior of the loaded contact. The traditional method based on a finite element modeling to describe the loaded contact characteristics (LCCs) is computationally expensive, while the use of Hertzian contact theory and the thread connection model results in a loss of computational accuracy. As a solution, this study aims to develop a semi-analytic modeling technique that can predict the LCCs of PRSM accurately and efficiently. To achieve this, the study first establishes a mathematical model using the spatial coordinate transformation theory to accurately describe the thread surface. The potential contact area is discretely divided using the differential geometry principle. Then, the linear deformation and nonlinear deformation are matched using node interpolation technique to obtain the accurate compliance of the potential contact area. Finally, the contact forces for all activated contact points are derived by a nonlinear iterative algorithm based on the influence coefficient method. The performance of the proposed model is verified by numerical examples. Further, the sensitivity weights of the geometric parameters to the LCCs are identified utilizing machine learning, and a preferred zone is also constructed to optimize these parameters. This study provides a valuable tool for evaluating and improving the LCCs of PRSM in engineering practice.

Log-aesthetic curves: Similarity geometry, integrable discretization and variational principles

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. In the framework of similarity geometry, those curves are characterized as invariant curves under the integrable flow on plane curves governed by the Burgers equation. They also admit a variational formulation leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formulation, we propose a discretization of these curves and the associated variational principle which preserves the underlying integrable structure. We finally present algorithms for generating discrete log-aesthetic curves for given G1 data based on similarity geometry. Our method is able to generate S-shaped discrete curves with an inflection as well as C-shaped curves according to the boundary condition. The resulting discrete curves are regarded as self-adaptive discretization and thus high-quality even with the small number of points. Through the continuous representation, those discrete curves provide a flexible tool for the generation of high-quality shapes.

Derived from the traditional principles of Islamic geometry, a methodology for generating non-periodic long-range sequences in one-dimension for 8-fold, 10-fold, and 12-fold rotational symmetries

This paper explores the use of traditional principles of Islamic geometry to generate the long-range order of different non-period sequences in one-dimension. The discovery of quasicrystals with forbidden long-range order has uncovered a range of puzzling non-periodic symmetries, triggering a variety of investigations into understanding their complicated structural logic. The discovery of historical patterns in Islamic architecture with similar quasi-periodic formations is providing new insights into understanding the mathematical properties of these complicated formations. This paper argues that the hierarchical self-similar logic underlying the construction process of the historical patterns can serve as the key ingredient for resolving the long-range order of the one-dimensional non-periodic sequences. A case study approach is employed to test the hypotheses. Three historical patterns with non-period symmetries were tested: 10-fold rotational symmetry, 8-fold rotational symmetry and 12-fold rotational symmetry. The results show that the methodology is successful in generating three different non-periodic sequences in one-dimension.

An effective stereo SLAM with high-level primitives in underwater environment

Visual simultaneous localization and mapping (SLAM) algorithms face challenges in complex underwater scenarios, such as turbidity, dynamism, and low texture, where point features are unreliable and can lead to weakened or even failed systems. To overcome these issues, high-level object features are considered due to their accuracy and robustness. In this paper, we introduce an effective object-level SLAM method that employs a stereo camera to enhance the navigation robustness of autonomous underwater vehicles and generates a detailed semantic map. Point features and object features are integrated to serve the proposed approach. We begin by detecting 2D objects in images using a state-of-the-art neural network, followed by obtaining 3D objects described by the general model through the principle of multi-view geometry and eventually constructing semantic landmarks. To account for object data association, we present an object match method that takes into consideration the stereo camera characteristics in a single stereo frame and a filter-based approach to track the landmarks in odometry. Experiments are also conducted using the KITTI dataset and our sequences collected from the pool and coast. The evaluation results indicate that the proposed method can improve the performance of ORBSLAM2 in terms of both navigation robustness and mapping information in underwater scenarios.

Point clouds segmentation of rapeseed siliques based on sparse-dense point clouds mapping.

In this study, we propose a high-throughput and low-cost automatic detection method based on deep learning to replace the inefficient manual counting of rapeseed siliques. First, a video is captured with a smartphone around the rapeseed plants in the silique stage. Feature point detection and matching based on SIFT operators are applied to the extracted video frames, and sparse point clouds are recovered using epipolar geometry and triangulation principles. The depth map is obtained by calculating the disparity of the matched images, and the dense point cloud is fused. The plant model of the whole rapeseed plant in the silique stage is reconstructed based on the structure-from-motion (SfM) algorithm, and the background is removed by using the passthrough filter. The downsampled 3D point cloud data is processed by the DGCNN network, and the point cloud is divided into two categories: sparse rapeseed canopy siliques and rapeseed stems. The sparse canopy siliques are then segmented from the original whole rapeseed siliques point cloud using the sparse-dense point cloud mapping method, which can effectively save running time and improve efficiency. Finally, Euclidean clustering segmentation is performed on the rapeseed canopy siliques, and the RANSAC algorithm is used to perform line segmentation on the connected siliques after clustering, obtaining the three-dimensional spatial position of each silique and counting the number of siliques. The proposed method was applied to identify 1457 siliques from 12 rapeseed plants, and the experimental results showed a recognition accuracy greater than 97.80%. The proposed method achieved good results in rapeseed silique recognition and provided a useful example for the application of deep learning networks in dense 3D point cloud segmentation.

Analytical Solution and Buckling of Hemi-Ellipsoidal Shell Structures of Revolution under Uniformly Distributed Load

This paper focuses on the analytical procedure for determining novel exact expressions for internal forces and displacements of hemi-ellipsoidal shells formed as an axisymmetric shell of revolution under uniformly distributed load such as imposed loads. The simplest form of expressions for solutions is derived based on the linear membrane theory under symmetrical loading that is formed as a shell of revolution. The results have been validated and have a good consistency with numerical solutions from the finite-element method (FEM) which are derived based on the principle of virtual work and differential geometry. The obtained analytical exact solution is only valid for small displacements or if the response does not exceed the linearity limit. In cases of large displacements, geometrical nonlinear finite-element analysis is recommended to determine the solution. The linearity limit determination is demonstrated, and the effects of shells’ geometry, thickness, and magnitude of applied loads are presented. Additionally, the linear buckling analysis has been performed. The study found that the size ratio, thickness, and support condition have a significant effect on the critical load of the first mode, and the hemispherical shells have the highest buckling resistance due to the geometry.

ANALYSIS OF STUDENTS' DIFFICULTIES IN LEARNING MATHEMATICS OF CIRCLE GEOMETRY IN FOURTH GRADE ELEMENTARY SCHOOL

The present study endeavors to comprehensively examine the obstacles faced by elementary school students when attempting to grasp fundamental mathematical concepts pertaining to circles. Specifically focusing on Grade 4 students at SDN Cipondoh 2, Tangerang City. The primary objective is to shed light on the difficulties encountered by students in three core areas: calculating circle area and circumference, understanding the complex concept of π (pi), and accurately sketching circles utilizing a compass. Various tools are employed to scrutinize the obstacles students encounter, including tasks such as calculating circle area and circumference, comprehending the intricate π (pi) concept, and accurately sketching circles using a compass. The findings indicate that students encounter hurdles in computing circle area and circumference due to the need to understand underlying formulas and calculation procedures. Additionally, comprehending the π (pi) concept presents a notable challenge in circle calculations. Moreover, students struggle with precise and symmetrical circle drawing using a compass. The research outcomes provide valuable insights for formulating effective pedagogical strategies to help students overcome challenges in comprehending and applying circle geometry principles.

A Novel Electromagnetic Positioning Prototype System With Simplified Receiver for Interventional Surgery Application

To meet the need of 3D positioning of surgical instru-ment in interventional surgery and the requirement of smaller size sensor for the narrow blood vessels, a new electromagnetic posi-tioning model is proposed with simplified receiver. Based on the electromagnetic theory and geometry principle, the electromag-netic field transmitter with groups of three orthogonal coils and the simplified receiver with smaller size than existing sensors are designed. Then, using Biot-Savart Law, the distance between re-ceiving end and geometric center of orthogonal coils is calculated, and the spatial coordinate of receiving end is computed with the spherical intersection formula. To further reduce the positioning error, a two-round accuracy improvement algorithm is designed for selecting the optimal topology of coil groups at the transmitter end and fitting correction of the calculated distance. We imple-ment the prototype system and perform both simulation experi-ments and actual experiments. The results show that our proposed new electromagnetic positioning method with simplified receiver has higher accuracy and stronger stability, compared with the tra-ditional positioning approach of three-coil transmitter and three-coil receiver.

Gravitation, and quantum theory, as emergent phenomena

There must exist a reformulation of quantum field theory, even at low energies, which does not depend on classical time. The octonionic theory proposes such a reformulation, leading to a pre-quantum pre-spacetime theory. The ingredients for constructing such a theory, which is also a unification of the standard model with gravitation, are : (i) the pre-quantum theory of trace dynamics – a matrix-valued Lagrangian dynamics, (ii) the spectral action principle of non-commutative geometry, (iii) the number system known as the octonions, for constructing a non-commutative manifold and for defining elementary particles via Clifford algebras, (iv) a Lagrangian with E 8 × E 8 symmetry. The split bioctonions define a sixteen dimensional space (with left-right symmetry) whose geometry (evolving in Connes time) relates to the four known fundamental forces, while predicting two new forces, SU(3) grav and U(1) grav . This latter interaction is possibly the theoretical origin of MOND. Coupling constants of the standard model result from left-right symmetry breaking, and their values are theoretically determined by the characteristic equation of the exceptional Jordan algebra of the octonions. The quantum-to-classical transition, precipitated by the entanglement of a critical number of fermions, is responsible for the emergence of classical spacetime, and also for the familiar formulation of quantum theory on a spacetime background.

Великая теорема Ферма с точки зрения физика

The article is about the fact that the extraordinary beauty and conciseness of the formulation of Fermat's Last Theorem make us look for its visual solution. Let's try to consider Fermat's theorem from the eyes of physicist. Perhaps from this positions Pierre de Fermat found a solution whose main ideas would fit schematically in the fairly wide margins of the book, in a few drawings. Skeptics continue to believe that Pierre de Fermat was probably mistaken. Meanwhile, consistent application of the basic principles of physics, geometry, and engineering make us think differently.

Algorithmic Color Methods of Media Arts

The paper describes a framework for systematically exploring the RGB color model (RGB color cube) for generating color palettes based on six algorithmic methods divided into two categories: Primitive and Derivative. The Primitive method is comprised of the RGB Linear method, the RGB Planar method, and the RGB Cuboid method. The Derivative method includes the RGB Multi-Linear method, the RGB Multi-Planar method, and the RGB Multi-Cuboid method. On a practical level, the goal of the paper is to introduce a systems approach to the notion of creating and applying algorithms to create a variety of color combinations and palettes in the field of Media Arts. The framework was implemented as a color design approach to the process of producing color palettes in the context of Media Arts while exploring the intersection of mathematical principles of geometry, color, and code. An investigation was conducted into how algorithmic color palettes generated from the methods correlate to culture, history, and other artistic practices. An application of the methodology and a discussion about how the framework can continue to evolve and be applied to other research areas and fields of applications, such as digital color literacy education, are presented.

프랙탈 기하학 차원에서 본 자하 하디드 건축작품의 공간형태특성에 관한 연구

This study utilizes fractal geometry, which is evaluated as a new alternative in the field of science, to interpret and apply complex and diverse architectural phenomena in the field of architectural design to create multi-dimensional space forms and respond to users' unpredictable and flexible demands. We want to explore the possibilities of whether it can be used with various algorithms that can be used.
 In this process, we tried to understand the concept, characteristics, and compositional principles of fractal geometry, and to interpret the creation process of spatial forms in Zaha Hadid's architectural works erected after the 2000s from the perspective of fractal geometry. In addition, we tried to explore the possibility of application to contemporary architecture by proposing the aspects of the architect's fractal characteristics through diagrams and analyzing the proportion of utilization of various algorithms to propose them as a mechanism for creating spatial forms.

Effect of wet ball milling on copper ore flotation by fractal geometry

ABSTRACT One of the important parameters in mineral processing is particles morphology, which is a function of the comminution mechanism. Particles morphology has considerable impacts in flotation process. Variations in particles morphology have been studied by different methods, including fractal geometry. This study aims to investigate the ball milling effect on particles flotation using fractal dimension, in which the calculations in the particles morphology has been made using the step-length method. The ore sample of under-investigation was a copper ore. The examined parameters include grinding time, ball load, and solid weight percentage. The collected fractal dimensions for particles ranged from 1 to 2, which is according to the principle of fractal geometry. Based on the results, low (30 minute) or high (50 minute) grinding time reduces fractal dimension and recovery because low grinding time is resulted in a coarse grain product while high grinding time is resulted in fine particles with spherical shapes. If balls load is high (8710 gr), predominant comminution force turns to an abrasive force are resulted in fine and spherical particles due to an increase in the accumulation volume. When the ball load decreased (4690 gr), the coarse particles were produced. Fractal dimensions and recovery are decreased in both cases. Low (40%) and high (60%) solid weight percentages are resulted in ineffective comminution, producing a coarse grain product. As a result, fractal dimensions and recovery are decreased. There is a direct correlation between recovery and fractal dimension.

Subscripto multiplex: A Riemannian symmetric positive definite strategy for offline signature verification

The human handwritten signature is considered to be a significant biometric trait. In the case of offline signatures, the problem is addressed as an image recognition task. On the other hand, the visual representation of symmetric positive definitive matrices, usually by means of the covariance descriptor of the image feature maps, forms a specific Riemannian manifold with a widespread usage and a favorable performance in a plethora of applications. Surprisingly, no records of offline-signature-verification-oriented research in the space of symmetric positive definitive matrix have been found up to now. In this work, we propose, for the first time in offline signature-verification literature, mapping of handwritten signature images in points of the tangent space of a connected symmetric positive definitive manifold for verification purposes. Furthermore, based on the principles of differential geometry, we address the notorious limited training problem of offline signature verification in this manifold by proposing two different feature augmentation methods. The efficiency of the proposed method is evaluated using three popular datasets of Western and Asian origin. Error rates against skilled and random forgery in both baselines as well augmentation scenarios are strong indicators of the informative and highly discriminative nature of symmetric positive definitive manifold oriented representation.

A Research on Autonomous Collision Avoidance under the Constraint of COLREGs

In this paper, a decision-making model suitable for the collision avoidance (CA) of numerous target ships (TSs) is proposed, based on the principle of ship collision avoidance geometry and the characteristics of numerous target ships’ collision avoidance at sea. To ensure that the collision avoidance behaviors of own-ship (OS) are subject to the International Regulations for Preventing Collisions at Sea (COLREGS), this paper gives full consideration to the requirements of COLREGS within the scope of CA action and the time of collision avoidance. A ship CA simulation is established based on the Mathematical Modeling Group (MMG) model. To optimize the CA decision-making model, the influence of hydrodynamic force on steering time required to reach the new course is integrated into the collision avoidance simulation system. The simulation results show that the method can quickly and effectively determine a collision avoidance decision under the complex situation of numerous target ships and static obstacles, and it can consider the unpredictable strategies used by other vessels.