Perspective Of Geometry Research Articles

Unified Attitude Determination Problem From Vector Observations and Hand–Eye Measurements

The hand-eye measurements have recently been proven to be very efficient for spacecraft attitude determination relative to an ellipsoidal asteroid. However, recent method does not guarantee full attitude observability for all conditions. This paper refines this problem by taking the vector observations into account so that the accuracy and robustness of the spacecraft attitude estimation can be improved. The vector observations come from many sources including visual perspective geometry, optical navigation and point clouds that frequently occur in aerospace electronic systems. Completely closed-form solutions along with their uncertainty descriptions are presented for the proposed problem. Experiments using our simulated dataset and real-world spacecraft measurements from NASA dawn spacecraft verify the effectiveness and superiority of the derived solution.

Unweyling Three Mysteries of Nordström Gravity

The paper re-examines Nordström's scalar theory of gravity (NG) – arguably the most convincing relativistic theory of gravity before the advent of General Relativity. It exists in two different formulations. In Nordström's original one (1913), NG appears to describe a scalar gravitational field on Minkowski spacetime. In Einstein and Fokker’s (1914) version, NG seems to be a spacetime theory: It reconceptualises gravitational effects as manifestations of non-Minkowskian inertial structure. Both variants of NG give rise to three contradictory verdicts on the status and validity of fundamental principles: the Weak Equivalence Principle, the existence of gravitational energy, and energy conservation. Given the putative equivalence of both variants of NG, this ambiguity seems paradoxical to the spacetime realist. I'll proffer a resolution from the perspective of integrable Weyl geometry: The paradoxes rest on the failure to recognise a more apposite spacetime setting for NG.

On energetic evaluation of robotic belt grinding mechanisms based on single spherical abrasive grain model

A tentative study from the perspective of abrasive grain geometry in this paper is conducted to investigate the specific energy and energy efficiency for clarifying the robotic belt grinding mechanisms. The energy efficiency model is established based on the friction coefficient model of the single spherical grain, then the experiments and simulation are implemented to energetically evaluate the microscale material removal mechanisms from the specific energy contributions. It has been demonstrated that the specific plowing energy is more predominant than both the specific cutting and sliding energy in robotic belt grinding, resulting in the energy efficiency ranges between 17 and 41 %. Both the large grain size and normal contact force can be taken as optimization strategies to maximize the energy efficiency for material removal.

Efficient topview person detector using point based transformation and lookup table

Nowadays due to big data revolution, image analytics is seen as a potential solution to solve different real-life problems. In this regard, one of the applications could be a person detection system. The overhead mounted camera with a wide-angle lens gives more coverage and visibility in occluded and cluttered environments than a traditional or frontal view. Person detection from the top view is a challenging task because there is variation in position, orientation, poses, body articulation and appearance of a person depending upon the position in the scene. To handle these issues an efficient method is proposed that uses different geometric transformations, concepts of perspective geometry and homography matrix as a pre-processing step. The composite transformation matrix is then used with perspective transformations to standardized the shape of the image patch containing person. At this stage, the histogram of oriented features is extracted along with our proposed five additional spatial features. These features are then fed to a linear SVM classifier for training and testing and finally, a simple effective clustering process is used to accumulate the votes of SVM to render a decision for localization of the Person in the image. To reduce the computational cost of these points based geometric and perspective transformations a Lookup table structure is used which contains pre-calculated positions of different perspective points against spatial co-ordinates. The use of lookup table and point based operations significantly reduced processing time up to 50% as compared to the previous approach which uses RHOG algorithm with image-based rotations, transformations, interpolations, and no lookup structure. The proposed method is efficient both in terms of computation and accuracy. The performance of the proposed algorithm is tested using our newly recorded dataset having more wide coverage of the scene. The performance of the developed technique shows an accuracy of 98% TDR with 3% FDR.

Road network generalization considering traffic flow patterns

ABSTRACT As one of the major concerns in cartographic generalization, road network generalization aims at maintaining the patterns of road networks across map scales. Previous methods define the pattern of road networks mainly from the perspectives of geometry and topology. However, for navigation purposes, traffic flow information is also important to generalize road networks. More specifically, road segments that have a proximity relationship in the traffic flow system should be retained together on small-scale maps to preserve the completeness of the driving route. In this regard, this study proposes an improved method for road network generalization that considers network geometry, topology, and traffic flow patterns. First, strokes are constructed from the road network data based on the ‘every best fit’ geometric principle. Then, the relationships among strokes are developed on the basis of traffic flow patterns, which are extracted from taxi trajectory data. The strokes are then selected in sequence based on the indicators of geometry, topology, and traffic flow. Our experimental results demonstrate that the proposed method can preserve both the ‘Good Continuity’ principle and the transport function relationship of roads after generalization.

Comparative Study of Partitioned Stator Memory Machines With Series and Parallel Hybrid PM Configurations

In this paper, the partitioned stator (PS) structure is extended to variable-flux memory machines, forming two newly emerged PS switched-flux memory machines (PS-SFMMs) with series and parallel hybrid magnet configurations. From the perspective of geometry, both two PS-SFMMs share identical outer stator and rotor segments, while two different types of permanent magnet (PM) arrangements are employed in the inner stationary part. Thus, the developed machines can inherent the geometric separation of the armature winding and PM excitations from the PS design, thus achieving acceptable torque capability, and excellent air-gap flux control. A comparative study between PS-SFMMs with series and parallel structures is established. First, the topologies and operating principle are introduced, respectively. In addition, the design tradeoffs and PM sizing of the two PS machines are revealed and optimized with a simplified magnetic circuit model. Then, the electromagnetic characteristics of PS-SFMMs with different magnetic circuits are investigated and compared with the finite-element (FE) method. The FE results are validated by the experiments on a parallel prototype.

Application Research of Design Geometry in the Design of High Speed Trains

The visual characteristics of high-speed trains in China are comparatively not at an advantage, while vehicle serialization and product identification remains to be a major problem. From the perspective of design geometry, This paper analyzes the mathematical composition of modeling curves in foreign excellent models, summarizes the geometric proportional relationship of the vehicle system, innovates a design and operation method of the vehicle body shape, and establishes a new generation of model design based on the geometric law of the line shape of vehicle body so as to enhance the viability of vehicle serialization, and provide a basic premise for design strategy research.

Non-Euclidean Geometry and Defected Structure for Bodies with Variable Material Composition

In the paper the relationship between pure geometrical concepts of the theory of affine connections, physical concepts related with non-linear theory of distributed defects and concepts of non-linear continuum mechanics for bodies with variable material composition is discussed. Distinguishing feature of the bodies with variable material composition is that their global reference shapes can not be embedded into Euclidean space and have to be represented as smooth manifolds with specific (material) connection and metric. The method for their synthesis based on the modeling of additive process are proposed. It involves specific boundary problem referred to as evolutionary problem. The statement of such problem as well as illustrative exact solutions for it are obtained. Because non-Euclidean connection is rarely used in continuum mechanics, it is illustrated from the perspective of differential geometry as well as from the point of view, adopted in the theory of finite incompatible deformations. In order to compare formal structures defined within the models of solids with variable material composition with their counterpart in non-linear theory of distributed defects, a brief sketch for latter is given. The examples for cylindrical and spherical non-linear problems are presented. The correspondences between geometrical structures that defines material connection, fields of related defect densities and evolutionary problems for bodies with variable material composition are shown.

Measuring road network topology vulnerability by Ricci curvature

Describing the basic properties of road network systems, such as their robustness, vulnerability, and reliability, has been a very important research topic in the field of urban transportation. Current research mainly uses several statistical indicators of complex networks to analyze the road network systems. However, these methods are essentially node-based. These node-based methods pay more attention to the number of connections between nodes, and lack of consideration for interactions, leading to the well-known node paradox problem, and their ability of characterizing the local and intrinsic properties of a network is weak. From the perspective of network intrinsic geometry, we propose a method for measuring road network vulnerability using a discrete Ricci curvature, which can identify the key sections of a road network and indicate its fragile elements. The results show that our method performs better than complex network statistics on measuring the vulnerability of a road network. Additionally, it can characterize the evolution of the road network vulnerability among different periods of time in the same city through our method. Finally, we compare our method with the previous method of centrality and show the different between them. This research provides a new perspective on a geometry to analyze the vulnerability of a road network and describes the inherent nature of the vulnerability of a road system from a new perspective. It also contributes to enriching the analytical methods of complex road networks.

4-particle amplituhedronics for 3-5 loops

Following the direction of 1712.09990 and 1712.09994, this article continues to excavate more interesting aspects of the 4-particle amplituhedron for a better understanding of the 4-particle integrand of planar N=4 SYM to all loop orders, from the perspective of positive geometry. At 3-loop order, we introduce a much more refined dissection of the amplituhedron to understand its essential structure and maximally simplify its direct calculation, by fully utilizing its symmetry as well as the efficient Mondrian way for reorganizing all contributing pieces. Although significantly improved, this approach immediately encounters its technical bottleneck at 4-loop. Still, we manage to alleviate this difficulty by imitating the traditional (generalized) unitarity cuts, which is to use the so-called positive cuts. Given a basis of dual conformally invariant (DCI) loop integrals, we can figure out the coefficient of each DCI topology using its dlog form via positivity conditions. Explicit examples include all 2+5 non-rung-rule topologies at 4- and 5-loop respectively. These results remarkably agree with previous knowledge, which confirms the validity of amplituhedron up to 5-loop and develops a new approach of determining the coefficient of each distinct DCI loop integral.

ВАРИАТИВНОСТЬ ИМЕН И ЛОКУСОВ БОЖЕСТВ ЯКУТСКОГО ПАНТЕОНА В ИХ ПЕРВООСНОВЕ

В условиях вызовов современной науки в сторону поиска фундаментальных инвариантов общенаучной картины мира, перспективным представляется синтез естественнонаучных построений и древнего мышления, зафиксированном в наследии олонхо. В таком ракурсе, при изучении традиционной культуры в целом, реконструкция стройной структуры якутского пантеона божеств, общепринятая классификация их имен и локусов, на сегодня почти сведено к минимуму. Следует также указать, что в якутской фольклористике остается вне поля исследований момент самого мифотворчества. Нельзя упускать из виду то, что на протяжении многих исторических эпох, передаваемая из поколения в поколение традиционная мифологическая ткань, преумножалась и перерабатывалась творческой активностью сказителей. Последняя же, в своей первичной основе складывалась суммой антропологических и психофизиологических констант, реально-исторических и фольклорно-мифологических нарративов, объективного и субъективного опыта сказителя, и наконец, особенностей внешних факторов в момент исполнения. В интересах повышения объективности исследования в работе предлагается междисциплинарный системный подход, где исходной точкой служит использование гумилевского метода дедукции – своеобразного взгляда «с высоты птичьего полета». Этот метод необходим для более адекватного разрешения проблемы разрозненности имеющихся в распоряжении науки источников. Целью исследования является выявление предпосылок (причин) констатации многовариативных трактовок имен и локусов небожителей пантеона предков. Используются широко распространенные в фольклористике методы обзора, анализа, синтеза, сравнения, а также применяются методы синергетики, экстраполяции, герменевтики. По предположению автора, многовариативность в своем корне исходит из естественной природы человека-сотворца, и для получения объективной картины классификации имен и локусов божеств пантеона предков необходим междисциплинарный синтез на стыке ряда дисциплин. Данная работа является логическим продолжением предыдущих статей автора «Модель мира саха: семантика в ракурсе геометрии форм (на материале якутского эпоса)»; «К вопросу структурной интерпретации локуса глав якутского пантеона в ракурсе геометрии форм». Планируются дальнейшие исследования в данном направлении. In the context of the challenges of modern science towards the search for fundamental invariants of the general scientific picture of the world, the synthesis of natural science constructions and ancient thinking, enshrined in the olonkho heritage, seems promising. In this perspective, when studying the traditional culture in general, the reconstruction of the harmonious structure of the Yakut pantheon of deities, the generally accepted classification of their names and loci, today is almost kept to a minimum. It should also be pointed out that in Yakut folklore studies, the moment of myth creation itself remains outside the field of research. We must not lose sight of the fact that over many historical epochs, the traditional mythological fabric transmitted from generation to generation has been multiplied and processed by the creative activities of narrators. The latter, in its primary basis, formed the sum of anthropological and psychophysiological constants, real-historical and folk-mythological narratives, the objective and subjective experience of the narrator, and finally, the features of external factors at the time of execution. In the interest of increasing the objectivity of the research, an interdisciplinary systems approach is proposed, where the starting point is the use of the Gumilyov deduction method - a peculiar bird's-eye view. This method is necessary for a more adequate resolution of the problem of the separation of sources available to science. The aim of the study is to identify the prerequisites (reasons) for the statement of multivariate interpretations of the names and loci of the celestials pantheon of the ancestors. The methods of review, analysis, synthesis, comparison, widely used in folklore, are used, and the methods of synergetics, extrapolation, and hermeneutics are also used. According to the author, multivariance in its root comes from the natural nature of human co-creator, and an interdisciplinary synthesis at the junction of a number of disciplines is necessary to obtain an objective picture of the classification of the names and loci of the ancestral pantheon deities. This work is a logical continuation of the previous articles of the author “Model of the Sakha World: semantics from the perspective of form geometry (based on the Yakut epos)”; "To the question of the structural interpretation of the locus of the heads of the Yakut pantheon from the perspective of the geometry of forms." Further research in this direction is planned.

Exploration of Javanese culture ethnomathematics based on geometry perspective

This study aims to explore the ethnomathematics of Javanese society using cultural elements of living equipment and technology systems that are based on geometry perspective. The method used in this study was exploration. Data collection was collected through document collections, interviews and observations. Life equipment and technology systems includes in the form of transportation equipment, housing, and clothing materials. Types of transportation equipment, housing, and clothing are artefacts. Mathematics elements are explored on horse-drawn carriages, Joglo houses and batik (Lereng, Ceplok, and Jlamprang motifs). This research is important to explore information related to ethnomathematics in Java that can be used for mathematics in schools. Batik motifs (Lereng, ceplok, and Jlamprang) are ethnomatematics related to lines and angles, triangles and quadrilaterals, circles. Ethnomatematics in Joglo homes can be associated with line and angle material, triangles and quadrilaterals, constructing flat side spaces, building curved, congruent and Pythagorean theorems. Ethnomatematics in horse-drawn carriages can be associated with constructing curved side spaces.

A Novel Self-Intersection Penalty Term for Statistical Body Shape Models and Its Applications in 3D Pose Estimation

Statistical body shape models are widely used in 3D pose estimation due to their low-dimensional parameters representation. However, it is difficult to avoid self-intersection between body parts accurately. Motivated by this fact, we proposed a novel self-intersection penalty term for statistical body shape models applied in 3D pose estimation. To avoid the trouble of computing self-intersection for complex surfaces like the body meshes, the gradient of our proposed self-intersection penalty term is manually derived from the perspective of geometry. First, the self-intersection penalty term is defined as the volume of the self-intersection region. To calculate the partial derivatives with respect to the coordinates of the vertices, we employed detection rays to divide vertices of statistical body shape models into different groups depending on whether the vertex is in the region of self-intersection. Second, the partial derivatives could be easily derived by the normal vectors of neighboring triangles of the vertices. Finally, this penalty term could be applied in gradient-based optimization algorithms to remove the self-intersection of triangular meshes without using any approximation. Qualitative and quantitative evaluations were conducted to demonstrate the effectiveness and generality of our proposed method compared with previous approaches. The experimental results show that our proposed penalty term can avoid self-intersection to exclude unreasonable predictions and improves the accuracy of 3D pose estimation indirectly. Further more, the proposed method could be employed universally in triangular mesh based 3D reconstruction.

Information Geometric Approach to Multisensor Estimation Fusion

Distributed estimation fusion is concerned with the combination of local estimates from multiple distributed sensors to produce a fused result. In this paper, we characterize local estimates as posterior probability densities, and assume that they all belong to a parametric family. Our starting point is to consider this family as a Riemannian manifold by introducing the Fisher information metric. From the perspective of information geometry, the fused density is formulated as an informative barycenter in the space of probability densities and sought by minimizing the sum of its squared geodesic distances from the local posterior densities. Under Gaussian assumptions, a geodesic projection (GP) method and a Siegel distance (SD) method in the information-geometric framework are proposed to tackle the problem. The GP method gives a fusion result in accord with the covariance intersection estimate but under an information-geometric criterion, while the SD method appears to achieve a better approximation of the informative barycenter. Numerical examples are provided to demonstrate the performance of the proposed estimation fusion algorithms.

A novel variational method for 3D viscous flow in flow channel of turbomachines based on differential geometry

ABSTRACT This paper presents a novel variational method for treating three-dimensional rotational Navier-Stokes equations in flow channel of turbomachines. The proposed method establishes a new semi-geodesic coordinate system on the central surface of blades. From the perspective of differential geometry, the system under concern is split into a set of membrane operator equations on two-dimensional manifolds and bending operator equations along hub circle. The third variable of the new coordinate system is approximated by the central difference scheme. We derive a new formulation of two-dimensional Navier-Stokes equations with three components on the manifolds in the variational sense. The well-posedness of the proposed variational formulation is rigorously justified.

Design of points, lines and planes in the reconstruction of occlusion and the risk control in implantation

Occlusion reconstruction involves various factors, such as joints, muscles, teeth and mental states. The relationship between implantation complications and occlusion, especially in the method of reducing the risk of implantation during occlusal reconstruction, is more and more emphasized by clinicians. This article mainly summarized the essentials of occlusal factors, such as mandibular position, cusp inclination and occlusal contacts, Spee's curve, Wilson curve and occlusal plane, in clinical treatment from the perspective of static geometry, dynamic functional movement in points, lines and planes, in order to accomplish precise occlusion reconstruction to achieve a stable and functional occlusion reconstruction, i.e. comfortable mandibular position, reasonable protrusive and lateral guidance, firm retrusive control and stable posterior supports, and to maintain the stability of implant.

Geometry of Thin Films

We study ray optics in the context of double-mirror systems, in the limit as the two mirrors approach one another (thin films). This leads to a novel set of differential equations on a mirror surface which have interesting structure as seen from the perspective of symplectic geometry and hamiltonian mechanics.

Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

We propose a new methodology to understand a stochastic process from the perspective of information geometry by investigating power-law scaling and fractals in the evolution of information. Specifically, we employ the Ornstein–Uhlenbeck process where an initial probability density function (PDF) with a given width and mean value y0 relaxes into a stationary PDF with a width ϵ, set by the strength of a stochastic noise. By utilizing the information length which quantifies the accumulative information change, we investigate the scaling of with ϵ. When , the movement of a PDF leads to a robust power-law scaling with the fractal dimension . In general when , is possible in the limit of a large time when the movement of a PDF is a main process for information change (e.g. ). We discuss the physical meaning of different scalings due to PDF movement, diffusion and entropy change as well as implications of our finding for understanding a main process responsible for the evolution of information.

An Information Geometry-Based Distance Between High-Dimensional Covariances for Scalable Classification

Modeling images/videos with covariance matrices has attracted increasing attentions in various vision tasks, especially in visual classification. For covariances-based visual classification, measuring the distances between covariances is one of the key issues and has been studied for decades. Since the space of covariances is a Riemannian manifold, the geometrical structure of covariances should be favorably considered when designing distance metrics. Although this problem has been widely studied, designing an effective and efficient metric between high-dimensional covariances (HDCOV) for scalable classification is still an open problem. In this paper, we present an information geometry-based distance (IGBD) to tackle this challenge from the perspective of information geometry. Our idea is based on the fact that each covariance can be viewed as a zero-mean Gaussian distribution, and thus the distances between covariances are measured by those between the corresponding Gaussian distributions. The core of our method is to project each distribution, in the form of a set of random samples, to a vector on the tangent space of a common, known distribution on the statistical manifold, based on Fisher information metric and maximum likelihood method. On the tangent space, the Euclidean norm can be used to measure the distances between those sets of projection vectors (or equivalently distributions). The proposed IGBD for HDCOV is computationally efficient and easily combined with a linear support vector machine, suitable for scalable visual classification. The experiments are conducted on various kinds and sizes of benchmarks, and results show the proposed method is efficient and the combination of HDCOV can achieve very competitive performance.

A new method for spatial structure detection of complex inner cavities based on 3D γ-photon imaging

This paper presents a new three-dimensional (3D) imaging method for detecting the spatial structure of a complex inner cavity based on positron annihilation and γ-photon detection. This method first marks carrier solution by a certain radionuclide and injects it into the inner cavity where positrons are generated. Subsequently, γ-photons are released from positron annihilation, and the γ-photon detector ring is used for recording the γ-photons. Finally, the two-dimensional (2D) image slices of the inner cavity are constructed by the ordered-subset expectation maximization scheme and the 2D image slices are merged to the 3D image of the inner cavity. To eliminate the artifact in the reconstructed image due to the scattered γ-photons, a novel angle-traversal model is proposed for γ-photon single-scattering correction, in which the path of the single scattered γ-photon is analyzed from a spatial geometry perspective. Two experiments are conducted to verify the effectiveness of the proposed correction model and the advantage of the proposed testing method in detecting the spatial structure of the inner cavity, including the distribution of gas–liquid multi-phase mixture inside the inner cavity. The above two experiments indicate the potential of the proposed method as a new tool for accurately delineating the inner structures of industrial complex parts.