Geometric Objects Research Articles

РОЗВИТОК ПРОСТОРОВОГО МИСЛЕННЯ УЧНІВ НА УРОКАХ КРЕСЛЕННЯ ЗАСОБАМИ ЦИФРОВИХ ТЕХНОЛОГІЙ

У статті досліджено шляхи і засоби цілеспрямованого розвитку просторового мислення учнів в умовах комп’ютерно-орієнтованого навчання креслення. Досліджено сутність та значення просторового мислення для всебічного розвитку особистості. З’ясовано можливості цифрових технологій (ЦТ) як засобу розвитку просторового мислення учнів на уроках креслення. Встановлено, що найбільш доцільним для розвитку просторового мислення учнів є використання комп’ютерних програмних засобів імітаційно-моделюючого типу.

Automatic far‐field camera calibration for construction scene analysis

AbstractThe use of cameras for safety monitoring, progress tracking, and site security has grown significantly on construction and civil infrastructure sites over the past decade. Localization of construction resources is a crucial prerequisite for many applications in automated construction management. However, most existing vision‐based methods perform the analysis in the image plane, overlooking the effect of perspective and depth. The manual and labor‐intensive process of traditional calibration techniques, as well as the busy and restrictive construction environment, makes this a challenging task. This study proposes a framework for automatic camera calibration with no manual intervention. The framework utilizes convolutional neural networks for geometrical scene analysis and object detection, which are used to estimate the location of horizon line, vertical vanishing point, as well as objects with known height distributions. This enables automatic estimation of camera parameters and retrieval of scale. The proposed framework is evaluated on images from two major construction projects in Melbourne, Australia. Results show that the proposed method achieves a minimum accuracy of 90% in estimating proximity of points on the ground and can facilitate further development of vision‐based solutions for safety and productivity analysis.

Flow parsing and biological motion

Flow parsing is a way to estimate the direction of scene-relative motion of independently moving objects during self-motion of the observer. So far, this has been tested for simple geometric shapes such as dots or bars. Whether further cues such as prior knowledge about typical directions of an object’s movement, e.g., typical human motion, are considered in the estimations is currently unclear. Here, we adjudicated between the theory that the direction of scene-relative motion of humans is estimated exclusively by flow parsing, just like for simple geometric objects, and the theory that prior knowledge about biological motion affects estimation of perceived direction of scene-relative motion of humans. We placed a human point-light walker in optic flow fields that simulated forward motion of the observer. We introduced conflicts between biological features of the walker (i.e., facing and articulation) and the direction of scene-relative motion. We investigated whether perceived direction of scene-relative motion was biased towards biological features and compared the results to perceived direction of scene-relative motion of scrambled walkers and dot clouds. We found that for humans the perceived direction of scene-relative motion was biased towards biological features. Additionally, we found larger flow parsing gain for humans compared to the other walker types. This indicates that flow parsing is not the only visual mechanism relevant for estimating the direction of scene-relative motion of independently moving objects during self-motion: observers also rely on prior knowledge about typical object motion, such as typical facing and articulation of humans.

A deep learning approach using synthetic images for segmenting and estimating 3D orientation of nanoparticles in EM images

Background and objective: Nanoparticles present properties that can be applied to a wide range of fields such as biomedicine, electronics or optics. The type of properties depends on several characteristics, being some of them related with the particle structure. A proper characterization of nanoparticles is crucial since it could affect their applications. To characterize a particle shape and size, the nanotechnologists employ Electron Microscopy (EM) to obtain images of nanoparticles and perform measures over them. This task could be tedious, repetitive and slow, we present a Deep Learning method based on Convolutional Neural Networks (CNNs) to detect, segment, infer orientations and reconstruct microscope images of nanoparticles. Since machine learning algorithms depend on annotated data and there is a lack of annotated datasets of nanoparticles, our work makes use of artificial datasets of images resembling real nanoparticles photographs.Methods: Our work is divided into three tasks. Firstly, a method to create annotated datasets of artificial images resembling Scanning Electron Microscope (SEM). Secondly, two models of convolutional neural networks are trained using the artificial datasets previously generated, the first one is in charge of the detection and segmentation of the nanoparticles while the second one will infer the nanoparticle orientation. Finally, the 3D reconstruction module will recreate in a 3D scene the set of detected particles.Results: We have tested our method with five different shapes of basic nanoparticles: spheres, cubes, ellipsoids, hexagonal discs and octahedrons. An analysis of the reconstructions was conducted by manually comparing each of them with the real images. The results obtained have been promising, the particles are segmented and reconstructed accordingly to their shapes and orientations.Conclusions: We have developed a method for nanoparticle detection and segmentation in microscope images. Moreover, we can also infer an approximation of the 3D orientation of the particles and, in conjunction with the detections, create a 3D reconstruction of the photographs. The novelty of our approximation lies in the dataset used. Instead of using annotated images, we have created the datasets simulating the microscope images by using basic geometrical objects that imitate real nanoparticles.

Odd length: Odd diagrams and descent classes

We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of [n]×[n] to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over descent classes of the symmetric groups.

The unit of cubic matrices in modeling of micro-relief

The apparatus of cubic matrices allows us to give a classification of the contiguous cuboid. Invariant theory is used for this purpose. A cubic matrix is a generalization of the concept of a square or rectangular matrix. Increasing the dimension of a square matrix by one requires a new definition of the product of three-dimensional matrices. It is necessary to search for new invariants for the operator of parallel transfer of the coordinate system and the operator of rotation around the origin. The contiguous cuboloid allows us to give a three-dimensional geometric model of the microrelief an order of magnitude more accurately than the contiguous paraboloid does. The geometrical structure of the contiguous coboloid is investigated by the method of sections. The curvature of the surface at the point of contact with the cuboid is estimated by the curvature tensor, which is analogous to the Riemann-Christoffel tensor. The curvature of the surface in the local area of the point of contact is estimated with certain assumptions. There is no more precise geometric object that defines the curvature of the surface.

Correlation between mathematical creative thinking ability and mathematical creative thinking disposition in geometry

Mathematics educators commonly believe that an open-ended approach could facilitate students to build their creative thinking ability and its disposition, especially in geometry. This research belongs to a quasi-experiment that used a non-equivalent control group design. The purpose of this research was to analyze the correlation between creative thinking ability and creative thinking disposition through an open-ended learning approach toward area and volume of flat geometrical objects problems. Two classes are selected as the sample, they are the control group and experiment group. The open-ended approach was used in the experiment group then direct instruction in the control group. The population is all of the eighth-grade students at one of the junior high school in Bandung, West Java, Indonesia. This study used to test and non-test as the research instruments. The test instrument was a creative thinking ability test while the non-test instrument was a questionnaire. These instruments were given after the learning process then analyzed by using Bivariate Correlation through SPSS. According to the analysis results, the finding is: the correlation coefficient between creative thinking ability and creative thinking disposition is at the middle level. The conclusion is there is a significant correlation between students’ creative thinking ability and students’ creative thinking disposition.

On smoothness of connection of spatial curve cyclographic projection segments

The correlation between smoothness of connection of segments of a spatial curve and the respective cyclographic projection is considered. The sufficient condition for smoothness at the point of connection of segments of cyclographic projection of a spatial curve described by a polynomial spline of order k +1 is formulated and verified. It has been demonstrated that despite the derivative of order k +1 of the spline function describing the spatial composite curve along with all the higher-order derivatives not being continuous, it is possible to connect segments of the cyclographic projection with smoothness , where r = k + i, i = 0,1,2,…. Achieving smoothness , through the formulated condition of sufficiency is demonstrated on numeric examples. The obtained results of the study can serve as the basis of development of effective algorithms of cyclographic formation of geometric objects applied in geometric optics, road surface form design, and optimal cutting tool trajectory design in pocket machining.

Investigation of Representation Theory for Algebraic and Geometric Applications

Understanding the algebraic and geometric structures that form in diverse mathematical areas depends heavily on the study of representation theory. The significance and uses of representation theory in both algebra and geometry are briefly discussed in this abstract.The primary goal of representation theory is to understand how linear transformations on vector spaces can represent abstract algebraic objects like groups, rings, and algebras. Representation theory offers a strong framework to analyse and interact with these structures using the methods and tools of linear algebra by linking algebraic structures with linear transformations.The representation theory has significant effects on algebra. Through the examination of the representations that go along with a group, it allows us to examine its composition and behaviour. One can learn more about the internal structures and underlying symmetries of groups by breaking representations down into irreducible parts. This has implications for number theory, combinatorics, and quantum physics, among other fields.Understanding symmetries and transformations of geometric objects in geometry depends critically on representation theory. The investigation of shape and space symmetry is made possible by representation theory, which links geometric objects with linear transformations. This has uses in a variety of disciplines, including physics' study of symmetry groups, differential geometry, and crystallography.A crucial area of research, representation theory has extensive uses in both algebra and geometry. It is indispensable in many branches of mathematics and opens up new directions for research and discovery because it can reveal the underlying structures and symmetries of abstract algebraic objects.

The Interactive Modeling Method of Virtual City Scene Based on Building Codes

For higher-level requirements of urban planning and management and the recent development of earth and city, it is urgent to establish protocols for the construction of three-dimensional digital city models. However, some problems still exist in the digital technology of the three-dimensional city model, such as insufficient precision of the three-dimensional model, not optimizing the scene and not considering the constraints of building codes. In view of those points, a method to interactively simulate a virtual city scene based on building codes is proposed in this paper. Firstly, some constraint functions are set up to restrict the models to adhere to the building codes, and an improved directional bounding box technique is utilized to solve the problem that geometric objects may intersect in a virtual city scene. The three-dimensional model invocation strategy is designed to convert two-dimensional layouts to a three-dimensional urban scene. A Leap Motion hardware device is used to interactively place the 3D models in a virtual scene. Finally, the design and construction of the three-dimensional scene are completed by using Unity3D. The experiment shows that this method can simulate urban virtual scenes that strictly adhere to building codes in a virtual scene of the city environment, but also provide information and decision-making functions for urban planning and management.

On the speed of algebraically defined graph classes

The speed of a class of graphs counts the number of graphs on the vertex set {1,…,n} inside the class as a function of n. In this paper, we investigate this function for many classes of graphs that naturally arise in discrete geometry, for example intersection graphs of segments or disks in the plane. While upper bounds follow from Warren's theorem (a variant of a theorem of Milnor and Thom), all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving an essentially tight lower bound for the number of graphs on {1,…,n} whose edges are defined using the signs of a given finite list of polynomials, assuming these polynomials satisfy some reasonable conditions. This in particular implies lower bounds for the speed of many different classes of intersection graphs, which essentially match the known upper bounds. Our general result also gives essentially tight lower bounds for counting containment orders of various families of geometric objects, including circle orders and angle orders. Some of the applications presented in this paper are new, whereas others recover results of Alon-Scheinerman, Fox, McDiarmid-Müller and Shi. For the proof of our result we use some tools from algebraic geometry and differential topology.

DYNAMICS IN FRACTAL SPACES

We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space.

Dominating set of rectangles intersecting a straight line

The Minimum Dominating Set (MDS) problem is one of the well-studied problems in computer science. It is well-known that this problem is $$\mathsf {NP}$$ -hard for simple geometric objects; unit disks, axis-parallel unit squares, and axis-parallel rectangles to name a few. An interesting variation of the MDS problem with rectangles is when there exists a straight line that intersects each of the given rectangles. In the recent past researchers have studied the maximum independent set, minimum hitting set problems on this setting with different geometric objects. We study the MDS problem with axis-parallel rectangles, unit-height rectangles, and unit squares in the plane. These geometric objects are constrained to be intersected by a straight line. For axis-parallel rectangles, we prove that this problem is $$\mathsf {NP}$$ -hard. When the objects are axis-parallel unit squares, we present a polynomial time algorithm using dynamic programming. We provide a polynomial time algorithm for unit-height rectangles as well. For unit squares that touch the straight line at a single point from either side of the straight line, we show that there is an $$O(n\log n)$$ -time algorithm.

The effect of realistic mathematics education on student achievement in 8th grades geometric objects teaching

The purpose of this study is to answer if there is a reasonable difference on academical success of students who get education with traditional and RME approach question on “Teaching geometrical objects to 8th grade students” subject. Study group consists of 47 students which contains 21 experimental and 16 control group from “Ordu Anadolu İmam Hatip High School Project School” in Altinordu, Ordu. Experimental and control group have same academical success level, as the school which this study has runned is a school which accepts students with an exam only. After the experimental and control groups were created, a 25 question pre-test was performed to understand the level of knowledge of the group regarding geometrical objects. The same test was performed on the same groups 8 weeks later as retention test. To determine opinions of the students in experimental group regarding RME and related learning activities, semi-structured interviews are conducted. The data obtained from the pretest, posttest and retention tests were analyzed with t-test for independent samples and t-test for dependent samples and variance analysis for mixed measurements with 0.05 significance level. According to the results, it is seen that learning activities prepared according to RME approach are much more effective than learning activities prepared according to the traditional approach on students’ academic success.

Metrics, Quantization and Registration in Varifold Spaces

This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold spaces, which allow to formulate and tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. We study in detail the construction of kernel metrics on varifold spaces and the resulting topological properties of those metrics and then propose a mathematical model for diffeomorphic registration of varifolds under a specific group action which we formulate in the framework of optimal control theory. A second important part of the paper focuses on the discrete aspects. Specifically, we address the problem of optimal finite approximations (quantization) for those metrics and show a $$\varGamma $$ -convergence property for the corresponding registration functionals. Finally, we develop numerical pipelines for quantization and registration before showing a few preliminary results for one- and two-dimensional varifolds.

Spectrum of a linear differential equation over a field of formal power series

In this paper we associate to a linear differential equation with coefficients in the field of Laurent formal power series a new geometric object, a spectrum in the sense of Berkovich. We compute this spectrum and show that it contains interesting information about the equation.

Robust, fast and flexible symmetry plane detection based on differentiable symmetry measure

Reflectional symmetry is a potentially very useful feature which many real-world objects exhibit. It is instrumental in a variety of applications such as object alignment, compression, symmetrical editing or reconstruction of incomplete objects. In this paper, we propose a novel differentiable symmetry measure, which allows using gradient-based optimization to find symmetry in geometric objects. We further propose a new method for symmetry plane detection in 3D objects based on this idea. The method performs well on perfectly as well as approximately symmetrical objects, it is robust to noise and to missing parts. Furthermore, it works on discrete point sets and therefore puts virtually no constraints on the input data. Due to flexibility of the symmetry measure, the method is also easily extensible, e.g., by adding more information about the input object and using it to further improve its performance. The proposed method was tested with very good results on many objects, including incomplete objects and noisy objects, and was compared to other state-of-the-art methods which it outperformed in most aspects.

Cognitive approach to analysis of regular packing of congruent objects on plane

New approach to investigate the two-dimensional regular packing of arbitrary geometric objects (GOs) using cognitive visualization is presented. GO correspond to congruent non-convex polygons with their associated coordinate system. The origins of these coordinate systems are accepted by object poles. The approach considered is based on cognitive processes that are forms of heuristic judgments. According to the first heuristic judgment, regular packing of congruent GOs on the plane have a honeycomb structure, that is, each GO contacts six neighboring GO, the poles of which are vertices of the pole hexagon in honeycomb construction of packing. Based on the visualization of the honeycomb constructions a second heuristic judgment is obtained, according to which inside the hexagon of poles there are fragments of three GOs. The consequence is a third heuristic judgment on the plane covering density with regular packings of congruent GOs. With the help of cognitive visualization it is established that inside the hexagon of poles there are fragments of exactly three objects. The fourth heuristic judgment is related to the proposal of a triple lattice packing for regular packing of congruent GOs.

Graph Deep Learning: State of the Art and Challenges

The last half-decade has seen a surge in deep learning research on irregular domains and efforts to extend convolutional neural networks (CNNs) to work on irregularly structured data. The graph has emerged as a particularly useful geometrical object in deep learning, able to represent a variety of irregular domains well. Graphs can represent various complex systems, from molecular structure, to computer and social and traffic networks. Consequent on the extension of CNNs to graphs, a great amount of research has been published that improves the inferential power and computational efficiency of graph-based convolutional neural networks (GCNNs). The research is incipient, however, and our understanding is relatively rudimentary. The majority of GCNNs are designed to operate with certain properties. In this survey we review of the state of graph representation learning from the perspective of deep learning. We consider challenges in graph deep learning that have been neglected in the majority of work, largely because of the numerous theoretical difficulties they present. We identify four major challenges in graph deep learning: dynamic and evolving graphs, learning with edge signals and information, graph estimation, and the generalization of graph models. For each problem we discuss the theoretical and practical issues, survey the relevant research, while highlighting the limitations of the state of the art. Advances on these challenges would permit GCNNs to be applied to wider range of domains, in situations where graph models have previously been limited owing to the obstructions to applying a model owing to the domains’ natures.

Fractal distribution of mineral species among the crystallographic point groups

Abstract Crystallographic data from 5289 IMA-approved mineral species in the RRUFF database were used to examine the distribution of species among the 32 crystallographic point groups. It is found that within each crystal system, minerals strongly prefer point groups with higher group orders. Within a crystal system, the abundance of minerals belonging to each point group approximately obeys a power law with respect to group order, the same mathematical formalism that describes objects with fractal geometry. In this framework, each crystal system has its own fractal dimension; crystal systems possessing threefold (or sixfold) symmetry elements (i.e., trigonal, hexagonal, isometric) have significantly lower fractal dimension (<2), while those with only one-, two-, or fourfold symmetry elements (triclinic, monoclinic, orthorhombic, tetragonal) have higher fractal dimension (>2). While higher symmetry is preferred within a crystal system, the opposite trend is observed when comparing between crystal systems, with more species preferring crystals systems with lower order symmetry elements than those with higher order symmetry elements at constant group order. The combination of these two competing trends leads to a complex distribution of minerals among the crystal systems, and to the monoclinic group 2/m, the orthorhombic group 2/m2/m2/m, and the triclinic group 1 being the three most popular point groups, respectively. The fractal behavior of symmetry distribution among minerals points toward universal scaling patterns not just in physical, geometric objects but also in the way that symmetry is incorporated into natural periodic structures.