Geometry & Graphics最新文献

{"title":"Geometric Modeling of Objects’ Thermal Characteristics by the Functional-Voxel Method","authors":"A. Plaksin, S. Pushkarev","doi":"10.12737/2308-4898-2020-25-32","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-25-32","url":null,"abstract":"In this paper the influence of objects’ thermal processes on their correspondence to a given geometry has been considered, and an alternative apparatus for geometric modeling of bodies’ temperature stress and thermal expansion after effect of a heat source, based on a functional-voxel approach, has been proposed as well. A discrete geometric model of temperature stress at a point of thermal loading in an isotropic heat-conducting body for a functional-voxel representation has been developed, allowing simulate a single action of a heat source to obtain local geometric characteristics of thermal stress in the body. This approach, unlike traditional approaches based on the FEM, allows apply the temperature load at the object’s point taken by itself. A discrete geometric model for expansion at the point of thermal loading in an isotropic heat-conducting body for a functional-voxel representation has been developed, which allows simulate the change of an object’s local geometric characteristics during the process of material expansion from a single effect of a heat source to obtain a value upon the body volume changing. This approach, unlike traditional approaches based on the FEM, allows simulate a change in the body’s surface geometry from thermal expansion at a point taken by itself without errors arising from calculations using a mesh. Have been proposed algorithms for functional-voxel modeling of temperature stress and expansion under distributed thermal loading. These algorithms allow construct a loading region of complex configuration based on the spatial distribution and scaling of the temperature stress’s geometric model for a single point of thermal loading, uniformly form a contour (surface) after material expansion, and obtain information about the change in products’ length (volume) based on information about each point of functional space. Has been presented an example of using the proposed approach for solving a processing tool’s correction problem based on the temperature in the cutting zone and material thermal reaction. The geometric model can be used to the automated design of a processing tool path for parts cutting on CNC machines.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"4 1","pages":"25-32"},"PeriodicalIF":0.0,"publicationDate":"2020-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82086761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computational Graphics in Solving of Non-Traditional Engineering Problems","authors":"Yu. Savel'ev, E.Yu. Cherkasova","doi":"10.12737/2308-4898-2020-33-44","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-33-44","url":null,"abstract":"Based on the published data, the essence of computational graphics has been laid down. Have been reported examples of new results obtained only through accurate computer constructions and measurements. The work content is a presentation of new ways to achieve the goal by solving non-traditional engineering problems. The author's method of projection with time-stamps, which, in fact, is a computer descriptive geometry, allows solve multi-parameter (not to be confused with multi-dimensional) problems with 9 variables [1–3; 13]. The author’s method of two-axis equal-sized evolvement [11; 12; 17] allows quantitatively measurements of solid angles. The addition of trigonometric functions (sinuses, sinusoids, etc.) can also be considered as a novelty [10; 11; 14]. At the junction of analytic (AG) and descriptive geometries have been calculated parameters of dodecahedron and has been given its mathematical description. In the traditional AG task, the required parameters have been calculated graphically, including a point’s speed of movement. Has been presented the author’s method for determining the instantaneous center in theoretical mechanics. For the first time, the equality of the angles of rotation for points and the link as a whole has been established, and a continuous centroid has been built. By decomposition of vectors a new way for summing up theirs vertical projections has been demonstrated. The developed method of projections with time-stamps allows simultaneously consider such parameters as spatial coordinates of moving objects (two or more) in time, their speeds and even sizes, including the variable ones. Has been shown the possibility for graphical programming while solving systems of equations, as well as for graphical solution of algebraic and stereometric problems. This publication aims to disseminate computer methods for engineering problems solving.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"16 1","pages":"33-44"},"PeriodicalIF":0.0,"publicationDate":"2020-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78280125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized Euler-Venn Diagrams for Fuzzy Sets","authors":"Yu. V. Mironova","doi":"10.12737/2308-4898-2020-34-43","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-34-43","url":null,"abstract":"The fuzzy set concept is often used in solution of problems in which the initial data is difficult or impossible to represent in the form of specific numbers or sets. Geo-information objects are distinguished by their uncertainty, their characteristics are often vague and have some error. Therefore, in the study of such objects is introduced the concept of \"fuzziness\" — fuzzy sets, fuzzy logic, linguistic variables, etc. The fuzzy set concept is given in the form of membership function. An ordinary set is a special case of a fuzzy one. If we consider a fuzzy object on the map, for example, a lake that changes its shape depending on the time of year, we can build up for it a characteristic function from two variables (the object’s points coordinates) and put a certain number in accordance with each point of the object. That is, we can describe a fuzzy set using its two-dimensional graphical image. Thus, we obtain an approximate view of a surface z = μ(x, y) in three-dimensional space. Let us now draw planes parallel to the plane. We’ll obtain intersections of our surface with these planes at 0 ≤ z ≤ 1. Let's call them as isolines. By projecting these isolines on the OXY plane, we’ll obtain an image of our fuzzy set with an indication of intermediate values μ(x, y) linked to the set’s points coordinates. So we’ll construct generalized Euler — Venn diagrams which are a generalization of well-known Euler — Venn diagrams for ordinary sets. Let's consider representations of operations on fuzzy sets A a n d B. Th e y u s u a l l y t a k e : μA B = min (μA,μB ), μA B = max (μA,μB ), μA = 1 − μA. Algebraic operations on fuzzy sets are defined as follows: μ A B x μ A x μ B x ( ) = ( ) + ( ) − −μ A (x)μ B (x), μ A B x μ A x μ B x ( ) = ( ) ( ), μ A (x) = 1 − μ A (x). Let's construct for a particular problem a generalized Euler — Venn diagram corresponding to it, and solve subtasks graphically, using operations on fuzzy sets, operations of intersection and integrating of the diagram’s bars.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"3 1","pages":"34-43"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83334519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New Problems of Descriptive Geometry","authors":"A. Girsh","doi":"10.12737/2308-4898-2020-18-33","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-18-33","url":null,"abstract":"Complex geometry consists of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually determines an open system in which a correctly posed problem may give no solution. Analytical geometry is an example of a closed system, in which the previously mentioned problem always gives a solution as a complex number, whose one of the parts may turn out to be zero. Development of imaginary solutions and imaginary figures is a new task for descriptive geometry. Degenerated conics and quadrics set up a new class of figures and a new class of descriptive geometry’s problems. For example, a null circle, null sphere, null cylinder, and a cone as a hyperboloid degenerated to an asymptote. The last ones necessarily lead to imaginary solutions in geometric operations. In this paper it has been shown that theorems formulated in one geometry are also valid in conjugate geometry as well, while the same figures of conjugated geometries visually look different. So imaginary points exist only by pairs, the imaginary circle is not round one, the centers of dissimilar circles’ similarity do not belong to the centerline and other examples. For solution, a number of problems on geometric relations, and operations with degenerated conics and quadrics, as well as several problems from 4D-geometry are proposed. Solutions for above mentioned problems are given in section 9. In this paper some examples of new problems for descriptive geometry have been considered. It has been shown that the new problems require access to a complex space. New figures consist of two parts, a real figure and a figure of its imaginary complement.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"27 1","pages":"18-33"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87584031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Imaginary Straight Lines in Cartesian Coordinate System","authors":"V. Korotkiy","doi":"10.12737/2308-4898-2020-5-17","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-5-17","url":null,"abstract":"A geometric model of imaginary conjugate straight lines a~b, allowing symbolic representation of these lines on the real coordinate plane xy is considered. In order to connect the algebraic and geometric representations of imaginary straight lines, it is proposed to use the “mark” formed by orthogonal d1 ⊥ d2 and main g1~g2 directions of the elliptic involution σ in the pencil V. The specification of two pairs of pulling apart each other real straight lines d1~d2, g1~g2 passing through V, uniquely defines the elliptic involution σ in the pencil V, therefore, the V(d1 ⊥ d2, g1~g2) mark completely defines a pair of imaginary double straight lines a~b of elliptic involution σ(V), that allows consider the mark as an “image” of these imaginary straight lines. When using a mark, it is required to establish a one-to-one correspondence between complex coefficients of imaginary double straight lines equations and a graphically given mark. The direct and inverse problems are solved in this paper. The direct one is creation a mark representing imaginary straight lines, given by its own equations. The inverse one is determination of coefficients for the equations of imaginary lines defined by the mark. The essence of the direct and inverse problems consists in establishing a oneto-one correspondence between the equations of imaginary double straight elliptic involutions σ in the pencil V, and a graphically given mark containing the orthogonal and main directions of this involution. To solve both the direct and inverse problems, the Hirsch theorem (A.G. Hirsch) is used, which establishes a one-to-one correspondence between the complex Cartesian coordinates for a pair of imaginary conjugated points and real coordinates of a special “marker” symbolically representing these points. Have been considered examples of solution for geometric problems involving imaginary lines. In particular, has been solved the problem of constructing a circle passing through a given point and touching imaginary lines defined by its mark V(d1 ⊥ d2, g1~g2). Has been proposed a graphical and analytical algorithm for determining the coefficients of equations of imaginary tangents, traced to a conic section from its inner point.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"9 1","pages":"5-17"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82733749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Construction of Receptor Geometric Models for Objects of Complex Technical Forms","authors":"E. VinTun, L. Markin","doi":"10.12737/2308-4898-2020-44-56","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-44-56","url":null,"abstract":"In this paper the question related to the use of receptor (voxel) method for geometric modeling to solve practical design problems has been considered. The use of receptor methods is effective in solving a certain class of problems, primarily the problems of automated layout. The complexity of this method’s practical use is due to the fact, that receptor geometric models are never the primary ones. They are formed based on parametric models specified by designer. Receptor models are the internal machine ones. The main problem that prevents the widespread use of the receptor method is the lack of universal methods for converting parametric models into the receptor ones. Available publications show that in solving practical problems various authors have developed their own methods for creating receptor models for objects of \"primitives\" and \"composition of primitives\" classes. Therefore, it is extremely urgent to solve the problem of developing a universal method of forming receptor models for objects of complex technical forms. The essence of the proposed method is the transformation of a solid-state model created in a CAD system into a receptor matrix. First in the physical one, in which the solid-state model is discretized into cubes with receptor sizes, and then in the mathematical one — a three-dimensional array with binary codes of zeros and ones. The creation of a physical receptor matrix is carried out by means of the CAD-system itself, allowing diagnose the belonging of a single receptor to a simulated object. The fact of intersection or non-intersection a given position by a single receptor is encoded by \"1\" and \"0\" respectively, and this information is transferred to a mathematical receptor model (3-dimensional binary array). This calculation procedure is programmed in the form of a macro, providing a given position of a single receptor and fixing the fact of its intersection with the solid-state model. Have been demonstrated examples for described method’s practical application, and has been carried out CPU time cost estimation for the construction of a physical receptor model depending on the receptor size and object geometric complexity. Actions on data transformation from a solid-state model to a receptor one have been implemented in the form of C# programs.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"76 1","pages":"44-56"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78261507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Testing Technologies in Assessment of Students’ Substantive Readiness to Study Higher Educational Institution’s Geometric and Graphic Disciplines","authors":"S. Ignat'ev, Z. Tret'yakova, A. Folomkin","doi":"10.12737/2308-4898-2020-65-75","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-65-75","url":null,"abstract":"Engineering practice cannot and will never be successful without spatial imagination and thinking, deep knowledge of geometric methods and ways for solution of engineering problems, comprehensive understanding of specificity and geometric features of technical objects and their images, understanding of labor organization impact on these processes etc. In this paper has been described the social and pedagogical investigation of question related to students’ substantive readiness to study higher educational institution’s graphic disciplines, and this question’s basis is contradiction between requirements to the initial level of students’ knowledge and skills and the level of students’ substantive readiness on the course of school training. In this paper have been performed analysis and generalization of existing pedagogical experience on issues related to graphic training. A method and a scheme for monitoring the degree of students’ competence on the school geometry course using Google forms have been proposed. Selective testing of students by the proposed way has been performed. Statistical processing of data obtained during this testing has been carried out, the density function of correct responses has been built, field of testing results correlation with total score of Unified State Exam (USE) results has been carried out, cluster analysis of obtained results has been performed by Varda method. Using the theory of latent variables and the Rush model an analysis of test tasks, aimed at optimizing the test structure, has been carried out. Based on the performed investigation, recommendations have been given for further research in this field, and measures are proposed on changing the learning process aimed at personalizing and building an individual learning trajectory in order to increase the educational process’s efficiency.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"45 1","pages":"65-75"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73478421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of Some Problems from a Course on Theoretical Mechanics Solved by Descriptive Geometry’s Methods","authors":"O. Nazarova","doi":"10.12737/2308-4898-2020-76-83","DOIUrl":"https://doi.org/10.12737/2308-4898-2020-76-83","url":null,"abstract":"In this paper the usability of descriptive geometry’s methods for solution problems related to theoretical mechanics is considered. Descriptive geometry emerged as a science intended for solution the problems of human activity’s different branches. In connection with development of modern graphical means, the value of descriptive geometry is greatly underestimated, but it is on descriptive geometry’s principles and laws that have been built algorithms used in graphic applications [30]. For a long time descriptive geometry has been used by scientists in various spheres of human activity [27]. In study of general technical disciplines, interdisciplinary connections play an important role, especially between theoretical disciplines, whose understanding is particularly difficult for students [35]. Understanding the possibility of using descriptive geometry’s methods, students can think over aspects of their use in the performance of tasks under study, for example, theoretical mechanics. Methods, which can be use while solving problems from the course of theoretical mechanics, are learned by students in the standard course of descriptive geometry and are not particularly difficult. Therefore, the graphical approach to solving problems of theoretical mechanics is accessible and understandable for majority of successful students. In this paper example problems from the course of theoretical mechanics on the topics \"Plane System of Forces\" and \"Spatial System of Forces\" have been considered. Graphical problem solving was performed using the image of force vectors with the help of orthogonal projections. For checking the correctness of graphicalcons.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"27 1","pages":"76-83"},"PeriodicalIF":0.0,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85585360","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Ensuring the Requirements of Ergonomics in the Automated the Layout of Equipment","authors":"Е ВинТун, E. VinTun, Л. Маркин, L. Markin","doi":"10.12737/article_5dce6e5ed88c68.81465629","DOIUrl":"https://doi.org/10.12737/article_5dce6e5ed88c68.81465629","url":null,"abstract":"The article discusses the solution to the problem of automating the design of layouts of various equipment, taking into account ergonomics, by which is meant the provision of service areas. The article describes the development of methods and algorithms that provide access to installation tools and workspace during installation and maintenance of already placed equipment. The solution method is geometric modeling of both the placed objects and the installation equipment necessary for its maintenance, as well as the trajectory of its movement to the service area. Thus, both the installation equipment and the movement paths are treated as composable objects, the intersection of which with other objects is unacceptable. As a modeling method, receptor-based geometric models that discretize the allocation space were used. The choice of receptor models is due to the fact that the solid-state model of all the instantaneous positions of the installation tool in the process of its delivery and operation is extremely complex from a geometrical point of view. The possibility of relatively easy to determine the fact of the intersection of all objects of the scene, described by receptor models, and is the rationale for the choice in our study of the receptor method of geometric modeling. Based on the receptor design model, a procedure has been developed for determining the trajectory of a mounting tool at a given operating point, as well as the formation of the space required for operation, or establishing the fact that it is impossible to service a particular object, which indicates an unsatisfactory (non-ergonomic) given design solution. In this study a feature of using receptor models is the use of 6-digit codes in the receptor matrices, which, with some complication of the modeling method, allows to obtain additional information about problem areas in the layout under study (impossibility of carrying a tool, impossibility of performing assembly operations, etc.). Algorithms for solving this problem, as well as a graphical shell that visualizes the results of computer-aided design, are implemented as C# programs.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"59 1","pages":"70-85"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76944373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Imaginary Points in Cartesian Coordinate System","authors":"Андрей Оттович Гирш, A. Girsh, Виктор Короткий, V. Korotkiy","doi":"10.12737/article_5dce651d80b827.49830821","DOIUrl":"https://doi.org/10.12737/article_5dce651d80b827.49830821","url":null,"abstract":"Geometric models are considered that allow symbolic representation of imaginary points on a real Cartesian coordinate plane XY. The models are based on the fact that through every pair of imaginary conjugate points A~B with complex coordinates x = a ± jb, y = c ± jd one unique real line m passes. For the image of imaginary points, it is proposed to use the graphic symbol m{OL} consisting of the line m passing through the imaginary points, the center O of the elliptic involution σ with imaginary double points A~B on the line m, and the Laguerre point L, from which the corresponding points involutions σ are projected by an orthogonal pencil of lines. According to A.G. Hirsch, the symbol m{OL} is called the marker of imaginary conjugate points A~B. A theorem is proved that establishes a one-to-one correspondence between the real Cartesian coordinates of the points O, L of the marker, and the complex Cartesian coordinates of the pair of imaginary conjugate points represented by this marker. The proved theorem allows us to solve both the direct problem (the construction of a marker depicting these imaginary points) and the inverse problem (the determination of the Cartesian coordinates of imaginary points represented by the marker). A graphical algorithm for constructing a circle passing through a real point and through a pair of imaginary conjugate points is proposed. An example of the graph-analytical determination of the Cartesian coordinates of imaginary points of intersection of two conics that have no common real points is considered.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"57 1","pages":"28-35"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77876859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}