Geometry & Graphics最新文献

{"title":"Graphic Training of Students Using Forms of Distance Learning","authors":"Zhanna Nazarova","doi":"10.12737/2308-4898-2023-59-66","DOIUrl":"https://doi.org/10.12737/2308-4898-2023-59-66","url":null,"abstract":"This article examines the experience of graphic training of students of a transport university in the conditions of distance learning caused by the COVID-19 pandemic by the example of studying the discipline \"Descriptive geometry and computer graphics\". The use of the BlackBoard electronic educational environment and the teaching and methodological developments of the department staff is shown both as a learning medium for students, including independent work, and as a means of teaching for the teaching staff. Not only the remote format of classes is considered, but also the so-called mixed format, when in-person laboratory or practical classes are held only in small groups, and streaming lectures are conducted with the use of an electronic educational environment. After the end of the pandemic, the mixed format is used in many educational organizations continuously or occasionally due to the withdrawal of a separate group to quarantine. Within the framework of the article, a comparative analysis was carried out, reflecting data on the quality of graphic training among students who studied before, during and after the pandemic, using the example of samples from among the students of the 2019 – 2022 sets. The results of the comparative analysis show that there were no significant fluctuations in one direction or another. This allows us to conclude that the measures taken to organize the educational process in the remote format of classes have helped to preserve the quality of graphic training of students. Recommendations on the organization of graphic training of students are given, which can be used by geometry teachers both in the field of higher education and secondary special education, after returning to the usual conditions of the educational process.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135957046","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":"Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres","authors":"Vladimir Vyshnyepolskiy, E. Zavarihina, D. Peh","doi":"10.12737/2308-4898-2021-9-3-12-29","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-3-12-29","url":null,"abstract":"The article deals with the geometric locations of points equidistant from two spheres. In all variants of the mutual position of the spheres, the geometric places of the points are two surfaces. When the centers of the spheres coincide with the locus of points equidistant from the spheres, there will be spheres equal to the half-sum and half-difference of the diameters of the original spheres. In three variants of the relative position of the initial spheres, one of the two surfaces of the geometric places of the points is a two-sheet hyperboloid of revolution. It is obtained when: 1) the spheres intersect, 2) the spheres touch, 3) the outer surfaces of the spheres are removed from each other. In the case of equal spheres, a two-sheeted hyperboloid of revolution degenerates into a two-sheeted plane, more precisely, it is a second-order degenerate surface with a second infinitely distant branch. \u0000 The spheres intersect - the second locus of the points will be the ellipsoid of revolution. Spheres touch - the second locus of points - an ellipsoid of revolution, degenerated into a straight line, more precisely into a zero-quadric of the second order - a cylindrical surface with zero radius. The outer surfaces of the spheres are distant from each other - the second locus of points will be a two-sheet hyperboloid of revolution. The small sphere is located inside the large one - two coaxial confocal ellipsoids of revolution. \u0000 In all variants of the mutual position of spheres of the same diameters, the common geometrical place of equidistant points is a plane (degenerate surface of the second order) passing through the middle of the segment perpendicular to it, connecting the centers of the original spheres. The second locus of points equidistant from two spheres of the same diameter can be either an ellipsoid of revolution (if the original spheres intersect), or a straight (cylindrical surface with zero radius) connecting the centers of the original spheres when the original spheres touch each other, or a two-sheet hyperboloid of revolution (if continue to increase the distance between the centers of the original spheres).","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78544316","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":"Introduction of Electronic Technology into Education","authors":"H. Damchaasuren","doi":"10.12737/2308-4898-2021-9-3-39-45","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-3-39-45","url":null,"abstract":"The development of information technology has given an important impetus to the development of many sectors of development, including education. One of the conditions for improving learning outcomes in terms of new approaches and requirements is the introduction of information technology. \u0000 The 21st century is called informational (knowledge-based, information technology, etc.). At this time of increasing information flow and rapid technological development, there is a need for cooperation and exchange of information and knowledge. In 2019, Mongolia was ranked 14th in Asia in the ICT Development Index in a keynote speech at the Mongolia International Digital User Conference. \u0000The use of active teaching methods improves the knowledge and skills of students. Active learning is learning that engages learners in the learning process and allows them to think about what they are doing and find ways to do it. Active learning is about helping students learn for themselves, not teaching them. Since the development of computer technology and the emergence of the Internet, scientists and educators in developed and developing countries of the world have conducted a wide range of experimental studies on the use of electronic technology and electronic materials in the learning process. Depending on the type of information technology used in training, it is divided into: e-learning, mobile learning, u-learning, blended learning, and more. The study mentioned in the article is a blended form of study, and in recent years, it has become commonplace in the best universities in the world to combine full-time education with online education at the same level. \u0000 The study of methods and ways of introducing electronic technology in education are of practical importance. In this article, we present the results of some studies carried out on the example of teaching the subject of engineering graphics at MGUNT.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90684504","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":"Spatial Geometric Cells — Quasipolyhedra","authors":"A. Efremov, T. Vereschagina, N. Kadykova, V. Rustamyan","doi":"10.12737/2308-4898-2021-9-3-30-38","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-3-30-38","url":null,"abstract":"Tiling of three-dimensional space is a very interesting and not yet fully explored type of tiling. Tiling by convex polyhedra has been partially investigated, for example, works [1, 15, 20] are devoted to tiling by various tetrahedra, once tiling realized by Platonic, Archimedean and Catalan bodies. The use of tiling begins from ancient times, on the plane with the creation of parquet floors and ornaments, in space - with the construction of houses, but even now new and new areas of applications of tiling are opening up, for example, a recent cycle of work on the use of tiling for packaging information [17]. Until now, tiling in space has been considered almost always by faceted bodies. Bodies bounded by compartments of curved surfaces are poorly considered and by themselves, one can recall the osohedra [14], dihedra, oloids, biconuses, sphericon [21], the Steinmetz figure [22], quasipolyhedra bounded by compartments of hyperbolic paraboloids described in [3] the astroid ellipsoid and hyperbolic tetrahedra, cubes, octahedra mentioned in [6], and tiling bodies with bounded curved surfaces was practically not considered, except for the infinite three-dimensional Schwartz surfaces, but they were also considered as surfaces, not as bodies., although, of course, in each such surface, you can select an elementary cell and fill it with a body, resulting in a geometric cell. With this work, we tried to eliminate this gap and described approaches to identifying geometric cells bounded by compartments of curved surfaces. The concept of tightly packed frameworks is formulated and an approach for their identification are described. A graphical algorithm for identifying polyhedra and quasipolyhedra - geometric cells are described.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81995420","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":"Features of Distance Learning in Geometric and Graphic Disciplines Using Methods of Constructive Geometric Modeling","authors":"E. Boyashova","doi":"10.12737/2308-4898-2021-9-3-46-56","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-3-46-56","url":null,"abstract":"The article is devoted to the peculiarities of teaching the discipline \"Descriptive geometry\" in the conditions of distance learning, it examines the application of information technologies in the educational process in geometric and graphic disciplines. Increasing the speed of information processes, reducing the number of hours for mastering the discipline. the conditions of distance learning set new tasks for teachers and dictate their requirements for teaching graphic disciplines and the use of teaching experience in a new reality; there is a need to introduce and develop new forms of education without losing the quality of education. Geometric-graphic disciplines occupy one of the important places in technical education, the complexity of the study of which lies in the development of a graphical representation of phenomena, objects and processes by methods of constructive geometric modeling. The knowledge and skills acquired by students contribute to the development of spatial, imaginative and rational thinking, which is necessary for future professional activities. Descriptive geometry is a discipline that is not easy to master on your own without a conscious understanding of the logic and sequence of geometric constructions, without deep knowledge of theoretical foundations and constant, repeated implementation of practical tasks. The acquisition of practical skills in mastering the methods of discipline has become more difficult in the current epidemiological situation. \u0000In modern conditions of distance learning, the use of the Simplex geometric modeling system made it possible to develop and propose a new concept of geometric-graphic interaction, which significantly reduced the time for completing and checking educational tasks in real time. The proposed technology reveals the deep informational essence of the studied discipline \"Descriptive Geometry\" and becomes a powerful research tool for students. The integration of traditional teaching methods in the graphic preparation of students with computer and communication facilities increases the possibilities of communication and improves the quality of teaching.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76195368","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":"The Origins of Formation of Descriptive Geometry","authors":"N. Sal'kov","doi":"10.12737/2308-4898-2021-9-3-3-11","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-3-3-11","url":null,"abstract":"The translation \"Descriptive geometry\" is not entirely accurate. In fact, the phrase should be translated as \"Narrative geometry\". Based on this translation, it can be confidently stated that the science under consideration serves not only as a theoretical basis for orthogonal projections, a special case of which are ordinary drawings, but also for any images – in this the author of the article fully agrees with such authorities as N.A. Rynin, N.F. Chetverukhin, V.O. Gordon, S.A. Frolov, N.A. Sobolev and many others. \u0000The paper considers the origins of one of the directions of geometry – descriptive geometry. The hypothesis is put forward that in reality descriptive geometry, or rather, its elements, was originally involved in ancient times, during the primitive communal system when making drawings on the walls of caves and rocks. Orthogonal projections were used in the ancient world and in the Middle Ages, and Gaspard Monge at the end of the XVIII century systematized all the existing disconnected developments on descriptive geometry, adding his own research. \u0000Most likely, geometry in general was the very first science that originated when our ancestors who lived in caves faced the problem of increasing the living area due to population growth. And descriptive geometry began to develop from the moment when the first artist depicted scenes from life on the cave wall: hunting, fishing, tribal wars, events that shocked people, etc. Ancient artists existed on all continents of the globe, except perhaps Antarctica, since rock carvings were found on all other continents. And the earliest was performed somewhere 25-30 thousand years ago. Thus, the hypothesis that the elements of descriptive geometry originated in the primitive communal system can be considered proven.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89667018","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":"The Main Reasons for Poor Assimilation of Descriptive Geometry","authors":"N. Sal'kov","doi":"10.12737/2308-4898-2021-9-2-3-11","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-2-3-11","url":null,"abstract":"Descriptive geometry is the most difficult subject studied by first-year students in technical universities. The paper considers the reasons for poor geometric knowledge among university graduates. It is determined that there are three main objective reasons. 1. Poor geometric training in high school, where they study planimetry and stereometry, but not all the knowledge that future students will need at the university is included for passing the Unified State Exam. Also, in high school, students do not develop the habit of thinking analytically, although when proving geometric theorems, this function has developed greatly. 2. Descriptive geometry has a completely different method, which differs from all the methods of disciplines that are studied at school, is the projection method, which develops spatial imagination. Exactly the method that gives any description in the daily activities of any person and in the work of any engineer. 3. Disadvantages of university textbooks on descriptive geometry. Each section of the textbook on descriptive geometry has been based on a particular geometric image since ancient times: a point, a straight line, a plane, etc. As a result, tasks that could be collected in their own section of the textbook (for example, all positional problems or all metric problems) are scattered throughout the entire body of the textbook. And from this there is an opinion that each of the tasks has its own unique solution algorithm. It is shown that with a systematic approach, all positional problems, as well as all metric ones, are solved, in principle, according to a single algorithm.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87908651","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":"Augmented Reality in Conducting Classes in Engineering and Technical Disciplines of Design","authors":"T. Musaeva, A. Urago","doi":"10.12737/2308-4898-2021-9-2-46-55","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-2-46-55","url":null,"abstract":"The article discusses the issue of increasing the efficiency of the educational process in engineering technical disciplines, in the study of which students acquire professional competencies in the design of industrial products. There are a number of problems when solving problems in which it is necessary to create drawings of realistic physical objects under study, as well as the correct representation of the 2D image format in 3D format. As a result, when mentally projecting a product onto different projection planes, typical errors occur. These errors are associated with human cognitive spatial reasoning abilities, as confirmed and described by experts in this subject area. \u0000In the article, in order to solve the indicated problem, it is proposed to create and introduce modern digital technologies of virtual and augmented reality into the educational process. The proposed technologies do not replace the teacher, but are considered only as additional means that will make it possible to more clearly and qualitatively disclose the thematic units of the discipline, which will help the student in mastering the material necessary for acquiring professional competencies. \u0000The process of creating a course requires solving a wide range of problems: \u00001. research of the possibilities of technologies and means of augmented reality; \u00002. selection and substantiation of the composition of technical means necessary in the process of studying theoretical material and performing practical tasks in the augmented reality environment; \u00003. creating an architecture of a model of interaction between a teacher and a student when studying a theoretical block of a discipline; \u00004. creating the architecture of the model of interaction between the teacher and the student when studying the practical block of the discipline; \u00005. development of the course structure; \u00006. development of a methodology for studying the discipline \"Engineering and Computer Graphics\" in the environment of augmented reality; \u00007. creation of a library of virtual objects of industrial products. \u0000The article briefly discusses only a part of the listed issues, namely 1 - 4.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84876217","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":"Constructing a G2-Smooth Compound Curve Based on Cubic Bezier Segments","authors":"V. Korotkiy","doi":"10.12737/2308-4898-2021-9-2-12-28","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-2-12-28","url":null,"abstract":"The theory and practice of forming composite G2-smooth (two-continuously differentiable) curves, used in technical design since the mid-60s of the 20th century, is still not reflected in any way in the curriculum of technical universities or in Russian textbooks in engineering and computer graphics. Meanwhile, such curves are used in modeling a wide variety of geometric objects and physical processes. \u0000The article deals with the problem of constructing a composite G2-smooth curve passing through given points and touching at these points pre-specified straight lines. To solve the problem, cubic Bezier segments are used. The main problem in constructing a smooth compound curve is to ensure the continuity of curvature at the joints of the segments. The article shows that for parametrized cubic curves, this problem is reduced to solving a quadratic equation. A software module has been compiled that allows one to construct a plane G2-smooth curve passing through predetermined points and tangent at these points with predetermined straight lines. The shape of the curve (“completeness” of its segments) is adjusted by the user in the dialog mode of the program module. \u0000Solved the problem of constructing a cubic curve smoothly connecting unconnected Bezier segments. An algorithm for constructing a Bezier segment with given tangents and given curvature at its boundary points is proposed. \u0000Some properties of the cubic Bezier segment are considered. In particular, it was shown that for the case of parallel tangents, the curvature at the end of a segment is determined by the position of only one control point (Theorem 1). Cases are considered when the curvature at the ends of the Bezier segment is equal to zero (Theorem 2). \u0000An approximation of a three-point physical spline is performed using Bezier segments. The approximation error was less than 2%, which is comparable to the error in processing the experimental data. \u0000A method is proposed for modeling a spatial G2-smooth curve passing through points set in advance in space and touching at these points arbitrarily oriented lines in space. \u0000The article is of an educational nature and is intended for an in-depth study of the basics of computational geometry and computer graphics.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88354808","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":"Investigation of Reflection from Curved Mirrors on a Plane in the Wolfram Mathematica","authors":"O. Suncov, L. Zhikharev","doi":"10.12737/2308-4898-2021-9-2-29-45","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-2-29-45","url":null,"abstract":"In this article, the study of the geometry of the flat shapes reflection from curved lines located in the plane of these shapes continues. The paper is devoted to a more detailed description of reflection from the analytical geometry point of view. In addition, the range of proposed tasks has been significantly expanded. \u0000An algorithm for reflecting zero-dimensional and one-dimensional objects from plane curves is compiled, and corresponding illustrations are given. \u0000For the first time, the authors have obtained equations that allow us to construct reflections of a point from second-order curves: a circle, an ellipse, a parabola and a hyperbola, as well as from high-order curves: Bernoulli lemniscates and cardioids [17], [24], [13], [25], [23], [22]. In addition, equations for the reflection results of one-dimensional objects: a segment and a circle, from the same plane curves were obtained. Similar studies are being conducted in the works [2], [1], [32], [28], [3], [4]. All equations are accompanied by blueprints of special cases of reflections obtained using the Wolfram Mathematica mathematical package [18], [19]. In addition, the application contains the source codes, which gives the reader to configure the reflection parameters themselves on condition having access this program, as well as visually assess the change in the reflection pattern when changing these parameters for various types of flat mirrors. \u0000This article demonstrates the possibilities that the obtained equations provide, and the prospects for further work, which consist in obtaining new equations of objects reflected from other flat mirrors.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90278244","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}