Geometry & Graphics最新文献

{"title":"Increasing the Visibility of Representation for Objects Studying in Descriptive Geometry","authors":"S. Ignat'ev, E. Muratbakeev, M. Voronina","doi":"10.12737/2308-4898-2022-10-1-44-53","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-10-1-44-53","url":null,"abstract":"Increasing the visibility of representation for objects studying in the process of solving basic problems related to descriptive geometry (DG) significantly improves the understanding and perception of the discipline by students. A team of authors representing St. Petersburg Mining University’s DG and Graphics Chair developed their own projects, which are interactive graphics that allow teachers to visualize the solution of some DG course’s problems directly during academic studies. Projects have been created by the Wolfram Mathematica (WM) system tools and divided into 7 sections: points, lines, planes, geometric elements’ mutual position; relative position of a straight line and a second order surface; relative position of a plane and a second order surface; relative position of second order surfaces; mutual position of revolution bodies and polyhedrons; relative position of polyhedrons; a tangent plane to a second-order surface. Difficulties that have arisen in the work, ideas and possibilities for further implementation of proposed principles for using computational WM algorithms when creating and employing these projects in the process of students’ geometric-graphic training are discussed. It is noted that WM allows you manipulate data and observe how the result is changing dynamically, that certainly makes it possible and is an advantage of WM for using it in students’ geometric-graphic teaching. Lecturers of DG and Graphics Chairs can: use the proposed in this work projects aimed at implement various graphic constructions for both preparation of academic studies and in real-time mode during them; create elements of information supply for the educational process; develop demonstration support for academic studies, that in turn will ensure the proper material assimilation by students.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130704520","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":"A Systematic Approach to the Study of Descriptive Geometry","authors":"N. Sal'kov","doi":"10.12737/2308-4898-2022-10-1-14-23","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-10-1-14-23","url":null,"abstract":"The issues of inconsistency of some terms used in descriptive geometry with the rules of GOST R 2.105-2019 ESCD \"General Requirements for Text Documents\", as well as related to the projection of surfaces, are considered. The structure of descriptive geometry’s textbooks is considered, according to which all textbooks of this discipline have been built up until recently. This structure forms a sequence in the study of descriptive geometry in almost all higher education institutes. Criticism of this structure, which has been using for teaching since time immemorial without significant changes what, in particular, presents significant difficulties in descriptive geometry studying, is given. \u0000In contrast with the existing structure, a new one is proposed (if the structure proposed near 50 years ago can be called new), but this new structure is almost nowhere used for release of information on descriptive geometry. This proposed structure is scientifically justified, practically verified and offered for universal use. \u0000In addition, the classification of positional and metric problems, and use of algorithms common for them is proposed. \u0000The proposed structure contains the following sections: 1. Basic concepts of descriptive geometry. 2. Set up of geometric figures in a drawing. 3. Positional problems. 4. Metric problems. 5. Conversion methods. 6. Involutes. \u0000The sections arranged in the proposed coherent system, considered separately from each other and not subject to mixing, make it possible to systematize and algorithmize the processes of descriptive geometry layout and studying.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115216145","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. Continuation","authors":"A. Girsh","doi":"10.12737/2308-4898-2022-9-4-3-10","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-9-4-3-10","url":null,"abstract":"Complex geometry is a synthesis of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually defines a non-closed system in which a correctly posed problem may not give a solution. Analytical geometry represents a closed system. In it, a correctly posed problem always gives solutions in the form of complex numbers, for each of which, one of the parts may be equal to zero. Finding imaginary solutions and imaginary figures formed by a set of such solutions is a new problem in descriptive geometry. Degenerated conics and quadrics, or curves and surfaces of higher orders, constitute a new class of figures and a new class of problems in descriptive geometry. For example, null-circle, null-sphere, null-cylinder, null-torus. In this paper the problem for studying the shape of second (conics, quadrics), third (conoid), and fourth (torus) order figures is posed. The latest suggest a meeting with new geometric properties of figures. Geometric operations are still immersed in the complex space E + M or real - imaginary. The examples under consideration continue a series of degenerated figures in which the null-circle splits into isotropic lines. Isotropic lines are taken as generators of surfaces. They form a cone of revolution and a hyperbolic paraboloid (an oblique plane).","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127618454","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":"Point Tools of Geometric Modeling, Invariant Relating to Parallel Projection","authors":"E. Konopatskiy, A. Bezditnyi","doi":"10.12737/2308-4898-2022-9-4-11-21","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-9-4-11-21","url":null,"abstract":"The purpose of this paper is to familiarize experts in geometric and computer modeling with specific tools for point calculus; demonstrate the possibilities of point calculus as a mathematical apparatus for modeling of multidimensional space’s geometric objects. In the paper with specific examples have been described the basic constructive tools for point calculus, having invariant properties relating to parallel projection. These tools are used to model geometric objects, including: affine ratio of three points of a straight line, intersection of two straight lines, intersection of a straight line with a plane, parallel translation and tangent to a curve. The theoretical foundations of point tools for geometric modeling, invariant relating to parallel projection, have been presented. For example, instead of traditional determination for straight lines intersection point by composing and solving a system of equations in coordinate form, zeroing of a moving triangle’s area is used. This approach allows to define geometric objects in multidimensional spaces keeping the symbolic representation of point equation, as well as to perform its coordinate-wise calculation at the last stage of modeling, which allows to significantly reduce computing resources in the process of solving the problems related to engineering geometry and computer graphics. \u0000The local results of the research presented in this paper, which served as examples for the use of point calculation constructive tools, are: definition of the cubic Bezier curve as a curve of one relation in point and coordinate form; determination of excessive parameterization of the plane and bypass arcs based on it; determination of the tangent to the spatial curve by differentiation the original curve with respect to a current parameter, followed by parallel transfer of the obtained segment to the tangency point; the general point equation for the torso surface has been obtained on account of its definition as a geometric place of tangents to its cusp edge, and examples for the construction of torso surfaces based on the cubic Bezier curve and a transcendental space curve have been presented.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131048988","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":"Functional-Voxel Modelling of Bezie Curves","authors":"A. Sycheva","doi":"10.12737/2308-4898-2022-9-4-63-72","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-9-4-63-72","url":null,"abstract":"The problem of this research is the impossibility of applying parametric functions in theoretical-multiple modeling, that significantly narrows the range of problems solved by analytical models and methods of computational geometry. To expand the possibilities of R-functional modeling application in the field of computer-aided design systems, it is proposed to solve the problem of finding an appropriate representation of parametric curves using functional-voxel computer models. \u0000The method of functional-voxel modeling is considered as a computer graphic representation of analytical functions’ areas on the computer. The basic principles and examples of combining R-functional and functional-voxel methods with obtaining R-voxel modeling have been presented. In this case, R-functional operations have been implemented on functional-voxel models by means of functional-voxel arithmetic. \u0000Based on the described approach to modeling of theoretical-multiple operations for the function area represented by graphical M-images, two approaches to construction a functional-voxel model of the Bezier curve have been proposed. The first one is based on the sequential construction of the curve’s interior by intersection a positive area of half-planes, which enumeration is performed by De Castiljo algorithm. This approach is limited by the convexity of the curve’s reference polygon. This problem’s solution has been considered. The second approach is based on the application of a two-dimensional function for local zeroing (FLOZ), i.e., a nil segment on the positive area of function values. By consecutive unification of such segments it is proposed to construct the required parametrically given curve. \u0000Some features related to operation and realization of the proposed approaches have been described and illustrated in detail. The advantages and disadvantages of described approaches have been highlighted. Assumptions about applicability of proposed algorithms for Bezier curve functional-voxel modeling in solving of various geometric modeling problems have been made.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116439864","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":"Overview of Geometric Ways to Increase the Constructions’ Specific Strength: Topological Optimization and Fractal Structures","authors":"L. Zhikharev","doi":"10.12737/2308-4898-2022-9-4-46-62","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-9-4-46-62","url":null,"abstract":"The paper is an overview of geometric methods for increasing the specific strength of parts and constructions. In the making of engineering knowledge it had been deduced by theoretical and empirical ways a number of rules for specifying the shape of bodies withstanding the loads applied to them. So, in construction, they prefer to use an I-beam instead of a beam with rectangular section, since the first one is able to withstand a large load with a similar mass and the same material, that is, with a certain loading scheme, the I-beam has a greater specific strength due to the features of its geometry. The basic principles of creating such a geometry have been considered in this paper. \u0000With the development of the theory of strength of materials, as well as methods for automatization of design and strength calculations, it became possible to create the shape of parts optimized for specific loads. Computer generation of such a form is called topological optimization. A lot of modern research has been devoted to the development and improvement of algorithms for topological optimization (TO). In this paper have been described some of TO algorithms, and has been presented a general analysis of optimized forms, demonstrating their similarity to fractals. \u0000Despite the rapid development of topological optimization, it has constraints, some of which can be circumvented by using fractal structures. In this study a new classification of fractals is presented, and the possibility of their use to create parts and constructions of increased specific strength is considered. Examples for successful application of fractal geometry in practice are also presented. \u0000The combination of principles for designing strong parts and fractal shaping algorithms will make it possible in the future to develop the structure of strong elements applicable to increase the constructions’ specific strength. Further research will be devoted to this.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115220259","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":"Content of the “Geometric Modeling” Course for the “Mathematics and Computer Science” Training Program","authors":"A. Zaharov, Y. Zakharova","doi":"10.12737/2308-4898-2022-9-4-35-45","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-9-4-35-45","url":null,"abstract":"In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization). \u0000The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well. \u0000Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented. \u0000The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students. \u0000The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126060629","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":"Loci Equidistant from Two Given Geometric Figures. Part 5: Loci Equidistant from a Sphere and a Plane","authors":"Vladimir Vyshnyepolskiy, E. Zavarihina, K. Egiazaryan","doi":"10.12737/2308-4898-2022-9-4-22-34","DOIUrl":"https://doi.org/10.12737/2308-4898-2022-9-4-22-34","url":null,"abstract":"In this paper have been investigated the loci equidistant from sphere and plane, and properties of obtained surfaces have been studied. Four options for possible mutual arrangement of plane and sphere have been considered: the plane passes through the center of the sphere; the plane intersects the sphere; the plane is tangent to the sphere; the plane passes outside the sphere. \u0000In all options of the mutual arrangement of the sphere and the plane, the loci are two surfaces - two coaxial confocal paraboloids of revolution. The general properties of the obtained paraboloids of revolution have been studied: foci and vertices positions, axes of rotation, the distance from the sphere center to the vertices of the paraboloids, the distance between the vertices of the paraboloids, and the position of the directorial planes have been defined. \u0000Have been derived equations for the surfaces of the loci equidistant from the sphere and the plane: various paraboloids of revolution. \u0000The loci in each of the four options for the possible mutual arrangement of the plane and the sphere are as follows. 1. The original plane passes through the sphere center – two coaxial confocal multidirectional paraboloids of revolution symmetric relative to the original plane. 2. The initial plane intersects the sphere – two coaxial confocal multidirectional but not symmetrical paraboloids of revolution, since the circle of intersection of the plane and the sphere does not coincide with the diameter of the sphere great circle. 3. The plane is tangent to the sphere – a paraboloid of revolution and a straight line (more precisely, a second order zero-quadric – a cylindrical surface with zero radius) passing through the tangency point of the plane and the sphere and the sphere center. 4. The plane passes outside the sphere – the equidistant loci will be two coaxial confocal unidirectional paraboloids of revolution.","PeriodicalId":426623,"journal":{"name":"Geometry & Graphics","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125754219","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}