The article discusses the Martin attractor (hopalong), which is a type of computational algorithm that allows generating visual structures that can have applications in graphic design or generative art. The available sources (the earliest references to Martin attractors in foreign and russian literature, modern ones - software systems for construction, web-pages of mathematical systems Maple and Wolfram), the mathematical model of the generalized Martin attractor and its special case – hopalong are analyzed. The influence of parameters on the image of hopalong is briefly considered, the irregular nature of the relationship between the image of hopalong and its parameters is shown, a number of hypotheses are put forward that require further verification. Known methods of hopalong coloring are considered and new ones are proposed. A new fractal approach to constructing hopalongs and studying hopalong as a hyperfractal is proposed. The essence of the fractal approach is briefly described, in which iterative calculations are performed for individual points of the image, with each such point corresponding to a strictly defined combination of parameter values. The construction of hopalong images based on the fractal approach is shown. The influence of the choice of statistical criterion on the hopalong image is shown. Two such criteria are shown - the average value of the jump length (the length of the trajectory divided by the number of iterations) and the area of the overall rectangle of the figure. Examples of hopalong images as sections of a hyperfractal object are shown. The methodology for further research of hopalongs for graphic design and generative art problems is outlined in general terms. Examples of problems that can be set for a student research work are given.
Polyhedrons, as one of the types of spatial forms, are of considerable practical interest and, due to this, are widely reflected in various fundamental and applied sciences, in engineering, architecture and design. Polyhedrons can be found in the designs of machine parts, machines and tools. They are also used in the design of complex technical structures, industrial and residential buildings, industrial design objects and jewelry. Convex polyhedrons are of the greatest practical interest. This study substantiates the need for a more in-depth study of the topic «Polyhedra» in the course «Engineering Graphics» for undergraduate students of some areas of Peter the Great St. Petersburg Polytechnic University (SPbPU) due to the specifics of their area of training and further use in practical activities. The article provides an analysis of the degree of disclosure of the topic of polyhedra in the course of stereometry in high school and in the discipline "Engineering graphics" studied at the university. The article shows the results of a statistical study conducted by the authors to control residual knowledge on the topic of «Polyhedra» in the school course of stereometry based on a survey of students from various areas of SPbPU. The purpose of the survey is to identify issues that require clarification and additions when studying the section «Descriptive Geometry» of the discipline «Engineering Graphics». The assessment of knowledge was carried out according to three selected criteria: knowledge of the main types of polyhedra, understanding the topology of regular polyhedra, previously acquired skills in constructing sections of polyhedra (cube) with a plane in an axonometric projection. An attempt was made to systematize convex polyhedra by their degree of regularity in a single classification, combining some provisions of the school course of stereometry and the section «Descriptive Geometry» of the discipline «Engineering Graphics» studied in higher educational institutions. A classification is presented with a detailed description of the designs of various polyhedral shapes and their main characteristics. The classification does not pretend to be a final version, and can be supplemented with polyhedra with other classification features, for example, by the shape of the faces or their inscribeability in a sphere. Separately considered are various types of prisms, pyramids and prismatoids, which are of the greatest practical interest and are studied in detail in the school course of stereometry and the section «Descriptive Geometry» of the discipline «Engineering Graphics» mastered at the University.
The article is devoted to the discussion of some aspects of computer graphics. The relevance of the research topic is due to the constant growth in demand for high-quality and efficient graphical visualization of data. In this regard, there is a need to understand how graphical systems work. The results of the research can be useful in creating various applications for scientific computing, computational modeling, and educational purposes. The purpose of the study is to implement an original algorithm for graphical construction of an ellipse and analyze the effectiveness of the proposed solution using standard graphical primitives in Python. The study attempts to propose original approaches to implementing individual basic elements of computer graphics and to investigate the effectiveness of the proposed algorithms in comparison with standard (library) solutions based on a specific programming language. Indeed, the quality of a graphical application's visualization directly depends on the algorithms chosen for manipulating graphical primitives. Different methods have their own advantages and disadvantages in terms of accuracy, performance, and application areas. This provides a fertile ground for exploring new ideas in the implementation of graphical objects and conducting a comparative analysis of such algorithms with standard solutions based on a specific programming language. The article focuses on two aspects of the research: the mathematical foundation of the proposed method and its computer implementation. It is well-known that ellipses offer greater flexibility in creating visual elements in gaming software. In particular, elliptic curves are used in image compression algorithms. Traditional compression methods based on rectangular blocks often produce imperfect results. Due to their anisotropy, ellipses allow for more accurate approximation of complex shapes in images. The proposed algorithm relies on the unique property of anisotropy, which allows for easy deformation of ellipses in the desired direction, enabling them to adapt more effectively to the specific features of the image being generated. The research described in the article clearly demonstrated that the ellipse drawing method proposed by the authors consistently outperforms the basic Python method on any number of curves in terms of runtime. The difference in the performance of the program runs allows us to make an assumption about the effectiveness of the original method in experiments with high computational load.
A number of topical problems related to geometric and graphic training of students of chemical directions of training a technical university in the study of the discipline "Engineering and computer graphics" are considered. Among these problems, one of the important is the weak basic graphic training of students, including due to the lack of manual drawing in secondary schools and professional educational organizations in the era of global digitalization and the active use of computer-aided design systems in the educational process. Key words: quality of education, engineering graphics, computer graphics, drawing, three-dimensional modeling, KOMPAS-3D, end-to-end graphic preparation. A study conducted in 2024 at the Federal State Budgetary Educational Institution of Higher Education "Yaroslavl State Technical University" (Yaroslavl) at the Department of Descriptive Geometry and Engineering Graphics among students of the Institute of Chemistry and Chemical Technology is described in order to determine the impact of manual drawing on the quality of graphic training of students in conjunction "sketching-manual drawing-computer graphics and three-dimensional modeling" and determine the place and role of manual drawing at present. As a result of the study, students are faced with difficulties when performing practical work both when sketching and manual mechanical engineering, and when using only software products for three-dimensional modeling and creating associative drawings and specifications. Methods have been developed to enhance the effectiveness of this discipline's educational process by harmoniously combining traditional manual drawing techniques with the creation of graphic documents in the KOMPAS-3D CAD system. Justifications are given for the need to form a regional model of end-to-end geometric and graphic training "School College - University”, which makes it possible to optimally approach the process of forming the content of educational disciplines at all levels of education and, thereby, improve the quality of training of university graduates.
The article highlights the problem of using situational tasks in the educational process, along with classical problems of descriptive geometry, taking into account the specifics of future professional activities. At the initial stage of studying descriptive geometry, it is important to acquire theoretical knowledge related to professional activities. Solving any graphical problem establishes interdisciplinary connections, which helps students develop a holistic perception of many practical problems from various fields of science and technology. When studying the topic «Straight Line and Point» it is suggested to use graphical models of engineering practice. Typical and control tasks are presented. Typical tasks are used to prepare for solving problems in practical classes after a specific lecture and after studying the corresponding section in the textbook. Control tasks are the main tasks that students must solve in their workbooks after completing a topic. Some of these tasks can be solved independently or with the help of a teacher in practical classes. The use of graphical generalizations of real objects in engineering practice allows future engineers to acquire theoretical knowledge and the fundamentals of professional practice. The results of this work are successfully used in the performance of calculation and graphic tasks, both in groups and individually. The results of the study lead to the conclusion that in order to acquire the competencies necessary to create and implement engineering solutions, a future specialist must have a solid theoretical foundation based on real-life professional experience.
The article is devoted to the creation of an effective method of mathematical scanning of surfaces, ensuring high accuracy of calculations of their areas. This problem is becoming particularly relevant in the context of the constant growth of requirements for the level of graphic competence of engineers and designers, as well as the active introduction of information technology and automation of project activities. An original method based on dividing a surface into a system of parallel lines and then constructing its sweep on a plane is described. This approach guarantees obtaining a reliable two-dimensional model that adequately reflects the actual configuration of the surface. The procedures for constructing sweeps for both classical deployable surfaces (cone, pyramid) and non-deployable surfaces (sphere, open torus, abstract surface of rotation) are described in detail. The potential application possibilities of the proposed method in industry, architecture and cartography are noted, however, specific practical examples are not given in the article. The results of experimental tests confirming the high accuracy and reliability of the developed methodology are presented. The potential of using this technique in the educational process of technical universities is discussed, where it can significantly enrich the programs of geometric and graphic training of students. The advantages of introducing specialized courses dedicated to new tools of automatic design and spatial modeling are considered. The main conclusions of the article are to identify promising areas for the development of mathematical methods of surface scanning, which will effectively integrate modern information technologies into the processes of professional education and scientific and technological progress. The work is intended for specialists dealing with the theory and practice of design, mathematics and engineering.
The digital direction of education development challenges both teachers to look for new ways to motivate students to study, to independently complete tasks and exercises, and students have received unlimited opportunities to receive answers and solutions in their arsenal, without making much effort to find and solve them. Open, accessible to everyone Internet, almost omnipotent artificial intelligence and other digital technologies are able to solve problems, write abstracts, build images. Students, as representatives of the digital generation, believe that the main thing in learning to give the right answer. However, the main thing in training is to "pass through yourself," conduct a mental analysis, build a structure in your head, etc. That is, the main thing in training is the process, and not the correct result itself. In this regard, the search for new tools and methods for the implementation of an adequate current reality of the training process is very relevant. Not only the introduction of digital technologies into the learning process, but also learning strategies aimed at developing a creative component aimed at involving students in the learning process, developing and maintaining interest in obtaining knowledge and education in general. This is what the main task of pedagogical teams is now. The reactive development and increasing use of artificial intelligence by students in the performance of tasks necessitates the development of motivational tools for the implementation of classroom work, which requires reinforcement of skills and knowledge. Focusing on the development of individual topics with the introduction of new data into the task condition and increasing the complexity of the task can serve as one of the means of changing the approach to the process of mastering the educational material. Stop teaching in old ways. It's time to change with a changing world
An important stage in the design of engineering networks is the development of network configurations that meet pre-defined conditions. The main thing for constructing optimization geometric models is to construct the shortest connecting lines for a given discrete set of points in space. The specified points have different weights. Such models also identify geometric factors and determine certain properties of the network configuration in the considered physical space. Geometric models define the image of the projected object and you can build various network tracing configurations that allow you to choose the optimal network. To solve this engineering problem, the article investigates and develops optimization geometric models on planes with Euclidean, orthogonal, polar metrics and algorithms. A unified algorithm for tracing extensive engineering networks on planes with different metrics has been developed. This goal is achieved by applying and generalizing the Steiner method. As you know, the Steiner problem related to the tasks of constructing a minimal spanning tree has not been solved in a general way. The practical solution to the tracing problem is to find the configuration of an extensive network with the smallest length, which has a greater impact on its cost. At the first stage of design, geometric models of engineering networks are developed, then optimization problems are solved, which are reduced to various generalizations of the problem J. Steiner [5,27]. The algorithm for solving the Steiner problem is based on dividing the given discrete points of the plane into a local subset, using the principle of least elongation and a comparative analysis of various geometric models, which allows you to select and build a network of the required configuration. A unified tracing algorithm is formulated for constructing the configuration of a Euclidean, orthogonal, and polar network. The complexity of solving the Steiner problem and engineering tracing problems is due to the fact that these problems belong to the extreme and to the class of NP-hard discrete optimization problems.
In the framework of solving the problem of approximating free-form surfaces by polyhedra with groups of congruent faces, an idea was proposed to predefine one or more groups of congruent triangles for any triangulation without limiting the flexibility of subsequent optimization. This led to the creation of the concept of αβ-triangulation. The article reveals theoretical aspects of αβ-triangulation in two-dimensional Euclidean space E2 , based on the intersection of elements from set theory, graph theory, and combinatorial topology. The paper presents a detailed study of this new mathematical model. The author introduces a precise definition of the notion of αβ-triangulation, formulates its main properties, and establishes important operations such as cutting and sewing that allow efficient transformation of the triangulation structure. A detailed description is given of the algorithm for constructing an αβ-triangulation from an arbitrary strongly connected triangulation, providing a universal approach to forming optimal structures for specific tasks. Furthermore, proof of consistency and independence of the introduced system of axioms is provided, which significantly strengthens the theoretical foundations of the model. Theoretical development of αβ-triangulation opens up broad prospects for solving problems in computational geometry, offering an effective tool for representing and processing complex forms of spatial objects. However, before practical application of this model, additional experiments and analysis of its efficiency compared to existing analogs are necessary. Thus, further research will determine the role and significance of the proposed model within modern technologies and methods of geometric modeling.
The coordinate method of constructing an electronic image and drawing in the model space of a computer is considered as an alternative to the classical method of two images based on projection. The currently generally accepted understanding of the coordinate method of image construction in axonometry is based on the projection of a system of spatial coordinates. The image itself is based on the coordinates of the points defining the object. The proposed coordinate method is based on a complete rejection of the use of projection. The transformation of the spatial coordinate system is performed by converting a simplex that uniquely defines the three-dimensional model space of a computer. The transformation of a simplex from three-dimensional to two-dimensional is carried out by computer software by visualizing the simplex with izometric and orthogonal views. The image of the full-scale detail is drawn within the boundaries of the transformed two-dimensional simplex in its local coordinate zone. Removing the boundaries of the local coordinate zone of the simplex allows you to start sizing on an electronic drawing, i.e. on orthogonal views of the drawn image. The method is based on the fundamental principles of modern image theory, linking the shape of an object and its image based on strict patterns and modern tools and digital technologies. The features of the method are: • using a three-dimensional simplex as a geometric finite shape. defining a three-dimensional model space of the world Cartesian rectangular coordinate system; • a three-dimensional wireframe model of a full-scale part of known overall dimensions is used as this geometric simplex. The created simplex is a local coordinate zone of a three-dimensional model space, where the coordinates of any point are known; • visualization of the simplex using isometric and orthogonal views by computer software naturally reduces the dimension of the three-dimensional model space to two-dimensional and can be considered as a result of space transformation; • drawing an image of a full-scale detail in the local coordinate zone of a simplex is the construction of an electronic image;• after drawing the image, the boundaries of the local coordinate zone are removed. The image appears in the model space in the coordinate system of the electronic model of the product. This allows you to start sizing up the electronic drawing; • the only initial condition for constructing an electronic image using the proposed coordinate method is to specify the overall size of the simplex;• separating image construction from dimensioning practically frees the created image from unpredictable shape adjustments due to the imposed dimensional dependencies.
Distance-related tasks are constantly encountered in various ways in industry, in transportation, in mathematical programming, and even in space. Basically, these tasks occur where there is a movement of geometric shapes. In this case, we are not talking about the distance between the centers of mass, as we find in physics, we are talking about the distance from the surface of one geometric shape to the surface of another. These tasks are especially pronounced in computer games, where, as shown in a couple of screenshots in the text of the article, secondary "inhabitants", the so-called Non-Player Character (NPC), are very effectively glued into the textures of the environment surrounding the player. The paper shows the general geometric position of the problem, geometric and mathematical solutions to the problem: the distance between two lines, between a line and a surface, between two surfaces, as well as the use of equidistant surfaces as intermediaries – all this using analytical and differential research tools. Then computer solutions are considered: linear search, linear sequential search, pendulum search, one-way search for a solution to the problem. Then the constraints that can be imposed on geometric shapes are considered. In the end, it talks about the simplicity of calculation in computer games: it is rather strange that, having in the arsenal of games such as those that occupy many gigabytes of memory on a hard drive, developers still have not been able to cope with such a small incongruity for the available computing speed in processors. This article will help them to overcome this disadvantage in the shortest possible time.
The article considers the methods of geometric modeling by means of NanoCAD of the following surfaces of the second order: a sphere, an ellipsoid of revolution and a triaxial ellipsoid, parts of which were widely used in classical architecture and continue to be used as prototypes of domed and large-span roofs in modern architecture, which determines the relevance of this work. On the other hand, ellipsoids, as well as spheres, are the simplest surfaces of the second order for modeling and are relatively easy to implement in various CAD systems, including domestic ones SOMPAS-3D and NanoCAD. Depending on the degree of symmetry of the surface, various options for its construction can be used. In particular, a triaxial ellipsoid can be constructed in at least three ways: using programming in the built-in nanoLisp or VBA languages; using blocks with subsequent stretching/compression along the selected directions; using the operation of "pulling by sections", having previously constructed these sections. In case of constructing an ellipsoid of revolution, it is possible to additionally add the operation of rotating a half arc of the ellipse relative to its diameter (there are two options for rotation - relative to the major axis or relative to the minor axis of the generating ellipse). The sphere can be created by any of the above methods, and, in addition, it is possible to use built-in primitives. The article analyzes the advantages and disadvantages of the algorithms of each of the three main methods for constructing a triaxial ellipsoid, as the least symmetrical of the three surfaces under consideration. The labor intensity, the quality of the obtained result and the required degree of user training for choosing one or another option for surface modeling are analyzed. Algorithms for constructing surfaces are also given, accompanied by illustrations of the resulting intermediate and final results. In conclusion, conclusions are made on the degree of appropriateness of using one or another method of construction in the educational process.
The article formulates the urgent problem of ensuring the required quality of engineering specialists and concludes that in order to solve this problem it is necessary to ensure the quality of basic engineering and geometric training in junior years during the study of the geometric and graphic course - Descriptive Geometry, Engineering and Computer Graphics. Definitions of engineering geometry, descriptive geometry, engineering graphics and computer graphics are given. It is noted that from the very first lesson on the geometric and graphic course, students study the basis of the engineering method - a constructive approach, according to which the solution of any problem consists of the stages of analyzing the conditions of the problem, synthesizing the solution (prototype production), validation, research. Examples of the application of engineering and geometric methods in various fields are given: in mechanical engineering and robotics, aircraft, shipbuilding and machine tool manufacturing; in instrument making, radio engineering and radio electronics; in thermal, nuclear, hydropower and electric power engineering; in geoinformation and space systems; in chemistry, chemical technology and biotechnology; in materials science and physical and chemical analysis; in medicine. Additionally, a special type of drawings is considered – nomograms. Examples of modern nomograms are given. Conclusions are made that all specialists in engineering specialties without exception must have knowledge, skills and abilities in working with geometric and graphic information. The fundamental basis for this is the basic geometric and graphic course. Improving the general level of engineering training is impossible without a corresponding increase in the level of basic engineering and geometric training of students. References are given to publications devoted to the analysis of the causes of weak engineering and geometric training, and publications that show ways to ensure the required quality of basic engineering and geometric training.
The article is devoted to the achievements of RTU MIREA students at the Olympiads in geometric-graphic and engineering disciplines held at major technical universities, including Bauman Moscow State Technical University, KNITU KAI named after A.N. Tupolev, R.E. Alekseev Nizhny Novgorod State Technical University, Omsk State Technical University, etc. The main parameters that should be taken into account when choosing an Olympiad are given. The existing All-Russian and International Student Olympiads in the disciplines of "Engineering Graphics" and "Computer Graphics" are described in detail. The system of selection and preparation of students for participation in these events is presented. Students who, as a rule, already have minimal competitive experience are invited to participate, for example, within the framework of the Moscow City or All-Russian Student Olympiad in Descriptive Geometry, Engineering and Computer Graphics. The system of development of intellectual abilities of students, developed at the Department of Engineering Graphics of RTU MIREA, shows its high efficiency, which is confirmed by victories in numerous Olympiads, regardless of the place where they are held. Since 2021, the RTU MIREA team has systematically taken first and prize places in both the team championship and the individual championship, and the number of places increases every year. In 2024 alone, RTU MIREA students took 5 first, 4 second and 1 third place in six different Olympiads. It is noteworthy that the team members are students of completely different fields of study, not directly related to engineering geometry and computer graphics.
Geometric and graphic training is an integral part of the education of technical students. It is aimed at developing skills in working with geometric and graphic models, which allows students to successfully solve problems related to design and construction. This article discusses the directions of development of geometric-graphic training in modern conditions, as well as its influence on the formation of professional competencies among students. Describes a study conducted in the 2022–2023 academic year at the Ural State University of Transport (Ekaterinburg) in order to identify the shortcomings of the existing methodology for geometric-graphic training and develop proposals for its adjustment. Modern pedagogical activities are based on information technologies, e-learning systems, and interactive teaching methods. Much is said about the requirements for students to successfully study the discipline (initial preparation, ability to work with information, quick acquisition of skills in specialized programs), but it is also worth highlighting the qualities necessary for a teacher to ensure a successful educational process: the ability to conduct an electronic training course , communication with students indirectly through special educational platforms, the ability to adapt to new challenges of the modern world (introduction of e-learning, the COVID-19 pandemic, presentation of material using examples understandable to modern students). Modern demands on engineers and designers are high - they need to have a wide range of knowledge and skills to effectively carry out design work. Competent use of graphics programs and applications, as well as the ability to work with 3D models are just some of the key competencies in this area. The development of geometric-graphic training allows students to master these skills already at the training stage, which gives them an advantage when searching for work and increases their competitiveness in the labor market.
Geometry by its nature is the most visual science of all that takes place in the life of mankind, and therefore the most visual for those who study at school or university – any book, even the most non-technical, contains drawings, and drawings, in turn, are directly related to geometry in general and descriptive geometry in general in particular. After all, geometry studies points, lines and surfaces, but if you look closely, any drawing consists of points, lines and surfaces. It is not difficult to notice, so it is difficult to refute. The clarity of the drawings for beginners to learn something is the most important help for understanding the text. From the clarity of the school with its drawings in textbooks, the first step is taken to abstraction: the study of exclusively points and flat geometric shapes that are included in planimetry. After studying the names of geometric shapes, the student begins to study the laws of construction, to study various theorems, to prove them, to apply these proofs to other, more complex theorems. Here comes into play a phenomenon that can be called heuristic thinking based on logical constructions. The article also shows the use of Olympiads – urban and All-Russian – to further improve the development of heuristic thinking among students in order to replenish the departments of geometric profiles of universities as a result, as well as replenish our country with scientists in the direction 2.5.1 — "Descriptive geometry and computer graphics. Digital product lifecycle support." And in other areas, too.
The concept of quality is considered in relation to graphic training in higher education. Among the wide variety of quality assurance issues, competent and result-oriented approaches to assessing the effectiveness (quality) of training are distinguished. It is noted that there are many questions for discussions, however, it is obvious that in the same case, the main role is played by monitoring and audit on the basis of a description of the expected learning outcomes. According to the Federal State Educational Standard, the graduate must be able to solve the “problems of design procedures” correctly, in accordance with the requirements of regulatory documents, to draw up design documentation, use computer and information technologies, and automated design tools. This is the basis of the notorious competency - based approach to training, in which the requirements of certain employers, and not the scientific justification of the existence of the necessary knowledge, skills that do not deny the possessions, are dominated. The situation is described in which the St. Petersburg professional community first encountered a similar approach to training. There is a contradiction between the principles of D.V. Manturov, declaring the need for wide and versatile training of modern engineers, and reality. The comprehensive impact on the quality of preparation of the level of basic training of applicants, the transformations of the organizational structure of universities, attempts to reduce screening in the undergraduate and specialty through the introduction of specific rating systems and technological cards that allow students to certify students (not fulfilling the requirements of the Program) only by passing the minimum level of competence . On the example of an elite educational group, recruited under the Rostec Wings program, the problems of ensuring the quality of graphic training, a tendency to reduce the intellectual level of students due to total gadgetization are shown. It is noted that in this situation to ensure the possible quality of students' training, independent cathedral rating systems that stimulate the educational activities of students should be developed.
The application of Polke's theorem in the search for a coordinate system for an electronic geometric model in the model space of a computer in 2D geometric modeling is considered. The possibility of creating an electronic geometric model in a system of axonometric axes in 2D modeling for scientific and educational purposes using a coordinate method is shown. It is possible to solve problems on axonometric coordinate planes that do not provide solutions in a rectangular coordinate system. In the computer model space, it has become possible to solve classical problems of descriptive geometry, the solution of which is associated only with the method of projecting space onto the projection plane. Secondary axonometry in the system of axonometric coordinate axes in 2D modeling has allowed us to solve a number of problems that do not have a solution in a rectangular coordinate system: • simulate the parallel (oblique) direction of the correspondence of two related shapes; • move the shape in space by rotating around the axonometric coordinate axes; • the construction of an arbitrary relationship of two affine corresponding figures with mutual perpendicularity of the axis of kinship and the direction of kinship; • switch to the coordinate solution method instead of projecting on the projection plane; • vased on the numerical equality of isometric coordinates with natural ones, it is possible to switch from one coordinate system to another right in the process of solving problems. A new reading of Polke's theorem expands the possibilities of the model space of personal computers for solving scientific and educational problems. However, a necessary condition for the implementation of these capabilities is the availability of isometric constructions by software. The possibility of learning how to create an electronic drawing from a full-scale part in the educational process is shown. In this case, it is advisable to use an isometric image as an electronic model, as it has visibility in a single-picture view and simplicity of drawing in a coordinate way. According to the constructed axonometric view, rectangular views are programmatically obtained using rectangular coordinates. A rectangular electronic drawing is formed from these types. If the purpose of its creation is to build a 3D geometric model of an object, then the construction can be continued, considering the created electronic drawing as the initial conditions for building a 3D model of the object
The article summarizes the experience of managing student scientific work at the Departments of graphics of IGEU and RTU MIREA. Examples of topics of scientific and applied interest, the development of which can be conducted by undergraduate students, are given, the current state of development on these topics is given. As a result of attracting students to research, the following scientific results were obtained: the concept of a hyperfractal was introduced, the construction of hyperfractal sections by arbitrarily oriented planes, spheres was considered, the construction of a hyperfractal by adding iterative formulas was considered; an algorithm for visualizing imaginary extensions of complex shapes using computer graphics technologies is proposed, a four-dimensional space is used to model the complex plane, which is displayed using a hyperepure in the form of two three-dimensional orthogonal projections; the use of ordinary Bezier curves of the third order to represent quadratic and cubic parabolas, rational Bezier curves to represent general conical curves and ordinary Bezier curves is shown, as well as the representation of cubic splines using Bezier curves; experimental and analytical methods for constructing trajectories of points rigidly connected to the analogue of the Reulot triangle rotated in a frame of a given shape are considered; the concept of quasi-monograniers is formulated and considered on numerous examples, geometric cells of quasi-monograniers are described, an approach to constructing geometric cells bounded by congruent sections of curved surfaces is formulated and demonstrated; software geometric algorithms for constructing a driven centroid, according to a given leading and location of the axes of rotation for non-circular meshes are developed and implemented. The topics were formulated and an information reserve was created for future promising areas of research work of students at the department.
This article is a continuation of the study of the process of reflection of various objects from curved mirrors. So, earlier in the works [18; 20], a geometric method of constructing the results of reflections was described, which was implemented mathematically in the article [38] using the principles of analytical geometry [6; 11–14; 30]. The obtained analytical equations of the reflection results were visualized in the Wolfram Mathematica [24] program with the ability to dynamically change the parameters of the mirror and the reflected object. However, in the listed works, only cases of reflection on the plane were considered. In this study, attention is paid to a more complex case — reflection in three-dimensional space. The article considered the reflection of a point from surfaces of the second order: a cylinder, a cone, a single-cavity and double-cavity hyperboloids, a sphere, elliptical and hyperbolic paraboloids, and from a torus — a surface of the fourth order. As before, the reflection result obtained in each of the cases is accompanied by a program code for Wolfram Mathematica, which allows the reader to independently simulate the reflection process with different initial parameters. In addition, the relationships between the results obtained were analyzed — both the relationships between the results of various three-dimensional reflections, and the relationship of the results of three-dimensional reflections with the results of similar plane reflections. In particular, on the basis of this, a hypothesis was formulated about the relationship between the curvature of the Gaussian mirror and the dimension of the object obtained as a result of reflection. Based on the results of the work, conclusions were drawn and prospects for further research were outlined. One of them is to obtain an analytical mechanism for describing complex geometric surfaces using a set of simpler objects. This feature will increase the efficiency of specialists when working with reflections from complex surfaces in areas such as aircraft construction (for creating aerodynamic surfaces and air ducts), medicine [40], shipbuilding [7; 31; 42], etc.
