In the present paper geometric locus of points (GLP) equidistant to a sphere and a plane is considered; the properties of the acquired surfaces are studied. Four possible cases of mutual location of a sphere and a plane are considered: the plane passing through the center of the sphere, the plane intersecting the sphere, the plane tangent to the sphere and the plane passing outside the sphere. GLP equidistant to a sphere and a plane constitutes two co-axial co-focused paraboloids of revolution. General properties of the acquired paraboloids were studied: the location of foci, vertices, axis and directing planes, distance between the sphere center and the vertices, the distance between the vertices. GLP for each case of mutual location of a plane and a sphere constitutes: in case one passes through the center of other, two co-axial co-focused oppositely directed paraboloids of revolution symmetrical with respect to the given plane; in case they intersect each other, two co-axial co-focused oppositely directed non-symmetrical paraboloids; in case they are tangent to each other, a paraboloid and a straight line passing through the tangency point; in case they have no common points, a pair of co-axial co-focused mutually directed paraboloids of revolution.
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.
The Department of Engineering Graphics of the RTU MIREA, has been holding the All-Russian Student Competition "Innovative Developments" as part of the All-Russian Student Olympiad in Descriptive Geometry, Engineering and Computer Graphics since 2008. This competition is a stage in the concept of a methodological system for the development of the student's intellectual abilities. The article describes the original system for evaluating works at the competition, the principle of forming the jury, the principles for selecting works for the competition. The goals of the competition were announced, namely: approbation and presentation of new ideas, including in the field of graphic disciplines; development of modern information technologies by students; development of cooperation between teams of university departments and production teams. The criteria for comparing past competitions with each other are described, the methodology for assessing the success of the competition, the success of competitions is analyzed by year. From 2008 to 2022 inclusive representatives of 24 universities presented 96 works for the competition. The article gives the titles, authors and scientific supervisors of the best projects - winners and prize-winners of competitions who scored 200 or more points, ranked by the points scored for the places taken by the students - participants in the competitions. The geography of participating universities is analyzed, universities are ranked by total achievements. There is a positive trend in the number of works submitted to the competition. It is noted that some works became the first step in serious scientific research, for example, “Spatial fractals” by L.A. Zhikharev, later became the topic of his dissertation, “Reflections from curvilinear mirrors in space and on a plane” by O.S. Suntsov are currently one of the registered areas of research at the department.
It is noted that geometric and graphical disciplines traditionally cause difficulties in studying elementary students. It is pointed out that the requirements for the quality of education and ensuring academic performance are mutually inverse and are in conflict with the limited number of teaching hours, which has been steadily declining for many years. Data are given on the number of hours allocated for the study of geometric and graphic disciplines in Russian universities. The main reasons for the problems of geometric and graphic training of students are listed. The first reason is an attempt to give knowledge and skills from three different sections - mathematics (descriptive geometry), computer science (computer graphics) and engineering (engineering graphics) in conditions of shortage of classroom hours. Provides information about the content of classical textbooks of descriptive geometry, engineering graphics. The second reason is the complexity of the development of spatial thinking. It is noted that information technologies that facilitate the understanding of images do not solve the problem of the development of spatial thinking. Information is given on the heterogeneity of students in terms of the psychophysiological features of spatial thinking in general and in terms of various types of mathematical thinking. The third reason is the poor initial preparation of students. Features and reasons are indicated. The main ways of increasing the motivation of students to study geometric and graphic disciplines are given. The fourth reason is the unpreparedness of students for independent work. Reasons are given. The main ways of increasing the efficiency of students' independent work are given. The shortcomings of automatic control in the form of testing and automatic verification of solutions of graphic tasks are considered in detail. The fifth reason is the difference in the requirements for geometric and graphic training for different educational areas, specialties and profiles. The presence in the geometric-graphical cycle of disciplines of an invariant (common for different specialties and training profiles) core and subject settings is noted. The sixth reason is the high requirements for lecturers of geometric and graphic disciplines. The problem of training lecturing staff of higher education in the field of geometric and graphic education is noted. Conclusions are made about the need to create a methodological training system that takes into account and solves these problems.
The paper considers a methodological system for the development of students' intellectual abilities in the process of preparing for Olympiads in descriptive geometry. In accordance with this system, a detailed calendar plan for conducting training sessions for the regional Olympiad in descriptive geometry is presented. The paper is also about the methodological system for preparing and holding a university, city and regional student Olympiads in graphic disciplines. The requirements for the choice of tasks for university Olympiads in descriptive geometry, instructions for holding university Olympiads are given. The paper provides information, including historical information, about the Moscow City Olympiad in descriptive geometry, engineering and computer graphics. To assess the level of organization of city and regional student Olympiads, in particular, the Moscow City Olympiads, a criterion and dependence for its determination are proposed. It was proposed the determination method of the composition of the university team for participation in the regional Olympiad in descriptive geometry is proposed based on four criteria: 1) the sum of points received for solving Olympiad tasks of previous years in the classroom; 2) the sum of points received for solving homework tasks; 3) the sum of points for two rounds of the University Olympiad; 4) integral indicator. The methodological system of preparation and holding of the All-Russian student Olympiads in graphic disciplines is considered. To assess the level of organization of the All-Russian Student Olympiad, a criterion is proposed, which is determined by the sum of four parameters: 1) the number of participating universities; 2) the number of participating students divided by ten; 3) the number of subjects of the Russian Federation (regions) represented at the Olympiad; 4) the number of federal districts of the Russian Federation represented at the Olympiad. The All-Russian Student Olympiads in graphic disciplines, which have been held by RTU MIREA since 1999 (until 2015, Moscow State University of Fine Chemical Technologies named after M.V. Lomonosov), are briefly considered. It was noted that 83 universities from the Russian Federation and one foreign university took part in these Olympiads.
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