In the 2017-18 and 2018-19 academic years, in Moscow State Academic Art Institute named after V.I. Surikov a two-year experiment for architecture students on determination variances in different approaches to teaching methods for geometric education was conducted. The experiment was caused by the fact that many teachers believe that if there are computers in the Institute, it is necessary to use graphic programs as soon as possible – immediately introduce students to work on the computer even without allocating hours for this. No one wants to prevent implementation of computers, but we must not forget some nuances related to high technology. As in any case connected with complex hardware unknown for future users, here at the University, at the beginning it is also necessary to teach students how to work with the graphics program itself, and only then allow them perform geometric problems. You can give such an example: put an untrained person at the control panel of interceptor missiles and force him to shoot down a border trespasser in combat conditions. They will notify us that we are engaged in voluntarism. However to put an untrained student at a computer and forcing him to solve a purely geometric problem immediately is not voluntarism. Is it? \nThe experiment had showed that a student, even if he knows how to solve a particular problem, is not ready at all to perform it immediately on a computer in a graphic software. He begin to lose a lot of time getting familiar with the program and only after obtaining at least a minimum of knowledge about working with this program becomes ready to start the task.

Quality of Geometric Education in Various Approaches to Teaching Methods

Previously, the method of rotating of flat geometric objects around curvilinear axes was described by us. The next step in the path of our research should be the development of methods for the automated creation of surfaces digital models obtained by the described rotation method. We have created models of surfaces, the axis and the forming curve of which are circles lying in the same plane. Several cases of mutual disposition for such circles were analyzed. Modeling was carried out using constructive techniques. Surfaces were created using the “surface by section” operation. The centers of such circular sections belong to the axis of rotation, if it is a circle. Using the special tools incorporated in the KOMPAS-3D program, we have cut the surfaces modeled in this way by planes, and obtained a number of flat sections. Taking into account the difficulties occurring during the study of such complex geometric objects by means of flat graphic constructions, as well as graphic computer modeling, we have realized the need to create a mathematical apparatus describing these objects’ shape. The required mechanism should be applicable to any pair of second-order curves interconnected as “axis — generatix”. We have considered an elementary example – the rotation of a point around a curve elliptical axis. In this paper a solution for the problem of finding a system of equations describing a set of point positions, which it will successively take when rotating around the elliptic axis, is presented. It is possible to apply a similar mathematical apparatus to axes having the form of other quadrics, for example, hyperbolas or parabolas, as well as to generatices consisting of more than one point, that is, to forming curves.

Mathematical Interpretation for a Method of Rotation of a Point Around a Second Order Curved Axis

Previously, we developed a constructive method for modeling surfaces of rotation with axes, which were second-order curves such as circle, ellipse, parabola and hyperbola [1]. We also described the principle of constructing a mathematical model [23] corresponding to this constructive technique [2], and expressed the method in mathematical form. In this paper, we applied the previously developed mathematical model that allows us to determine the trajectory of rotation of a point around an elliptical axis to some special cases of the location of this point and identified the features of each of them. We applied the previously accepted terminology and the system of designating points, straight and curved lines involved in the search for circular trajectories of rotation of points. We analyzed the cases of the location of the generating point on the coordinate axes. We determined in mathematical form the trajectory of the point located in these positions. This entry is represented as systems of parametrically given equations. The article also describes a step-by-step algorithm used to find the equation of a circle, which is the trajectory of rotation of a point around an elliptic axis. We applied this algorithm to various positions of the generating point relative to the elliptic axis foci. We applied the previously developed criteria for selecting near and far centers of rotation relative to one of the focuses of the ellipse. The results of these mathematical studies will be used in the future to create a computer program capable of generating digital 3D-models of surfaces formed by the rotation of arbitrary sets forming points around the curves of the axes of the second order.

A Mathematical Description of the Rotation of a Point Around an Elliptic Axis in Some Special Cases

The Main Reasons for Poor Assimilation of Descriptive Geometry

Among specialists prevails the primitive view, according to Prof. G.S. Ivanov, on descriptive geometry only as on a \"grammar of a technical language\", as it characterized V.I. Kurdyumov in the XIX Century. If in the century before last his definition was actual, although many contemporaries had a different opinion, then a century and a half later this definition became outdated, especially since have been revealed the close relationships of descriptive geometry with related sections: analytical, parametric, differential geometry, etc., and descriptive geometry became an applied mathematical science. In this paper it has been shown that an image is obtained as a result of display (projection). In this connection, according to prof. N.A. Sobolev, \"All visual images – documentary, geometrographic, and creative ones – are formed on the projection principle\". In other words, they belong, in essence, to descriptive geometry. Thus, all made by hand creative images – drawings, paintings, sculptures – can be attributed with great confidence to descriptive geometry as a theory of images. These creative images, of course, have a non-obvious projection origin, nevertheless, according to Prof. N.A. Sobolev, \"They, including the most abstract fantasies, are essentially the projection ones\". Further in the paper it has been shown which disciplines apply some or other of graphic models, and has been considered a number of drawings belonging to different textbooks, in which graphic models are present. Thus, clearly, and also referring to the authorities in the area of images and descriptive geometry, it has been proved that each of the mentioned textbooks has a direct or indirect connection with descriptive geometry, and descriptive geometry itself is present in all textbooks, at least, in the technical and medical ones.

The Phenomenon of Descriptive Geometry Existence in Other Student Courses

The textbook contains theoretical information on morphemics and word formation of the modern Russian language; Glossary of terms, plans of practical classes, tasks and exercises for them; tasks for self-control, options for tests and tests; schemes and samples of analysis of language units, a list of scientific and educational literature; questions for the exam. Prepared in accordance with the Federal state educational standard of higher education in directions of preparation "Pedagogical education", "Philology", in accordance with the approximate program of the course "Modern Russian language" and is intended for students enrolled in the profile "Russian language and literature, Russian language and foreign language", "national Philology", for students of the specialist degree, students majoring in "Russian language and literature", master of Philology, as well as for foreign students studying Russian language.

Инна Ковтуненко

Papers

Modern Russian language: Morphemics. Word-formation