The Origins of Formation of Descriptive Geometry

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. The 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. Most 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.

Spatial Geometric Cells — Quasipolyhedra

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.

Features of Distance Learning in Geometric and Graphic Disciplines Using Methods of Constructive Geometric Modeling

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. In 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.

Introduction of Electronic Technology into Education

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. 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. The 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. 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.

Geometric Locations of Points Equally Distance from Two Given Geometric Figures. Part 4: Geometric Locations of Points Equally Remote from Two Spheres

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. 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. 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).

Intersection Operation on a Complex Plane

Two plane algebraic curves intersect at the actual intersection points of these curves’ graphs. In addition to real intersection points, algebraic curves can also have imaginary intersection points. The total number of curves intersection points is equal to the product of their orders mn. The number of imaginary intersection points can be equal to or part of mn. The position of the actual intersection points is determined by the graphs of the curves, but the imaginary intersection points do not lie on the graphs of these curves, and their position on the plane remains unclear. This work aims to determine the geometry of imaginary intersection points, introduces into consideration the concept of imaginary complement for these algebraic curves in the intersection operation, determines the form of imaginary complements, which intersect at imaginary points. The visualization of imaginary complements clarifies the curves intersection picture, and the position of the imaginary intersection points becomes expected.

Wolfram Mathematica Functional Possibilities for Curved Lines and Surfaces Visualization

The paper states that the current conditions in which the education system is located, and the rapid development of IT require constant improvement of methodological materials, taking into account the full capacity of the current software. Examples of programs necessary for preparation of methodological materials for high-quality classes are given. The purpose of the paper was to identify the Wolfram Mathematica didactic potential when conducting classes in the disciplines of the geometric and graphic profile at a technical high educational institution. In this paper has been performed analysis of literature sources both domestic and foreign ones on the Wolfram Mathematica system application in science and teaching of various disciplines. It has been shown that the program use scope is very wide, in fact, it is comprehensive and requires additional and in-depth study. Examples of Wolfram Mathematica using in mathematics, physics, chemistry, geometry, robotics, virology, and the humanities are given. In the paper have been provided examples for pedagogical design of simulation models for an electronic course on descriptive geometry in the Moodle system. An example of code written in the Wolfram Mathematica is provided. Interactive models developed during the design are presented, which allow the user to change the constructed curves and surfaces’ parameters. Have been defined some functional capabilities of the system, and has been revealed the Wolfram Mathematica didactic potential for teaching geometric and graphic disciplines. Have been considered other authors’ similar models, which can be used in the educational process to increase the clarity of the material presented in the classroom. In conclusion it is pointed out that interactive visualization in the "Descriptive Geometry" discipline, together with classical working practices, significantly enriches the content of geometric education.

Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis

In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.

Approximation of Physical Spline with Large Deflections

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.

Approximation of Physical Spline with Large Deflections

Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.