Showing papers in "Geometry & Graphics in 2019"

Modern Approaches to Products Design in the Process of Students Teaching in Computer Graphics

Abstract: Using the computer graphics tools in the design allows improve the design quality and speed, as well as provide the qualitative front end engineering design. In this paper the problem related to improvement of quality in engineering training for students of technical high educational institutions, that today is considered as one of the main tasks for the system of higher professional education. A method proposed by authors allows solve this problem in the frame of students training in disciplines of graphic cycle, and directed to introducing the computer technologies in the teaching process of students. This method provides the development of students’ professional skills in the area of products and electronic devices design, and in the front end engineering design. In such a case, design is regarded from the standpoint of project-process approach. Project-process approach is a combination of interrelated projects implemented in the frame of process. A process is considered as a group of projects aimed at achievement of a planned result – a design object model. Design objects models in the area of mechanical engineering, electronics and electric power engineering, are presented as drawings, schemes and 3D models. That is why the emphasis is upon models and drawings building by means of CAD software (Autodesk AutoCAD, Autodesk Inventor, Kompas-Grafik, Kompas 3D). A high level of students’ competences can be achieved by modernization the educational content so that from their first teaching year the students could see the relation of learned graphical disciplines with their future professional occupation and prospects of production development and project activities. State-oftheart software for devices design provides for students an opportunity to extent their possibilities in the learning of graphical disciplines’ courses. The project-process approach is necessary for identifying and studying the relationship between the design as a process and the reformative human activity in general.

Training Procedure in Graphic Disciplines for Students of Technical High Educational Institutions

Abstract: In connection with transition of the higher education system to the bachelor's degree, a number of difficulties has arisen e in the student’s educational schedule. The question is especially acute for disciplines of general education, and a rather large reduction in classroom time has happened. Taking into account the rather low level of students training in the area of drawing, and sometimes its complete absence, due to the abolition of the subject of drawing from the mandatory school program, teachers are faced with more and more tasks on the formation of educational and methodical complexes [3; 6; 15–17; 23]. In this paper has been considered the issue related to complex training in the area of graphic disciplines for students of technical high educational institutions. These disciplines, as is known, are the basis of many special engineering disciplines, such as machine parts, theory of mechanisms and machines and so on. The fundamental component is the design documentation, the possession of which is necessary for a future engineer. Studying the general course of descriptive geometry and engineering graphics is not enough, training should be carried out taking into account the professional orientation and relevant competencies. However, despite the global automation in all areas, it is not necessary to completely abandon the traditional methods of training, for example, in descriptive geometry’s section. This course allows develop students' spatial reasoning [12–14; 19]. Work on descriptive geometry’s tasks gives for a student the opportunity to more clearly understand the projection principles, methods of drawings transformation, formation of complex surfaces, obtaining of visual images by constructing of axonometric projections or performing of a technical drawing. Another question in our opinion is the order for studying the section of engineering graphics, which is more appropriate to study using modern graphics programs [10; 11].

General Principles for Formation of Ruled Surfaces. Part 2

Abstract: In this paper the consideration related to formation of ruled surfaces with a single method for their set up that was proposed in the first part of the work, is continued. In the first part of the work were considered more than twenty variants for set up of ruled surfaces, including four set ups by guide lines, overall known in the literature, for example, in the books of S.A. Frolov, A.V. Bubennikov, M.Ya. Gromov. However, the set up of ruled surfaces with the help of guide lines was proposed in a new aspect – as a constituent of a single method for setting absolutely all ruled surfaces, taking place in science and industry, and with additional variants. Besides, have been proposed equation systems, which solution leads to generation of equation for the given ruled surface. New set ups of ruled surfaces have included eighteen examples, which is the main material of the work’s first part. Also was proposed a table in which have been put all possible variants for set up of geometric figures’ guiding lines to obtain ruled surfaces. Of course, the proposed variants of guiding lines’ combination were presented in the enlarged form. In the proposed paper have been considered new, not presented before, variants for set up of ruled surfaces. Have been presented 19 examples, including the ones with one or two guiding planes, as well as when the guiding line belongs to one of guiding surfaces. Such surfaces can be considered as ruled surfaces of smooth transition. As in the first part of the work, the equation systems leading to the equation of the set ruled surface are proposed.

General Principles for Formation of Ruled Surfaces. Part 1

Abstract: Probably, it is impossible to find such industry where the ruled surfaces would not be used. They are used in agriculture, in the heavy and light industries, in construction, in aircraft manufacturing, and in military art. Ruled surfaces are used in the design of wings, tail and partially fuselage of aircraft, car bodies, in the project engineering of slopes and embankments of auto-roads, abutments of bridge supports, transitions from a vertical quay to inclined walls of embankments, various hydraulic structures, towers, masts, cooling towers, vaults and arches, overlaps of pavilions, circuses, stadiums and other building structures, as well as in the calculation of solar exposure. This paper deals with the formation of ruled surfaces in a single method of their definition. A number of examples for definition of ruled surfaces have been presented. These examples show that in general for definition of ruled surfaces it is required to have three guides and three geometric conditions characterizing the position of a rectilinear generator with respect to each of the guides. Both surfaces and lines can act as guides. The plane is selected separately from other surfaces. The geometric conditions are the intersection with the guide line and the tangent or intersection at a certain sharp angle with the guide surface. The table of 19 variants for guides has been given. An attempt to classify surfaces does not even consider in this paper since it is impossible to classify ruled surfaces, even within its class, due to the lack of a criterion showing their belonging to one or another species. It can be concluded that the classification of surfaces may be used only for educational purposes and in cases where the surface name is obvious.

Loci of Points Equally Spaced From Two Given Geometrical Figures. Part 3

Abstract: The loci (L) equally spaced from a sphere and a straight line, and from a conic surface and a plane, are considered. The following options have been considered. The straight line passes through the center of the sphere (a = 0), at the same time completely at spheres’ positive radiuses a surface of rotation is obtained, forming which the parabola is, and a rotation axis – this straight line. The parabola’s top forms the biggest parallel on the site points of intersection of the parabola’s forming with the rotation axis. Let's call such paraboloid a perpendicular paraboloid of rotation. The straight line crosses the sphere, but does not pass through the center (0 < a < R/2) – a perpendicular paraboloid, at that the surface is also completely obtained at radiuses’ positive values. The straight line is tangent to the sphere (a = R/2) – a surface which projections are parabolas, lemniscates and circles, and a piece from a tangency point to the sphere center – at radiuses positive values; a beam from the sphere center, perpendicular to this straight line – at radiuses negative values, at that the beam and the piece belong to one straight line. The straight line lies out of the sphere (α > R/2) – two different surfaces, having the general properties with a hyperbolic paraboloid, are obtained, one of which is obtained at radius positive values, and another one – at radius negative values. It has been noticed that loci, equally spaced from a sphere and a straight line, and from a cylinder and a point, coincide at equal radiuses and distances from axes to points and straight lines if to take into account the surfaces obtained both at positive, and negative values of radiuses. Locus, equally spaced from the conic surface of rotation and the plane, are two elliptic conic surfaces which in case 7.4.1 degenerate in the conic surfaces of rotation. In cases 7.4.3 and 7.4.4 one elliptic conic surface degenerates in a plane and a parabolic cylinder respectively.

Modern Means of Graphic Disciplines Teaching for Extramural Students

Abstract: When studying ''Engineering and Computer Graphics'' discipline, extramural students are faced with a number of difficulties. Age groups of these students differ from full-time students from behind a greater number of students related to more age categories. Also, unlike full-time students, the level of extramural students’ primary education is higher, but it has been acquired for a long time, and knowledge, in the vast majority of cases, leaves much to be desired. In addition to the described differences it is possible to report a lesser amount of free time, which an extramural student can use for his independent work because of his primary employment’s strained activity timetable. An important moment that plays a key role in discipline understanding is the complexity of \"Engineering and Computer Graphics\" subject itself, which requires drawing skills (in the school some students did not even have such a discipline) and the ability for spatial thinking. In the presented paper have been considered features on age groups, primary education and drawing skill level for two streams of extramural students learning on the \"Railway Operations\" specialty in the Russian University of Transport (MIIT). In view of students’ contingent peculiarities the use of modern teaching tools in the process of studying \"Engineering and Computer Graphics\" discipline has been suggested as a method for enhancement of effectiveness for understanding of educational material. As an illustration of obtained theoretical concepts has been presented a plan for carrying out a laboratory work on \"Engineering and Computer Graphics\" discipline using modern teaching techniques. In the process of performing the laboratory work, modern teaching tools are used, and after its completion the trainees receive a useful solid piece (a stand for a smartphone) made on a 3D printer, obtained with the help of a three-dimensional model prepared by students, that increases the efficiency of received material’s understanding.

Modeling Approximating the 16-Point Compartment the Response Surface With Respect To the Solution of the Inhomogeneous Heat Equation

Abstract: The paper proposes a computational method for solving differential equations of mathematical physics by approximating the desired solution using geometric objects of multidimensional space passing through predetermined points. The essence of the method is to simulate an approximating geometric object of a multidimensional affine space constructed on a regular multidimensional network of points. In this case, the response function values satisfying the solution of the original differential equation are calculated at the nodal points of the network. Modeling of approximating geometric object is carried out by means the arcs of algebraic curves passing through predetermined points. It should be noted that taking into account the boundary conditions does not require changes in the geometric algorithm or point equations. It is sufficient to use the necessary coordinates of the nodal boundary points corresponding to the boundary conditions of the solution of the differential equation. To achieve the required accuracy of the solution of differential equations, it is sufficient to compact the reference network of points. Under such conditions, it is possible to use as a single geometric object to approximate the solution of the differential equation, and composite, based on the simulation of multidimensional contours on a regular network of points of multidimensional space. A geometric classification of differential equations depending on the number of parameters determining the approximating geometric object in multidimensional space is proposed. An example of solving the inhomogeneous heat equation by means of an approximating response surface passing through 16 predetermined points is given. In this case, the required approximating compartment of the response surface passes through 3 straight lines that correspond to the boundary conditions and satisfies the solution of the original differential equation at the nodal points of the 16-point network. A comparison of the results of solving the inhomogeneous heat equation approximated by a 16-point compartment of the response surface with the reference compartment of the surface obtained by the method of separating variables is also presented.

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

Abstract: 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.

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

Abstract: 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.

Cyclographic Interpretation and Computer Solution of One System of Algebraic Equations

Abstract: The subject of this study is an algebraic equation of one form and a system of such equations. The peculiarity of the subject of research is that both the equation and the system of equations admit a cyclographic interpretation in the operational Euclidean space, the dimension of which is one more than the dimension of the subspace of geometric images described by the original equations or system of equations. The examples illustrate the advantages of cyclographic interpretation as the basis of the proposed solutions, namely: it allows you to get analytical, i.e. exact solutions of the complete system of equations of the considered type, regardless of the dimension of the subspace of geometric objects described by the equations of the system; in the geometric version of the solution of the system (the Apollonius and Fermat problems), no application of any transformations (inversions, circular transforms, etc.) is required, unlike many existing methods and approaches; constructive and analytical solutions of the system of equations, mutually complementary, are implemented by available means of graphic CAD and computer algebra. The efficiency of cyclographic interpretation is shown in obtaining an analytical solution to the Fermat problem using a computer algebra system. The solution comes down to determining in the operational space the points of intersection of the straight line and the 3-α-rotation cone with the semi-angle α = 45° at its vertex. The cyclographic images of two intersection points in the operational space are the two desired spheres in the subspace of given spheres. A generalization of the proposed algorithm for the analytical solution of the Fermat problem for n given (n – 2)-spheres in (n – 1)-dimensional subspace. It is shown that in this case the analytical solution of the Fermat problem is reduced to determining the intersection points of the straight line and the (n – 1)-α-cone of rotation in the operational n-dimensional Euclidean space.

Graphic Algorithms for Constructing a Quadric, Given Nine Points

Abstract: The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of any number of points of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J.H. Engel’s well-known algorithm, which makes it possible to define the ninepoint quadric metric. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem. The article discusses two well-known graphical algorithms for constructing a quadric (the Rohn — Papperitz algorithm and the J.H. Engel algorithm) and proposes a simplified version of the J.H. algorithm. For its constructive implementation using computer graphics. All algorithms allow you to determine the set of points and the set of flat sections of the surface of the second order, given by nine points. The Rohn — Papperitz algorithm, based on the spatial configuration of Desargues, is best suited for its implementation on an axonometric drawing using 3D computer graphics. Algorithm J.H. Engel allows you to solve a problem on the plane. The proposed simplified constructive version of the algorithm J.H. Engel is supplemented with an algorithm for constructing the principal axes and symmetry planes of a quadric, given by nine points. The construction cannot be performed with a compass and a ruler, since this task reduces to finding the intersection points of two second-order curves with one known general point (third degree task). For its constructive solution, a computer program is used that performs the drawing of a second order curve defined by an arbitrarily specified set of five points and tangents (both real and imaginary). The proposed graphic algorithm can be considered as an alternative to the algebraic solution of the problem.

Geometric Model for Generation of Contour- Parallel Lines’ Family for Cutting Tool’s Path Automated Computation

Abstract: In this paper has been proposed a geometric model for forming problem of contour-parallel lines (equidistant lines) for a flat contour with an island, and has been obtained the problem’s analytical solution, which is relevant for computer-aided design of cutting tools processing pocket surfaces on CNC machines. The proposed geometric model is based on cyclograph mapping of space on a plane. Beyond the analytical solution the geometric model differs from the known algebraic models and their solutions for considered forming problem also by the fact that it allows obtain a more complete and evident representation on the relationship and interaction for all its geometric components at the stages of 3D computer visualization. A 3D geometric model based on a cyclograph mapping of space has been proposed for obtaining the families of equidistant lines for connected and multiply connected regions with closed contours taken as a basis for pocket surfaces modeling. An algorithm for the analytical solution of the problem related to equidistant families generation is getting from the geometric model. All stages of the analytical solution are accompanied by a figurative representation of geometric objects and their relations in the geometric model’s virtual electronic space. The proposed in this paper algorithm for the case of a doubly connected polygonal region can be used as a basis for generation of equidistant families for multiply connected polygonal regions. The presence of the analytical solution for the problem related to equidistant families generation simplifies greatly the automated calculation of the tool path and preparation of control programs for pocket surfaces manufacturing on CNC machines. Have been presented an example and algorithm providing support for working capacity of the proposed geometric model for considered forming problem.

Golden Section and Golden Rectangles When Building Icosahedron, Dodecahedron and Archimedean Solids Based On Them

Abstract: A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.

Discrete Geometric Models for Estimating the Degree of Shading in Solar Energy

Abstract: Geometric simulation and its software for estimating the efficiency of deployment of solar panels on spacecraft and solar concentrators on the ground are considered in this work. Both the physical and mathematical set up of the problem for estimating the energy efficiency of solar panels, taking into account their shading both by each other and by other elements of a space station has been described in this paper. It has been shown that the known methods for mechanization and automation of such calculations are focused on objects of relatively simple geometric shapes (such as buildings), and are inefficient for objects of complex and diverse geometric shape, characteristic both for spacecraft themselves and their solar panels. Therefore, to solve this problem, a receptor (voxel) geometric model digitizing the computational space has been chosen. The receptor model’s uniqueness is that comparing the values of receptor codes allows easy determine the intersection of objects. Has been described a developed receptor geometric model for estimating the effective area of solar panels, taking into account their shading when the object (for example, a spacecraft) is illuminated by a flow of solar energy from a given direction. The essential difference between the developed receptor geometric model and the classical one is that the former is multiform, i.e. uses not the 2-digit code (0 and 1), but the 4-digit one (0, 1, 2 and 3). Has been demonstrated a software implementation of the described geometric model in C#, and a graphical shell developed for this problem, allowing see the obtained results’ numerical values. Have been provided examples of its implementation in solving of practical problems. The results of verification for the described receptor geometric model have been demonstrated. All this allows speak about efficiency of using receptor geometric models both in singular computation calculations and for creating the appropriate algorithmic, mathematical support and software for the corresponding CAD systems.

Geometric-Graphic Student Olympiad in St. Petersburg

Abstract: Two geometric-graphic Olympiads are held in St. Petersburg: the urban Olympiad in descriptive geometry, initiated by BSTU “VOENMECH” since 1979, and the Olympiad called “Engineering Computer Graphics”, conducted by LETI and ITMO. The peculiarity of the Olympiad in descriptive geometry is its democracy. Its content and organization features are supervised by the professional community, which is united by the section “Geometry, Graphics, Design” of the House of Scientists named after M. Gorky. Competition tasks are developed not only by the organizers. Accepted and suggestions of participants. The content of the Olympiad eventually changes, contributing to its development. Thus, at the suggestion of a number of participants, a comprehensive task was introduced to know the main sections of the course, the task of composition of the task. Despite the withdrawal of the course of descriptive geometry from a number of standards, the fundamentals of this discipline are kept up to date with engineering graphics, which ensures participation in the Olympiad of 7–10 leading technical universities of the city. Olympiad in engineering computer graphics can be attributed to the problem: the level of tasks, focused exclusively on the bachelor degree; on the principles of organization (problem bank of tasks, features of the appeals process); authoritarian chairman of the jury. As a result, it was boycotted by universities, which, unlike the winners, show decent results at All-Russian Olympiads. Among the All-Russian Olympiads, the Olympiad held by MIT stands out. The organizers managed to create a complex competition, which included the ability to solve interesting applied problems on an orthogonal drawing, possession of tools for creating three-dimensional models and drawings of technical products. Given the experience of MIT, the need to create in St. Petersburg an alternative computer graphics competition that is not purely instrumental in nature, the GUT organized an Olympiad called “Total Drawing”. This competition, held under the direction of the chairman of the jury of Professor D.Voloshinov, is gaining popularity. The article discusses and analyzes the principles of organization and the content of these competitions, offers for their modernization and development.

Visualization of Regular Polyhedrons during Their Formation

Abstract: In this work the automated formation of surfaces correct to convex polyhedrons of Platon and two regular not convex star-shaped polyhedrons of Kepler-Poinsot by the kinematic method. Researches on realization of a goal were carried out in the environment of AutoCAD with use of the programs developed in the functional Autolisp programming language which is built in AutoCAD. The AutoLisp language and the AutoCAD environment are chosen for achievement of a goal as they allow showing bodies in the movement. The technique of formation of electronic models of the polyhedrons necessary for performance of visualization of polyhedrons is stated. The model is a set of compartments of a surface, issued in the form of the block. The user function in the AutoLisp language which identifier is team in the environment of AutoCAD is developed for each model. Each compartment was placed in the drawing layer which is taken away for it. When developing the user functions were taken into account to a possibility of the AutoCAD environment – the available teams for formation of surfaces. The user functions in the AutoLisp language for formation of the studied surfaces in the environment of AutoCAD are made by the defrosting method of the block containing surface compartments. In the course of \"defrosting\" of layers with compartments on the screen of the monitor process of formation of a surface is shown – drawings of compartments of a surface appear one by one. The last drawing is an image of a surface. The user functions in the AutoLisp language for formation of the studied surfaces in the environment of AutoCAD are made. The fragment of the program by training of one side of a tetrahedron is given Drawings of elements of surfaces of all regular polyhedrons of Platon and star-shaped polyhedrons of Kepler-Poinsot are provided in initial situation and in the course of stage-by-stage formation of these surfaces, the programs received in the environment of AutoCAD with use in the AutoLisp language. Drawings of elements of surfaces of all regular polyhedrons of Platon and star-shaped polyhedrons of Kepler-Poinsot are provided in initial situation and in the course of stage-by-stage formation of these surfaces, the programs received in the environment of AutoCAD with use in the AutoLisp language. The possibility of formation of surfaces of regular polyhedrons is shown by a kinematic method: the movement rectilinear forming on the directing lines as which edges of polyhedrons are used.

Imaginary Points in Cartesian Coordinate System

Abstract: Geometric models are considered that allow symbolic representation of imaginary points on a real Cartesian coordinate plane XY. The models are based on the fact that through every pair of imaginary conjugate points A~B with complex coordinates x = a ± jb, y = c ± jd one unique real line m passes. For the image of imaginary points, it is proposed to use the graphic symbol m{OL} consisting of the line m passing through the imaginary points, the center O of the elliptic involution σ with imaginary double points A~B on the line m, and the Laguerre point L, from which the corresponding points involutions σ are projected by an orthogonal pencil of lines. According to A.G. Hirsch, the symbol m{OL} is called the marker of imaginary conjugate points A~B. A theorem is proved that establishes a one-to-one correspondence between the real Cartesian coordinates of the points O, L of the marker, and the complex Cartesian coordinates of the pair of imaginary conjugate points represented by this marker. The proved theorem allows us to solve both the direct problem (the construction of a marker depicting these imaginary points) and the inverse problem (the determination of the Cartesian coordinates of imaginary points represented by the marker). A graphical algorithm for constructing a circle passing through a real point and through a pair of imaginary conjugate points is proposed. An example of the graph-analytical determination of the Cartesian coordinates of imaginary points of intersection of two conics that have no common real points is considered.

General Principles for Formation of Ruled Surfaces. Part 3

Abstract: We continue to consider the formation of ruled surfaces with a single method of their formation. In the first and second parts have been introduced more than forty options for specifying surfaces. These formations with the help of guide lines and surfaces are considered in a new aspect – as formation in science and production of all of ruled surfaces. In this paper, we consider new options for specifying ruled surfaces. Generalized the task for torso surface. If in textbooks on descriptive geometry torso surface is given as 1∞ straight lines, tangent to the spatial line, the proposed version of the are considered three guides: two curves (surface) plus a plane touching to both curves (surfaces). It is shown that three guides are also required to set screw ruled surfaces. The concept of a limit ruled surface is introduced to determine the region of existence of ruled surfaces. The table of the simplest geometrical figures for obtaining congruences is given. A number of examples of congruences obtained by using two guides are given. All these examples once again confirmed the validity of the law of assignment of ruled surfaces using three guides and three geometric conditions characterizing the ratio of the forming line to these three guides. The three geometric conditions are the contact of the forming line to the guide surface and the intersection of the forming line with the guide line. The proposed task of ruled surfaces can be used in the consideration of ruled surfaces in lectures on descriptive geometry and other geometric disciplines.

Visualization of Linear Shifts of Nodal Points during Implementation of Instantaneous States of Various Configurations of an Android Robot Arm

Abstract: During planning the movement of an android robot arm in an organized space, there is a need in reducing calculation time of the trajectory in the space of generalized coordinates. The indicated time significantly depends on calculation time the vector of increments of the generalized coordinates at each step of calculations in the synthesis of movements along the velocity vector. In this paper, geometric studies were carried out based on the visualization of patterns of changes in the average displacement of the nodal points of the hand mechanism of an android robot while implementing instantaneous states. On the basis of the geometric analysis of the indicated displacements, a method is proposed which makes it possible to reduce the time of iterative search for the increment vector of generalized coordinates. Also images are shown of multiple positions of arm mechanism links on the frontal and horizontal projections when implementing instantaneous states. This images allows to make a graphic interpretation of manipulator mechanism maneuverability at each point of the configuration space. Hypersurfaces in four-dimensional space are used to establish the analytical dependencies reflecting the relationship of the average displacement of manipulator mechanism nodal points and the generalized coordinates that defining the positions of the manipulator configurations. For this purpose, the equations of interpolating polynomials located in three mutually perpendicular planes are used. Based on these three interpolating polynomials, a third-order hypersurface equation is obtained, which reflects the interrelation of geometric and kinematic parameters. The article also presents the results of virtual modeling of android robot hand mechanism movement, taking into account the position of the restricted area in the AutoCAD system. The results of calculations using the obtained analytical dependencies showed a reduction in the calculation time of test tasks. The conducted studies can be used in the development of intelligent motion control systems for autonomously functioning android robots in an organized environment without the participation of a human operator.

A Geometrical Model Producing a Surface Equivalent To the Working Surface of the Gear Tool «Hob»

Abstract: Existing mathematical models for calculating worm gearing [34; 38] are quite complex and do not always provide an opportunity to quickly and accurately obtain the desired result [1; 3; 24–26]. A simpler way to find a suitable gearing option that satisfies the task is using computer simulation methods and computer graphics, and in particular solid modeling algorithms [4; 5; 30–33; 36; 37]. This information can be entered into the computer in order to simulate control of the movement of the cutting tool. Ultimately, this boils down to the problem of analytic description and computer representation of curves and surfaces in three-dimensional space [18–20]. Despite the diversity and good development of the calculation methods, and the analysis of the geometrical parameters of the worm gear, there is a lack of means and methods for displaying the process of forming the working surfaces of the worm gear elements [28; 29; 41]. There are no computer algorithms for obtaining the producing surfaces of a worm cutter, which are obtained by a tool with a modified producing surface. A change in the geometric shape of the tool producing surface will lead to a change in the working surfaces of the worm wheel and turns of the worm, which may lead to an improvement in their contact. This article shows the application of the developed methods and algorithms of geometric and computer modeling, which are designed to form the helical surface of the turns of the worm and the teeth of the worm wheel. Their use will speed up the process of calculating intermediate adjustments of machines used for cutting worm gears, bypassing complex mathematical calculations that, under conditions of aging of the gear-cutting machine fleet, their wear and inevitable reduction in the accuracy of their kinematic chains. This can be achieved only by applying a deliberate modification of the contacting surfaces, which reduces the sensitivity of the worm gear to the manufacturing errors of its elements, which allows to maintain the quality of the gears produced at a sufficiently high level.

Analytical Dependences of the Kinematic Forming Primary Surfaces of the Worm Gear

Abstract: The run-in method for obtaining the screw surface of a worm is based on the use of the worm gearing principle. In this case, the shaping surface (cutting tool) and the workpiece constitute a gear pair [4; 7]. The use of geometric modeling methods [8; 9] to simulate the process of shaping the working surface is based on the relative movement of intersecting objects in the form of a “workpiece-tool” system. This allows to obtain the necessary geometrical model that accurately reproduces the geometric configuration of the surfaces of the teeth of spatial gears [14; 15], where the producing surface of the tool moves in the selected reference system and its position at an arbitrary time is determined by a certain parameter, the motion parameter. The position of the cutting tool at the beginning and at the end of each pass is calculated using parametric equations, which make it possible to calculate the tool path for accurate processing of spatially complex surfaces [16–19]. In the process of mechanical action of a tool on a solid (workpiece), shaping occurs, which consists in the movement of the tool relative to the workpiece [30; 31]. The use of modern methods of three-dimensional computer graphics allows us to improve and accelerate the process of designing technological operations of tooth profiling, providing the final forms of the surfaces of the teeth in the form of visual and accurate computer-based solid-state models [39; 40]. The method is based on a virtual representation of the process of shaping in the form of intersection of solid-state 3D models of two objects (tools and workpieces), which generally perform a screw relative motion. As a result, the working surfaces of the teeth are formed as the envelopes of the tool producing surface [32–34]. For the formation of fission surfaces, mathematical dependences were obtained, which allow one to describe the mutual motion of a worm, a worm gear and a disk cutter [35–37]. These analytical dependences make it possible to simulate the virtual process of forming the side surfaces of the worm gearing elements [1–3; 5; 6]

Approximation of Linear Sets in the Plane

Abstract: A few general lines in the ordinary Euclidean plane are said to be line generators of a plane linear set. To be able to say that every line of the set belongs to one-parametrical line set we have to find their envelope. We thus create a pencil of lines. In this article it will be shown that there are a finite number of pencils in one linear set. To find a pencil of lines the linear parametrical approximation is applied. Almost all of problems concerning the parametrical approximation of figure sets are well known and deeply developed for any point sets. The problem of approximation for non-point sets is an actual one. The aim of this paper is to give a path to parametrical approximation of linear sets defined in plane. The sets are discrete and consist of finite number of lines without any order. Each line of the set is given as y = ax + b. Parametrical approximation means a transformation the discrete set of lines into completely continuous family of lines. There are some problems. 1. The problem of order. It is necessary to represent the chaotic set of lines as well-ordered one. The problem is solved by means of directed circuits. Any of chaotic sets has a finite number of directed circuits. To create an order means to find all directed circuits in the given set. 2. The problem of choice. In order to find the best approximation, for example, the simplest one it is necessary to choose the simplest circuit. Some criteria of the choice are discussed in the paper. 3. Interpolation the set of line factors. A direct approach would simply construct an interpolation for all line factors. But this can lead to undesirable oscillations of the line family. To eliminate the oscillations the special factor interpolation are suggested. There are linear sets having one or several multiple points, one or several multiple lines and various combinations of multiple points and lines. Some theorems applied to these cases are formulated in the paper.

“Engineering and Computer Graphics” Discipline in the System of Higher Military Education

Abstract: In this paper features for creation of educational process in a military higher education institution when studying “Engineering and Computer Graphics” discipline are revealed. Military education is a part of the Russian Federation’s education system. In conditions of the Armed Forces modernization and development of new methods and ways for conduct of operations the young officers’ perfection acquires a big significance. Requirements applicable to military specialists reflect the concept of educational activity in general – possession of strong theoretical knowledge and formed practical skills at the tasks solution. The big part in the system of development for military engineering education is assigned to practical orientation of training. Future officer has to understand the processes for design, production and operation of cars and mechanisms with varying complexity, therefore be able to work with design documentation of any kind. In the course of “Engineering and Computer Graphics” discipline studying cadets are learned to read and carry out drawings, to develop their technical support, and also to design and model both two, and three-dimensional objects on a plane and in space. The efficiency of graphic training in a greater degree depends on educational activity’s organization. Application of education traditional forms in combination with innovative practice and methods, development of the system of didactic tools focused on increase in educational process’s intensity is the most optimal one for achievement of training maximum results. During realization of the tasks set by the state for training of competent military specialists, the educational process based on principles of personally focused training with developing orientation has been organized by “Engineering and Computer Graphics” discipline teachers of Military Academy of Troops Air Defense of Russian Federation Armed Forces. The developed system of didactic tools enhances the intensity and productivity of cadets’ educational activity, helps to cultivate professional qualities of future military specialists.

About Building of Models for Objects in Space of Four and More Dimensions in Educational Process

Abstract: In this paper the visibility concept in the context of modeling of multidimensional spaces’ objects is clarified. It is concluded that such model’s visibility should be defined as unambiguity and completeness of information presented in this model and consistent with the student’s experience in the area of modeling a space of higher dimension (3D) by elements of spaces of lower dimension (2D). Such possibilities are presented by the generalized complex drawing. Examples for objects 4D-modeling using two 3D or three 2D flat projections are presented, some properties of the 4D generalized drawing are listed. The solution of problems with 4D-objects is considered on the example of 4D-pyramid section construction, and deploying its lateral surface. It is shown that to simplify the solution of these problems is required a system allowing automatically perform repetitive sequences of constructions. A list of elementary constructions is presented, and a method for recording of composite constructions and based on them algorithms for problems solving is shown. It is demonstrated that a 3D-scan of 4D-pyramid’s lateral surface, constructed with 2D drawing, can be imported into CAD as a 3D-model. The deploying of the 4D-cone’s lateral surface is considered. The resulting scan’s surface 3D-model imported into CAD is shown. Cases are indicated when a multidimensional space’s object 3D-model may be more visual than a flat one. As an example, 2D-models for imaginary continuations of lines and circles of the complex plane (simulated by Euclidean 4D-space) are presented. Two 3D-projections for imaginary continuations of a circle with a real radius as 3D-space surfaces are shown. It is noted that in order to combine in an educational course the multidimensional space’s objects modeling and work in CAD the tasks on designing of complex technical surfaces by means of output in multidimensional space are suitable. A brief review of sources is given, in which theoretical foundations and the use of key geometrical methods for surfaces construction are considered; an example of a surface constructed by a progressive key method and imported into CAD is shown. The concept of a product’s electronic model (PEM) is described, in which the modeled object’s 3D-simulator as its visual representation is combined with numerous 2D-layers, which elements automatically perform geometrical and graphical calculations in spaces of any dimensions, and control 3D-model’s dimensions and shape through constructive and parametric links. Conclusions are drawn about the possibility of visual multidimensional modeling in the educational process, the advantages of using a complex drawing for solving of problems with multidimensional objects, the need to use special systems of constructive geometric modeling that automate repetitive sequences of constructions. It is also concluded that multidimensional objects’ 2D-models can and should be directly involved in the PEM formation.

About the Science «Geometry of Technical Objects»

Abstract: The work proposes the development of the science \"Geometry of technical objects\" as an interdisciplinary science, which is a part of the sciences \"Geometry\" and \"Design of technical objects\". The object of the science is the geo metrical design, and its subject is the determination of the geometry of assembly units and parts in terms of the functions they perform. The co mponents of science are examined: tasks; laws, regularities, principles and rules; methods and terminology. The existing science of \"Geometry\", part of mathematics, explores spatial structures and relationships, as well as their generalizations, but does not consider the relationship between geometry and functions of real technical objects. There are many developments that are associated with the geometry (especially the shape) of important details of specific types sorockyj units, however, the studies that discuss the General questions of geometry invariant with respect to the types (names) of Assembly units available, but their number is negligible and they can be presented as General guidelines. The exceptional importance and the possibility of a common approach allow us to offer the formation of specific geometry, which refers to real technical objects (Assembly units and flying). Geometry plays a fundamental role in the functioning of technical objects, so it is especially important to look for common approaches to its (geometry) disclosure. The geometry of technical objects can also be considered as an opportunity to transform the main content of graphic disciplines in technical universities.

Reflection from Curved Mirrors in a Plane

Abstract: Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.

Competence Approach in Teaching the Topic "Tangent Plane and Normal"

Abstract: Qualified presentation of the topic \"Tangent Plane and Surface Normal\" in terms of competence approach is possible with the proper level for students' attention focusing on both intra-subject and inter-subject relations of descriptive geometry. Intra-subject connections follow from the position that the contingence is a particular (limit) case of intersection. Therefore, the line of intersection of the tangent plane and the surface, or two touching surfaces, has a special point at the tangency point. It is known from differential geometry [1] that this point can be nodal, return, or isolated one. In turn, this point’s appearance depends on differential properties of the surface(s) in this point’s vicinity. That's why, for the competent solution of the considered positional problem account must be also taken of the inter-subject connections for descriptive and differential geometry. In the training courses of descriptive geometry tangent planes are built only to the simplest surfaces, containing, as a rule, the frames of straight lines and circles. Therefore, the tangent plane is defined by two tangents drawn at the tangency point to two such lines. In engineering practice, as such lines are used cross-sections a surface by planes parallel to any two coordinate planes. That is, from the standpoints for the course of higher mathematics, the problem is reduced to calculation for partial derivatives. Although this topic is studied after the course of descriptive geometry, it seems possible to give geometric explanation for computation of partial derivatives in a nutshell. It also seems that the study of this topic will be stimulated by a story about engineering problems, which solution is based on construction of the tangent plane and the normal to the technical surface. In this paper has been presented an example for the use of surface curvature lines for programming of milling processing for 3D-harness surfaces.

Ensuring the Requirements of Ergonomics in the Automated the Layout of Equipment

Abstract: The article discusses the solution to the problem of automating the design of layouts of various equipment, taking into account ergonomics, by which is meant the provision of service areas. The article describes the development of methods and algorithms that provide access to installation tools and workspace during installation and maintenance of already placed equipment. The solution method is geometric modeling of both the placed objects and the installation equipment necessary for its maintenance, as well as the trajectory of its movement to the service area. Thus, both the installation equipment and the movement paths are treated as composable objects, the intersection of which with other objects is unacceptable. As a modeling method, receptor-based geometric models that discretize the allocation space were used. The choice of receptor models is due to the fact that the solid-state model of all the instantaneous positions of the installation tool in the process of its delivery and operation is extremely complex from a geometrical point of view. The possibility of relatively easy to determine the fact of the intersection of all objects of the scene, described by receptor models, and is the rationale for the choice in our study of the receptor method of geometric modeling. Based on the receptor design model, a procedure has been developed for determining the trajectory of a mounting tool at a given operating point, as well as the formation of the space required for operation, or establishing the fact that it is impossible to service a particular object, which indicates an unsatisfactory (non-ergonomic) given design solution. In this study a feature of using receptor models is the use of 6-digit codes in the receptor matrices, which, with some complication of the modeling method, allows to obtain additional information about problem areas in the layout under study (impossibility of carrying a tool, impossibility of performing assembly operations, etc.). Algorithms for solving this problem, as well as a graphical shell that visualizes the results of computer-aided design, are implemented as C# programs.

Parquetting the Surface of a Parabolic Concentrator of a Solar Photovoltaic Thermal Module According to Given Differential- Geometric Requirements

Abstract: The article discusses the geometric aspects of the design and creation of parabolic-type solar radiation concentrators. Practical methods of geometric design and manufacturing of concentrators of this kind are presented. Parabolic type concentrator is the main part of the solar photovoltaic thermal installation. Its effectiveness depends on the quality factors of the geometric shaping of the working surface, composed of a set of parquet components, linked to each other on the basis of differential geometric requirements. The distribution of illumination in the focal spot of such a concentrator, made by parquet based on the constructive connection of individual elements, makes it possible to obtain acceptable results. However, there is considerable potential for improving performance by providing a smoother and more uniform illumination of the photodetector. To ensure the specified accuracy and smoothness of the rim of the surface at the stages of designing and manufacturing the device, two methods are proposed: orthogonal and fan-shaped geometric parquetting of the surface of a parabolic concentrator with the ability to pre-set the required shape accuracy for given rim geometrical characteristics. Parquetting with given differential requirements for the surface, in turn, provides for two methods for calculating parquet elements: first, by the minimum number of curvilinear elements followed by stitching, taking into account the differential conditions; the second is based on the maximum number of flat elements, the multiplicity of which provides acceptable smooth surface properties. In this paper, we consider the first method for cases of orthogonal and fan parquet. On the example of a parabolic concentrator, the implementation of the considered method is presented, which provides for the possibility of controlling the geometric smoothness of the concentrator surface in order to ensure optimal distribution of concentrated solar radiation in the focal region. The output characteristics of photovoltaic and thermal converters of solar energy, which are in the focus of such a concentrator, become optimal, and the installation itself will operate in nominal mode.

Approximate Solution for Squaring the Circle Problem

Abstract: It is known that squaring the circle (the problem consisting in construction of a square with the same area as a given circle), together with duplication of cube and angle trisection, is one of the most famous unsolv able problems of constructive geometry for construction with compass and straightedge. The solution of squaring the circle problem is reduced to the straightening of the circle, that is, to the construction of a segment equal in length to the circle, and its insolvability is connected with the pi character transcendence. In this paper, the limiting case of one of Christian Huygens theorems, which establishes an estimate for length of circumference of a circle through perimeters of regular polygons inscribed in circle and circumscribed about it, is proved. On this basis has been proposed and justified an approximate method for squaring the circle problem solving, which allows consistently construct arbitrarily exact solutions of the problem. We will approximate an arc of a circle whose radius is a multiple of the given circle’s radius, with the help of a segment which is parallel to a shrinking it chord, and then will increase or decrease this segment in the required number of times, so that the resulting segment’s length would be approximately equal to half of the given circle’s circumference. The approximation accuracy will be the higher the smaller arc of the circle we will consider. But possibilities of real tools are limited, and not suitable for both too small and too large drawing scales. In order to cope with this problem, an algorithm for scaled approximation has been proposed, in which it is sufficient to increase (or reduce) the drawing fragment, so that all the time sta y within the sheet of the same size. Perhaps this approach will be useful for other constructions, including the exact ones, where it is necessary to come to very large or vice versa very small drawings’ dimensions.