Pub Date : 2019-12-02DOI: 10.12737/article_5dce6cf7ae1d70.85408915
Вячеслав Васильевич Юрков, V. Yurkov
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.
{"title":"Approximation of Linear Sets in the Plane","authors":"Вячеслав Васильевич Юрков, V. Yurkov","doi":"10.12737/article_5dce6cf7ae1d70.85408915","DOIUrl":"https://doi.org/10.12737/article_5dce6cf7ae1d70.85408915","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"15 1","pages":"60-69"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89750260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-02DOI: 10.12737/article_5dce5e528e4301.77886978
К. Панчук, K. Panchuk, Е. Любчинов, E. Lyubchinov
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.
{"title":"Cyclographic Interpretation and Computer Solution of One System of Algebraic Equations","authors":"К. Панчук, K. Panchuk, Е. Любчинов, E. Lyubchinov","doi":"10.12737/article_5dce5e528e4301.77886978","DOIUrl":"https://doi.org/10.12737/article_5dce5e528e4301.77886978","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"54 1","pages":"3-14"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77531542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-02DOI: 10.12737/article_5dce66dd9fb966.59423840
И. Антонова, I. Antonova, И. Беглов, I. Beglov, Елена Борисовна Соломонова, E. Solomonova
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.
{"title":"A Mathematical Description of the Rotation of a Point Around an Elliptic Axis in Some Special Cases","authors":"И. Антонова, I. Antonova, И. Беглов, I. Beglov, Елена Борисовна Соломонова, E. Solomonova","doi":"10.12737/article_5dce66dd9fb966.59423840","DOIUrl":"https://doi.org/10.12737/article_5dce66dd9fb966.59423840","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"488 1","pages":"36-50"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77051155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-02DOI: 10.12737/article_5dce6b81e2a808.81762326
Федор Притыкин, F. Pritykin, Василий Хомченко, V. Khomchenko, Анна Янишевская, A. Yanishevskaya, Валерий Небритов, V. Nebritov
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.
{"title":"Visualization of Linear Shifts of Nodal Points during Implementation of Instantaneous States of Various Configurations of an Android Robot Arm","authors":"Федор Притыкин, F. Pritykin, Василий Хомченко, V. Khomchenko, Анна Янишевская, A. Yanishevskaya, Валерий Небритов, V. Nebritov","doi":"10.12737/article_5dce6b81e2a808.81762326","DOIUrl":"https://doi.org/10.12737/article_5dce6b81e2a808.81762326","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"38 1","pages":"51-59"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81980965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-12-02DOI: 10.12737/article_5dce6084f1ac94.09740392
Сергей Синицын, S. Sinitsyn, Дмитрий Семенович Стребков, D. Strebkov, Владимир Анатольевич Панченко, V. Panchenko
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.
{"title":"Parquetting the Surface of a Parabolic Concentrator of a Solar Photovoltaic Thermal Module According to Given Differential- Geometric Requirements","authors":"Сергей Синицын, S. Sinitsyn, Дмитрий Семенович Стребков, D. Strebkov, Владимир Анатольевич Панченко, V. Panchenko","doi":"10.12737/article_5dce6084f1ac94.09740392","DOIUrl":"https://doi.org/10.12737/article_5dce6084f1ac94.09740392","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"34 1","pages":"15-27"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74077954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-15DOI: 10.12737/ARTICLE_5D2C1502670779.58031440
Виктор Короткий, V. Korotkiy
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.
{"title":"Graphic Algorithms for Constructing a Quadric, Given Nine Points","authors":"Виктор Короткий, V. Korotkiy","doi":"10.12737/ARTICLE_5D2C1502670779.58031440","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C1502670779.58031440","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85196359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-15DOI: 10.12737/ARTICLE_5D2C24F391D6B6.68532534
С. Рязанов, S. Ryazanov
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.
{"title":"A Geometrical Model Producing a Surface Equivalent To the Working Surface of the Gear Tool «Hob»","authors":"С. Рязанов, S. Ryazanov","doi":"10.12737/ARTICLE_5D2C24F391D6B6.68532534","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C24F391D6B6.68532534","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"76 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87051694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-15DOI: 10.12737/ARTICLE_5D2C187251B6C8.21632403
Михаил Лепаров, M. Leparov
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.
{"title":"About the Science «Geometry of Technical Objects»","authors":"Михаил Лепаров, M. Leparov","doi":"10.12737/ARTICLE_5D2C187251B6C8.21632403","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C187251B6C8.21632403","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83829924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-15DOI: 10.12737/ARTICLE_5D2C170AB37810.30821713
Николай Сальков, N. Sal'kov
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.
{"title":"General Principles for Formation of Ruled Surfaces. Part 3","authors":"Николай Сальков, N. Sal'kov","doi":"10.12737/ARTICLE_5D2C170AB37810.30821713","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C170AB37810.30821713","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73998791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-08-15DOI: 10.12737/ARTICLE_5D2C350BAF0B28.40160405
Сергей Александрович Абросимов, S. Abrosimov, Дмитрий Евгеньевич Тихонов-Бугров, D. Tikhonov-Bugrov, К. Глазунов, K. Glazunov
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.
{"title":"Geometric-Graphic Student Olympiad in St. Petersburg","authors":"Сергей Александрович Абросимов, S. Abrosimov, Дмитрий Евгеньевич Тихонов-Бугров, D. Tikhonov-Bugrov, К. Глазунов, K. Glazunov","doi":"10.12737/ARTICLE_5D2C350BAF0B28.40160405","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C350BAF0B28.40160405","url":null,"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.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80719175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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