Pub Date : 2021-07-06DOI: 10.12737/2308-4898-2021-9-1-3-18
V. Korotkiy, Igor' Vitovtov
Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a "nature-like" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials ("cubic spline"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.
{"title":"Approximation of Physical Spline with Large Deflections","authors":"V. Korotkiy, Igor' Vitovtov","doi":"10.12737/2308-4898-2021-9-1-3-18","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-1-3-18","url":null,"abstract":"Physical spline is a resilient element whose cross-sectional dimensions are very small compared to its axis’s length and radius of curvature. Such a resilient element, passing through given points, acquires a \"nature-like\" form, having a minimum energy of internal stresses, and, as a consequence, a minimum of average curvature. For example, a flexible metal ruler, previously used to construct smooth curves passing through given coplanar points, can be considered as a physical spline. The theoretical search for the equation of physical spline’s axis is a complex mathematical problem with no elementary solution. However, the form of a physical spline passing through given points can be obtained experimentally without much difficulty. In this paper polynomial and parametric methods for approximation of experimentally produced physical spline with large deflections are considered. As known, in the case of small deflections it is possible to obtain a good approximation to a real elastic line by a set of cubic polynomials (\"cubic spline\"). But as deflections increase, the polynomial model begins to differ markedly from the experimental physical spline, that limits the application of polynomial approximation. High precision approximation of an elastic line with large deflections is achieved by using a parameterized description based on Ferguson or Bézier curves. At the same time, not only the basic points, but also the tangents to the elastic line of the real physical spline should be given as boundary conditions. In such a case it has been shown that standard cubic Bézier curves have a significant computational advantage over Ferguson ones. Examples for modelling of physical splines with free and clamped ends have been considered. For a free spline an error of parametric approximation is equal to 0.4 %. For a spline with clamped ends an error of less than 1.5 % has been obtained. The calculations have been performed with SMath Studio computer graphics system.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88792089","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 : 2021-07-06DOI: 10.12737/2308-4898-2021-9-1-38-44
I. Antonova, E. Solomonova, N. Kadykova
In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.
{"title":"Mathematical Description for a Particular Case of Ellipse Focus Quasi-Rotation Around an Elliptical Axis","authors":"I. Antonova, E. Solomonova, N. Kadykova","doi":"10.12737/2308-4898-2021-9-1-38-44","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-1-38-44","url":null,"abstract":"In this paper is provided mathematical analysis related to a particular case for a point quasi-rotation around a curve of an elliptical axis. The research complements the previous works in this direction. Has been considered a special case, in which the quasi-rotation correspondence is applied to a point located at the elliptical axis’s focus. This case is special, since the quasi-rotation center search is not invariant and does not lead to determination of four quasi-rotation centers, as in the general case. A constructive approach to the rotation center search shows that any point lying on the elliptical axis can be the quasi-rotation center. This feature leads to the fact that instead of four circles, the quasi-rotation of a point lying in the elliptical axis’s focus leads to the formation of an infinite number of circle families, which together form a channel surface. The resulting surface is a Dupin cyclide, whose throat circle has a zero radius and coincides with the original generating point. While analyzing are considered all cases of the rotation center location. Geometric constructions have been performed based on previously described methods of rotation around flat geometric objects’ curvilinear axes. For the study, the mathematical relationship between the coordinates of the initial set point, the axis curve equation and the motion trajectory equation of this point around the axis curve, described in earlier papers on this topic, is used. In the proposed paper has been provided the derivation of the motion trajectory equation for a point around the elliptic axis’s curve.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"134 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77388891","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 : 2021-07-06DOI: 10.12737/2308-4898-2021-9-1-28-37
S. Ignat'ev, A. Folomkin, E. Muratbakeev
The paper states that the current conditions in which the education system is located, and the rapid development of IT require constant improvement of methodological materials, taking into account the full capacity of the current software. Examples of programs necessary for preparation of methodological materials for high-quality classes are given. The purpose of the paper was to identify the Wolfram Mathematica didactic potential when conducting classes in the disciplines of the geometric and graphic profile at a technical high educational institution. In this paper has been performed analysis of literature sources both domestic and foreign ones on the Wolfram Mathematica system application in science and teaching of various disciplines. It has been shown that the program use scope is very wide, in fact, it is comprehensive and requires additional and in-depth study. Examples of Wolfram Mathematica using in mathematics, physics, chemistry, geometry, robotics, virology, and the humanities are given. In the paper have been provided examples for pedagogical design of simulation models for an electronic course on descriptive geometry in the Moodle system. An example of code written in the Wolfram Mathematica is provided. Interactive models developed during the design are presented, which allow the user to change the constructed curves and surfaces’ parameters. Have been defined some functional capabilities of the system, and has been revealed the Wolfram Mathematica didactic potential for teaching geometric and graphic disciplines. Have been considered other authors’ similar models, which can be used in the educational process to increase the clarity of the material presented in the classroom. In conclusion it is pointed out that interactive visualization in the "Descriptive Geometry" discipline, together with classical working practices, significantly enriches the content of geometric education.
{"title":"Wolfram Mathematica Functional Possibilities for Curved Lines and Surfaces Visualization","authors":"S. Ignat'ev, A. Folomkin, E. Muratbakeev","doi":"10.12737/2308-4898-2021-9-1-28-37","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-1-28-37","url":null,"abstract":"The paper states that the current conditions in which the education system is located, and the rapid development of IT require constant improvement of methodological materials, taking into account the full capacity of the current software. Examples of programs necessary for preparation of methodological materials for high-quality classes are given. The purpose of the paper was to identify the Wolfram Mathematica didactic potential when conducting classes in the disciplines of the geometric and graphic profile at a technical high educational institution. In this paper has been performed analysis of literature sources both domestic and foreign ones on the Wolfram Mathematica system application in science and teaching of various disciplines. It has been shown that the program use scope is very wide, in fact, it is comprehensive and requires additional and in-depth study. Examples of Wolfram Mathematica using in mathematics, physics, chemistry, geometry, robotics, virology, and the humanities are given. In the paper have been provided examples for pedagogical design of simulation models for an electronic course on descriptive geometry in the Moodle system. An example of code written in the Wolfram Mathematica is provided. Interactive models developed during the design are presented, which allow the user to change the constructed curves and surfaces’ parameters. Have been defined some functional capabilities of the system, and has been revealed the Wolfram Mathematica didactic potential for teaching geometric and graphic disciplines. Have been considered other authors’ similar models, which can be used in the educational process to increase the clarity of the material presented in the classroom. In conclusion it is pointed out that interactive visualization in the \"Descriptive Geometry\" discipline, together with classical working practices, significantly enriches the content of geometric education.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88357386","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 : 2021-07-06DOI: 10.12737/2308-4898-2021-9-1-19-27
A. Girsh
Two plane algebraic curves intersect at the actual intersection points of these curves’ graphs. In addition to real intersection points, algebraic curves can also have imaginary intersection points. The total number of curves intersection points is equal to the product of their orders mn. The number of imaginary intersection points can be equal to or part of mn. The position of the actual intersection points is determined by the graphs of the curves, but the imaginary intersection points do not lie on the graphs of these curves, and their position on the plane remains unclear. This work aims to determine the geometry of imaginary intersection points, introduces into consideration the concept of imaginary complement for these algebraic curves in the intersection operation, determines the form of imaginary complements, which intersect at imaginary points. The visualization of imaginary complements clarifies the curves intersection picture, and the position of the imaginary intersection points becomes expected.
{"title":"Intersection Operation on a Complex Plane","authors":"A. Girsh","doi":"10.12737/2308-4898-2021-9-1-19-27","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-9-1-19-27","url":null,"abstract":"Two plane algebraic curves intersect at the actual intersection points of these curves’ graphs. In addition to real intersection points, algebraic curves can also have imaginary intersection points. The total number of curves intersection points is equal to the product of their orders mn. The number of imaginary intersection points can be equal to or part of mn. The position of the actual intersection points is determined by the graphs of the curves, but the imaginary intersection points do not lie on the graphs of these curves, and their position on the plane remains unclear. This work aims to determine the geometry of imaginary intersection points, introduces into consideration the concept of imaginary complement for these algebraic curves in the intersection operation, determines the form of imaginary complements, which intersect at imaginary points. The visualization of imaginary complements clarifies the curves intersection picture, and the position of the imaginary intersection points becomes expected.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84182741","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 : 2021-03-04DOI: 10.12737/2308-4898-2021-8-4-47-60
N. Sal'kov
In the 2017-18 and 2018-19 academic years, in Moscow State Academic Art Institute named after V.I. Surikov a two-year experiment for architecture students on determination variances in different approaches to teaching methods for geometric education was conducted. The experiment was caused by the fact that many teachers believe that if there are computers in the Institute, it is necessary to use graphic programs as soon as possible – immediately introduce students to work on the computer even without allocating hours for this. No one wants to prevent implementation of computers, but we must not forget some nuances related to high technology. As in any case connected with complex hardware unknown for future users, here at the University, at the beginning it is also necessary to teach students how to work with the graphics program itself, and only then allow them perform geometric problems. You can give such an example: put an untrained person at the control panel of interceptor missiles and force him to shoot down a border trespasser in combat conditions. They will notify us that we are engaged in voluntarism. However to put an untrained student at a computer and forcing him to solve a purely geometric problem immediately is not voluntarism. Is it? The experiment had showed that a student, even if he knows how to solve a particular problem, is not ready at all to perform it immediately on a computer in a graphic software. He begin to lose a lot of time getting familiar with the program and only after obtaining at least a minimum of knowledge about working with this program becomes ready to start the task.
{"title":"Quality of Geometric Education in Various Approaches to Teaching Methods","authors":"N. Sal'kov","doi":"10.12737/2308-4898-2021-8-4-47-60","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-8-4-47-60","url":null,"abstract":"In the 2017-18 and 2018-19 academic years, in Moscow State Academic Art Institute named after V.I. Surikov a two-year experiment for architecture students on determination variances in different approaches to teaching methods for geometric education was conducted. The experiment was caused by the fact that many teachers believe that if there are computers in the Institute, it is necessary to use graphic programs as soon as possible – immediately introduce students to work on the computer even without allocating hours for this. No one wants to prevent implementation of computers, but we must not forget some nuances related to high technology. As in any case connected with complex hardware unknown for future users, here at the University, at the beginning it is also necessary to teach students how to work with the graphics program itself, and only then allow them perform geometric problems. You can give such an example: put an untrained person at the control panel of interceptor missiles and force him to shoot down a border trespasser in combat conditions. They will notify us that we are engaged in voluntarism. However to put an untrained student at a computer and forcing him to solve a purely geometric problem immediately is not voluntarism. Is it? \u0000The experiment had showed that a student, even if he knows how to solve a particular problem, is not ready at all to perform it immediately on a computer in a graphic software. He begin to lose a lot of time getting familiar with the program and only after obtaining at least a minimum of knowledge about working with this program becomes ready to start the task.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"67 1","pages":"47-60"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73156072","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 : 2021-03-04DOI: 10.12737/2308-4898-2021-8-4-24-34
A. Ivaschenko, D. Vavanov
The presented paper is devoted to classification questions of fourth-order spatial curves, obtained as a result of intersection of non-degenerate second-order surfaces (quadrics) from the point of view of the forms of the original quadrics generating this curve. At the beginning of the paper is performed a brief historical overview of appearance of well-known and widely used curves ranging from ancient times and ending with the current state in the theory of curves and surfaces. Then a general analysis of the influence of the shape parameters and the relative position of original surfaces on the shape of the resulting curve and some of its parameters (number of components, presence of singular points, curve components flatness or spatiality) is carried out. Curves obtained as a result of intersection of equitype surfaces are described in more detail. The concept of interacting surfaces is introduced, various possible cases of the forms of the quadrics generating the curve are analyzed. A classification of fourth-order curves based on the shape parameters and relative position of second-order surfaces is proposed as an option. Illustrations of the resulting curve shapes with different shape parameters and location of generating quadrics are given. All surfaces and curves are considered in real affine space, taking into account the possibility of constructing them using descriptive geometry methods. Possible further research directions related to the analysis of the curves under discussion are briefly considered. In addition, are expressed hypotheses related to these curves use in the process of studying by students of technical universities the courses in analytical geometry, descriptive geometry, differential geometry and computer graphics. The main attention is paid to forms, therefore a wide variability of the surface shape in the framework of its described equation has been shown, provided by various values of numerical parameters.
{"title":"General Analysis of the Shape of Two Similar Second-Order Surfaces’ Intersection Line","authors":"A. Ivaschenko, D. Vavanov","doi":"10.12737/2308-4898-2021-8-4-24-34","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-8-4-24-34","url":null,"abstract":"The presented paper is devoted to classification questions of fourth-order spatial curves, obtained as a result of intersection of non-degenerate second-order surfaces (quadrics) from the point of view of the forms of the original quadrics generating this curve. At the beginning of the paper is performed a brief historical overview of appearance of well-known and widely used curves ranging from ancient times and ending with the current state in the theory of curves and surfaces. Then a general analysis of the influence of the shape parameters and the relative position of original surfaces on the shape of the resulting curve and some of its parameters (number of components, presence of singular points, curve components flatness or spatiality) is carried out. Curves obtained as a result of intersection of equitype surfaces are described in more detail. The concept of interacting surfaces is introduced, various possible cases of the forms of the quadrics generating the curve are analyzed. A classification of fourth-order curves based on the shape parameters and relative position of second-order surfaces is proposed as an option. Illustrations of the resulting curve shapes with different shape parameters and location of generating quadrics are given. All surfaces and curves are considered in real affine space, taking into account the possibility of constructing them using descriptive geometry methods. Possible further research directions related to the analysis of the curves under discussion are briefly considered. In addition, are expressed hypotheses related to these curves use in the process of studying by students of technical universities the courses in analytical geometry, descriptive geometry, differential geometry and computer graphics. The main attention is paid to forms, therefore a wide variability of the surface shape in the framework of its described equation has been shown, provided by various values of numerical parameters.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"24 1","pages":"24-34"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89555682","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 : 2021-03-04DOI: 10.12737/2308-4898-2021-8-4-74-84
S. Ignat'ev, A. Folomkin, E. Muratbakeev
The possibilities of Wolfram Mathematica (WM), which is a package of symbolic mathematics, are endless. In this paper are investigated the possibilities of WM software product contextualization in the process of geometric-graphic teaching of students; is considered the experience for using a set of WM computational algorithms in teaching practice. The source codes and output data of our own WOLFRAM projects are offered, which are illustrative, in the form of interactive graphs, visualization of the solution for one of DG course’s basic problems on construction of points A, B, C and D, given by coordinates in space; construction a plane of general position passing through points A, B and C; determining positions of these points and the constructed plane in space; and also on determining the point D belonging to this plane. Have been presented examples of several of the existing WOLFRAM projects in the form of interactive graphs used in teaching. The study of applied issues and the ability to translate a professional problem into mathematical language, working in WM, provides a student with an opportunity to assimilate the necessary competencies. Interactive visualization of solving DG problems, which can be carried out using WM, together with traditional problem solving, enriches the content of geometric education, introduces new opportunities in organization of the educational process, allows stimulate, maintain and increase students' interest in studying the discipline and further research in the area of DG. However, the ability to work in WM cannot replace the fundamental knowledge obtained by students of a technical high educational institute from the traditional course, but only complements the DG course by WM acquaintance.
{"title":"Descriptive Geometry Problems Visualization by Wolfram Mathematica","authors":"S. Ignat'ev, A. Folomkin, E. Muratbakeev","doi":"10.12737/2308-4898-2021-8-4-74-84","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-8-4-74-84","url":null,"abstract":"The possibilities of Wolfram Mathematica (WM), which is a package of symbolic mathematics, are endless. In this paper are investigated the possibilities of WM software product contextualization in the process of geometric-graphic teaching of students; is considered the experience for using a set of WM computational algorithms in teaching practice. The source codes and output data of our own WOLFRAM projects are offered, which are illustrative, in the form of interactive graphs, visualization of the solution for one of DG course’s basic problems on construction of points A, B, C and D, given by coordinates in space; construction a plane of general position passing through points A, B and C; determining positions of these points and the constructed plane in space; and also on determining the point D belonging to this plane. Have been presented examples of several of the existing WOLFRAM projects in the form of interactive graphs used in teaching. The study of applied issues and the ability to translate a professional problem into mathematical language, working in WM, provides a student with an opportunity to assimilate the necessary competencies. \u0000Interactive visualization of solving DG problems, which can be carried out using WM, together with traditional problem solving, enriches the content of geometric education, introduces new opportunities in organization of the educational process, allows stimulate, maintain and increase students' interest in studying the discipline and further research in the area of DG. However, the ability to work in WM cannot replace the fundamental knowledge obtained by students of a technical high educational institute from the traditional course, but only complements the DG course by WM acquaintance.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"31 1","pages":"74-84"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79072309","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 : 2021-03-04DOI: 10.12737/2308-4898-2021-8-4-35-46
S. Ryazanov, M. Reshetnikov
Spatial helical gears, worm gears with a cylindrical worm, globoid gears, etc., are widely used in most of modern engineering products [1-3; 37; 42]. Cylindrical worm gears are actively used in the creation of metalworking equipment (push mechanisms of rolling mills, presses, etc.), in lifting and transport machines, in drives and kinematic chains of various machine tool equipment where high kinematic accuracy is required (dividing machine tools, adjustment mechanisms), etc. In a worm gear a cylindrical worm or its cylindrical helical surface can be cut by various technological methods [49-51], but no matter how the shaping of the worm gear elements’ working surfaces is carried out, the worm wheel is cut with a gear cutting tool, whose producing surface coincides with the worm thread’s lateral surface [19; 22; 23]. In this regard, the working surface of the cylindrical worm wheel’s tooth, even with a non-orthogonal arrangement of axes, is an envelope of a one-parameter family of surfaces that gives a linear contact, which presence makes it possible to transfer a large load using a worm gear. For high-quality manufacturing of worm gears, it is necessary to design and manufacture a productive gear cutting tool - an accurate worm cutter, whose shaping (working) surface must be identical to the profiled worm’s shaping (working) surface [24-27; 54]. One of the most important tasks in the implementation of worm gearing is the problem of jamming of the cylindrical worm and the worm wheel’ contacting surfaces. This problem is excluded by relieving the contacting surfaces’ profile along the contact line. Considering that any violations of contacting surfaces’ geometric parameters affect the change in their geometric characteristics, the tasks of accurately determining the adjustment parameters of the technological equipment, used for shaping the worm and worm wheel, enter into in the foreground of the worm gearing elements production. In modern conditions of plant and equipment obsolescence, and in particular, of gear cutting machines used for worm gears manufacture, these machines physical wear, implies an inevitable decrease in the accuracy of their kinematic chains. Therefore, in order to maintain the produced gears’ quality at a sufficiently high level, it is necessary to use deliberate modification of contacting surfaces when calculating the worm gearing’s geometric parameters; such modification reduces the worm gear sensitivity to manufacturing and mounting errors of its elements [28-31].
{"title":"Calculation of the coordinates of the modified profile of the generating surface of the gear cutting tool","authors":"S. Ryazanov, M. Reshetnikov","doi":"10.12737/2308-4898-2021-8-4-35-46","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-8-4-35-46","url":null,"abstract":"Spatial helical gears, worm gears with a cylindrical worm, globoid gears, etc., are widely used in most of modern engineering products [1-3; 37; 42]. Cylindrical worm gears are actively used in the creation of metalworking equipment (push mechanisms of rolling mills, presses, etc.), in lifting and transport machines, in drives and kinematic chains of various machine tool equipment where high kinematic accuracy is required (dividing machine tools, adjustment mechanisms), etc. \u0000 In a worm gear a cylindrical worm or its cylindrical helical surface can be cut by various technological methods [49-51], but no matter how the shaping of the worm gear elements’ working surfaces is carried out, the worm wheel is cut with a gear cutting tool, whose producing surface coincides with the worm thread’s lateral surface [19; 22; 23]. In this regard, the working surface of the cylindrical worm wheel’s tooth, even with a non-orthogonal arrangement of axes, is an envelope of a one-parameter family of surfaces that gives a linear contact, which presence makes it possible to transfer a large load using a worm gear. \u0000 For high-quality manufacturing of worm gears, it is necessary to design and manufacture a productive gear cutting tool - an accurate worm cutter, whose shaping (working) surface must be identical to the profiled worm’s shaping (working) surface [24-27; 54]. \u0000 One of the most important tasks in the implementation of worm gearing is the problem of jamming of the cylindrical worm and the worm wheel’ contacting surfaces. This problem is excluded by relieving the contacting surfaces’ profile along the contact line. Considering that any violations of contacting surfaces’ geometric parameters affect the change in their geometric characteristics, the tasks of accurately determining the adjustment parameters of the technological equipment, used for shaping the worm and worm wheel, enter into in the foreground of the worm gearing elements production. \u0000 In modern conditions of plant and equipment obsolescence, and in particular, of gear cutting machines used for worm gears manufacture, these machines physical wear, implies an inevitable decrease in the accuracy of their kinematic chains. Therefore, in order to maintain the produced gears’ quality at a sufficiently high level, it is necessary to use deliberate modification of contacting surfaces when calculating the worm gearing’s geometric parameters; such modification reduces the worm gear sensitivity to manufacturing and mounting errors of its elements [28-31].","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"17 1","pages":"35-46"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81434184","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 : 2021-03-04DOI: 10.12737/2308-4898-2021-8-4-13-23
S. Abdurahmanov
A brief historical excursion into the graphics of geometry of multidimensional spaces at the paper beginning clarifies the problem – the necessary to reduce the number of geometric actions performed when depicting multidimensional objects. The problem solution is based on the properties of geometric figures called N- simplexes, whose number of vertices is equal to N + 1, where N expresses their dimensionality. The barycenter (centroid) of the N-simplex is located at the point that divides the straight-line segment connecting the centroid of the (N–1)-simplex contained in it with the opposite vertex by 1: N. This property is preserved in the parallel projection (axonometry) of the simplex on the drawing plane, that allows the solution of the problem of determining the centroid of the simplex in its axonometry to be assigned to a mechanism which is a special Assembly of pantographs (the author's invention) with similarity coefficients 1:1, 1:2, 1:3, 1:4,...1:N. Next, it is established, that the spatial location of a point in N-dimensional space coincides with the centroid of the simplex, whose vertices are located on the point’s N-fold (barycentric) coordinates. In axonometry, the ends of both first pantograph’s links and the ends of only long links of the remaining ones are inserted into points indicating the projections of its barycentric coordinates and the mechanism node, which serves as a determinator, graphically marks the axonometric location of the point defined by its coordinates along the axes х1, х2, х3 … хN.. The translational movement of the support rods independently of each other can approximate or remote the barycentric coordinates of a point relative to the origin of coordinates, thereby assigning the corresponding axonometric places to the simplex barycenter, which changes its shape in accordance with its points’ occupied places in the coordinate axes. This is an axonograph of N-dimensional space, controlled by a numerical program. The last position indicates the possibility for using the equations of multidimensional spaces’ geometric objects given in the corresponding literature for automatic drawing when compiling such programs.
{"title":"Use of Mechanisms Marking Centers of Simplexes in Their 2-Dimensional Projections as Axonographs of Multidimensional Spaces","authors":"S. Abdurahmanov","doi":"10.12737/2308-4898-2021-8-4-13-23","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-8-4-13-23","url":null,"abstract":"A brief historical excursion into the graphics of geometry of multidimensional spaces at the paper beginning clarifies the problem – the necessary to reduce the number of geometric actions performed when depicting multidimensional objects. \u0000The problem solution is based on the properties of geometric figures called N- simplexes, whose number of vertices is equal to N + 1, where N expresses their dimensionality. The barycenter (centroid) of the N-simplex is located at the point that divides the straight-line segment connecting the centroid of the (N–1)-simplex contained in it with the opposite vertex by 1: N. This property is preserved in the parallel projection (axonometry) of the simplex on the drawing plane, that allows the solution of the problem of determining the centroid of the simplex in its axonometry to be assigned to a mechanism which is a special Assembly of pantographs (the author's invention) with similarity coefficients 1:1, 1:2, 1:3, 1:4,...1:N. \u0000Next, it is established, that the spatial location of a point in N-dimensional space coincides with the centroid of the simplex, whose vertices are located on the point’s N-fold (barycentric) coordinates. In axonometry, the ends of both first pantograph’s links and the ends of only long links of the remaining ones are inserted into points indicating the projections of its barycentric coordinates and the mechanism node, which serves as a determinator, graphically marks the axonometric location of the point defined by its coordinates along the axes х1, х2, х3 … хN.. \u0000The translational movement of the support rods independently of each other can approximate or remote the barycentric coordinates of a point relative to the origin of coordinates, thereby assigning the corresponding axonometric places to the simplex barycenter, which changes its shape in accordance with its points’ occupied places in the coordinate axes. This is an axonograph of N-dimensional space, controlled by a numerical program. The last position indicates the possibility for using the equations of multidimensional spaces’ geometric objects given in the corresponding literature for automatic drawing when compiling such programs.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"9 1","pages":"13-23"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79926138","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 : 2021-03-04DOI: 10.12737/2308-4898-2021-8-4-61-73
N. Sal'kov, N. Kadykova
Among specialists prevails the primitive view, according to Prof. G.S. Ivanov, on descriptive geometry only as on a "grammar of a technical language", as it characterized V.I. Kurdyumov in the XIX Century. If in the century before last his definition was actual, although many contemporaries had a different opinion, then a century and a half later this definition became outdated, especially since have been revealed the close relationships of descriptive geometry with related sections: analytical, parametric, differential geometry, etc., and descriptive geometry became an applied mathematical science. In this paper it has been shown that an image is obtained as a result of display (projection). In this connection, according to prof. N.A. Sobolev, "All visual images – documentary, geometrographic, and creative ones – are formed on the projection principle". In other words, they belong, in essence, to descriptive geometry. Thus, all made by hand creative images – drawings, paintings, sculptures – can be attributed with great confidence to descriptive geometry as a theory of images. These creative images, of course, have a non-obvious projection origin, nevertheless, according to Prof. N.A. Sobolev, "They, including the most abstract fantasies, are essentially the projection ones". Further in the paper it has been shown which disciplines apply some or other of graphic models, and has been considered a number of drawings belonging to different textbooks, in which graphic models are present. Thus, clearly, and also referring to the authorities in the area of images and descriptive geometry, it has been proved that each of the mentioned textbooks has a direct or indirect connection with descriptive geometry, and descriptive geometry itself is present in all textbooks, at least, in the technical and medical ones.
{"title":"The Phenomenon of Descriptive Geometry Existence in Other Student Courses","authors":"N. Sal'kov, N. Kadykova","doi":"10.12737/2308-4898-2021-8-4-61-73","DOIUrl":"https://doi.org/10.12737/2308-4898-2021-8-4-61-73","url":null,"abstract":"Among specialists prevails the primitive view, according to Prof. G.S. Ivanov, on descriptive geometry only as on a \"grammar of a technical language\", as it characterized V.I. Kurdyumov in the XIX Century. If in the century before last his definition was actual, although many contemporaries had a different opinion, then a century and a half later this definition became outdated, especially since have been revealed the close relationships of descriptive geometry with related sections: analytical, parametric, differential geometry, etc., and descriptive geometry became an applied mathematical science. In this paper it has been shown that an image is obtained as a result of display (projection). In this connection, according to prof. N.A. Sobolev, \"All visual images – documentary, geometrographic, and creative ones – are formed on the projection principle\". In other words, they belong, in essence, to descriptive geometry. Thus, all made by hand creative images – drawings, paintings, sculptures – can be attributed with great confidence to descriptive geometry as a theory of images. These creative images, of course, have a non-obvious projection origin, nevertheless, according to Prof. N.A. Sobolev, \"They, including the most abstract fantasies, are essentially the projection ones\". Further in the paper it has been shown which disciplines apply some or other of graphic models, and has been considered a number of drawings belonging to different textbooks, in which graphic models are present. Thus, clearly, and also referring to the authorities in the area of images and descriptive geometry, it has been proved that each of the mentioned textbooks has a direct or indirect connection with descriptive geometry, and descriptive geometry itself is present in all textbooks, at least, in the technical and medical ones.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"32 1","pages":"61-73"},"PeriodicalIF":0.0,"publicationDate":"2021-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88248541","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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