Pub Date : 2018-11-14DOI: 10.12737/ARTICLE_5BC457ECE18491.72807735
Евгений Конопацкий, E. Konopatskiy, Антон Крысько, A. Krys’ko, А. Бумага, A. Bumaga
In this paper have been proposed theoretical bases for formation one-dimensional contours of the first order smoothness, passing through k in advance given points, including requirements to the contour in general, and the contour’s arcs in particular, as well as representations for tangents in extreme and intermediate contour sections, which determine the contour arc shape. Based on this theoretical material have been developed computational algorithms for simulation of closed (A5, A6) and open (A1-A4) contours according to postulated conditions, which allow form irregular composite curves and surfaces with different degree of complexity, docked together on the first order smoothness. The proposed computational algorithms can also be used to construct contours of higher orders smoothness using arcs of the same ratio curves. For analytical description of computational algorithms for one-dimensional contours simulation is used the mathematical apparatus of BN-calculation (Balyuba – Naidysh point calculation). The obtained algorithms have been presented in a point form, which is a symbolic form. For transition from point equations to a system of parametric equations, it is necessary to perform a coordinate-by-coordinate calculation, which can be presented geometrically as population of projections on the global coordinate system’s axes. As an example has been presented a computational algorithm that provides the use a system of parametric equations instead of symbolic point recording. The proposed algorithms have been successfully used for computer modeling and prediction for the impact of geometric shape imperfections on the strength and stability of engineering structures’ thin-walled shells. In particular, a numerical study method for a stress-strain state of steel vertical cylindrical reservoirs with regard to imperfections of theirs geometric shapes has been proposed.
{"title":"Computational Algorithms For Modeling of One-Dimensional Contours Through k In Advance Given Points","authors":"Евгений Конопацкий, E. Konopatskiy, Антон Крысько, A. Krys’ko, А. Бумага, A. Bumaga","doi":"10.12737/ARTICLE_5BC457ECE18491.72807735","DOIUrl":"https://doi.org/10.12737/ARTICLE_5BC457ECE18491.72807735","url":null,"abstract":"In this paper have been proposed theoretical bases for formation one-dimensional contours of the first order smoothness, passing through k in advance given points, including requirements to the contour in general, and the contour’s arcs in particular, as well as representations for tangents in extreme and intermediate contour sections, which determine the contour arc shape. Based on this theoretical material have been developed computational algorithms for simulation of closed (A5, A6) and open (A1-A4) contours according to postulated conditions, which allow form irregular composite curves and surfaces with different degree of complexity, docked together on the first order smoothness. The proposed computational algorithms can also be used to construct contours of higher orders smoothness using arcs of the same ratio curves. For analytical description of computational algorithms for one-dimensional contours simulation is used the mathematical apparatus of BN-calculation (Balyuba – Naidysh point calculation). The obtained algorithms have been presented in a point form, which is a symbolic form. For transition from point equations to a system of parametric equations, it is necessary to perform a coordinate-by-coordinate calculation, which can be presented geometrically as population of projections on the global coordinate system’s axes. As an example has been presented a computational algorithm that provides the use a system of parametric equations instead of symbolic point recording. The proposed algorithms have been successfully used for computer modeling and prediction for the impact of geometric shape imperfections on the strength and stability of engineering structures’ thin-walled shells. In particular, a numerical study method for a stress-strain state of steel vertical cylindrical reservoirs with regard to imperfections of theirs geometric shapes has been proposed.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79823678","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 : 2018-11-14DOI: 10.12737/article_5bc45918192362.77856682
Л. Жихарев, L. Zhikharev
One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.
{"title":"Fractal Dimensionalities","authors":"Л. Жихарев, L. Zhikharev","doi":"10.12737/article_5bc45918192362.77856682","DOIUrl":"https://doi.org/10.12737/article_5bc45918192362.77856682","url":null,"abstract":"One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79221143","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 : 2018-11-14DOI: 10.12737/ARTICLE_5BC454948A7D90.80979486
Я. Кокарева, Y. Kokareva
Ruled surfaces have long been known and are widely used in construction, architecture, design and engineering. And if from the technical point of view the developable surfaces are more attractive, then architecture and design successfully experiment with non-developable ones. In this paper are considered non-developable ruled surfaces with three generators, two of which are curvilinear ones. According to classification, such surfaces are called twice oblique cylindroids. In this paper has been proposed an approach for obtaining of twice oblique cylindroids by immersing a curve in a line congruence of hyperbolic type. Real directrixes of such congruence are a straight line and a curve. It has been proposed to use helical lines (cylindrical and conical ones) as a curvilinear directrix, and a helical line’s axis as the straight one. Then the congruence’s rectilinear ray will simultaneously intersect the helical line and its axis. Congruence parameters are the line’s pitch and the guide cylinder or cone’s radius. The choice of the curvilinear directrix is justified by the fact that the helical lines have found a wide application in engineering and architecture. Accordingly, the helical lines based surfaces can have a great potential. In this paper have been presented parametric equations of the considered congruences. The congruence equations have been considered from the point of view related to introducing a new curvilinear coordinate system. The obtained system’s coordinate surfaces and coordinate lines have been also studied in the paper. To extract the surface, it is necessary to immerse the curve in the congruence. To synthesize the equations has been used a constructive-parametric method based on the substitution of the immersed line’s parametric equations in the congruence equations according to a special algorithm. In the paper have been presented 5 examples for the synthesis of ruled surfaces equations such as the twice oblique cylindroid and their visualization. The method is universal and algorithmic, and therefore easily adaptable for the automated construction of surfaces with variable parameters of both the congruence and the immersed line.
{"title":"Synthesis of Equations For Ruled Surfaces With Two Curvilinear And One Rectangular Directrixes","authors":"Я. Кокарева, Y. Kokareva","doi":"10.12737/ARTICLE_5BC454948A7D90.80979486","DOIUrl":"https://doi.org/10.12737/ARTICLE_5BC454948A7D90.80979486","url":null,"abstract":"Ruled surfaces have long been known and are widely used in construction, architecture, design and engineering. And if from the technical point of view the developable surfaces are more attractive, then architecture and design successfully experiment with non-developable ones. In this paper are considered non-developable ruled surfaces with three generators, two of which are curvilinear ones. According to classification, such surfaces are called twice oblique cylindroids. In this paper has been proposed an approach for obtaining of twice oblique cylindroids by immersing a curve in a line congruence of hyperbolic type. Real directrixes of such congruence are a straight line and a curve. It has been proposed to use helical lines (cylindrical and conical ones) as a curvilinear directrix, and a helical line’s axis as the straight one. Then the congruence’s rectilinear ray will simultaneously intersect the helical line and its axis. Congruence parameters are the line’s pitch and the guide cylinder or cone’s radius. The choice of the curvilinear directrix is justified by the fact that the helical lines have found a wide application in engineering and architecture. Accordingly, the helical lines based surfaces can have a great potential. In this paper have been presented parametric equations of the considered congruences. The congruence equations have been considered from the point of view related to introducing a new curvilinear coordinate system. The obtained system’s coordinate surfaces and coordinate lines have been also studied in the paper. To extract the surface, it is necessary to immerse the curve in the congruence. To synthesize the equations has been used a constructive-parametric method based on the substitution of the immersed line’s parametric equations in the congruence equations according to a special algorithm. In the paper have been presented 5 examples for the synthesis of ruled surfaces equations such as the twice oblique cylindroid and their visualization. The method is universal and algorithmic, and therefore easily adaptable for the automated construction of surfaces with variable parameters of both the congruence and the immersed line.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90783812","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 : 2018-11-14DOI: 10.12737/ARTICLE_5BC45CBCCFBE67.89281424
Михаил Юрьевич Куприков, M. Kuprikov, Л. Маркин, L. Markin
In this paper have been considered questions related to automation of the layout for products with high layout density (primarily means of transport). It has been demonstrated how an aircraft’s geometric shape is formed on the basis of infrastructural and layout constraints. Influences of aerodynamic and internal layouts on the aircraft’s geometric shape have been described, taking into account mass-inertia characteristics of units placed in it. The layout’s reverse problem (when a required layout space is initial data for the aircraft’s geometric shape under hard infrastructure restrictions) has been presented. A project task of finding the rational parameters for the aircraft’s geometric shape as the task of multi-criterion discrete optimization has been described in a generalized form. It has been demonstrated that this task can be formulated as a search for the vector of design parameters as a multitude of admissible variants for design-and-engineering solutions. In the paper has been described a physical task formulation for automated layout as a system of restrictions on objects allocation indoor (required orientation, mutual compatibility, serviceability etc.). Mathematical task formulation for automated layout as the optimization problem has been described too. Since the allocation task is a classical geometric problem, it is necessary to develop appropriate geometric models for its solving. It has been shown that this process’s complexity is due to the complexity related to computer representation of information about geometric shape for layout objects of modern transport, especially the aerospace one. In this paper it has been shown that the abundance of models used in modern applied geometry and allowing describe geometric shapes for objects of any complexity, does not provide any solution for automated layout tasks. Possibilities of modern software have been also shown, and the reasons not allowing its direct use in the tasks of automated layout have been proved. The layout task’s mathematical formulation has been described as an optimization problem, specifying its objective function, limitations and efficiency criteria. Has been justified an approach (receptor methods and apparatus of normal equations) allowing, while creating geometric models for the automated layout, go from exhaustive options for allocation of layout objects to intellectual algorithms for automated allocation As has been shown in the paper, fractal theory is a good mathematical tool for study of rigid bodies’ surface geometry and mechanisms influencing on the obtaining surface structure.
{"title":"Geometric Aspects of Aircraft Automated Layout Design","authors":"Михаил Юрьевич Куприков, M. Kuprikov, Л. Маркин, L. Markin","doi":"10.12737/ARTICLE_5BC45CBCCFBE67.89281424","DOIUrl":"https://doi.org/10.12737/ARTICLE_5BC45CBCCFBE67.89281424","url":null,"abstract":"In this paper have been considered questions related to automation of the layout for products with high layout density (primarily means of transport). It has been demonstrated how an aircraft’s geometric shape is formed on the basis of infrastructural and layout constraints. Influences of aerodynamic and internal layouts on the aircraft’s geometric shape have been described, taking into account mass-inertia characteristics of units placed in it. The layout’s reverse problem (when a required layout space is initial data for the aircraft’s geometric shape under hard infrastructure restrictions) has been presented. A project task of finding the rational parameters for the aircraft’s geometric shape as the task of multi-criterion discrete optimization has been described in a generalized form. It has been demonstrated that this task can be formulated as a search for the vector of design parameters as a multitude of admissible variants for design-and-engineering solutions. In the paper has been described a physical task formulation for automated layout as a system of restrictions on objects allocation indoor (required orientation, mutual compatibility, serviceability etc.). Mathematical task formulation for automated layout as the optimization problem has been described too. Since the allocation task is a classical geometric problem, it is necessary to develop appropriate geometric models for its solving. It has been shown that this process’s complexity is due to the complexity related to computer representation of information about geometric shape for layout objects of modern transport, especially the aerospace one. In this paper it has been shown that the abundance of models used in modern applied geometry and allowing describe geometric shapes for objects of any complexity, does not provide any solution for automated layout tasks. Possibilities of modern software have been also shown, and the reasons not allowing its direct use in the tasks of automated layout have been proved. The layout task’s mathematical formulation has been described as an optimization problem, specifying its objective function, limitations and efficiency criteria. Has been justified an approach (receptor methods and apparatus of normal equations) allowing, while creating geometric models for the automated layout, go from exhaustive options for allocation of layout objects to intellectual algorithms for automated allocation As has been shown in the paper, fractal theory is a good mathematical tool for study of rigid bodies’ surface geometry and mechanisms influencing on the obtaining surface structure.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88113672","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 : 2018-11-14DOI: 10.12737/ARTICLE_5BC453447DB654.91666264
Юрий Поликарпов, Yu. V. Polikarpov
A brief journey into the history of industrial revolutions has been presented. It is noted that our society has entered the third industrial revolution’s era. In this regard, the main consequences of the third industrial revolution have been noted. The stages of development for design methods and the basic science providing the design process have been considered. The historical necessity and significance of Gaspar Monge’s descriptive geometry appearance has been considered as well. Modern products design approaches using CAD systems are described. It is stated that design has again returned to three-dimensional space, in fact prior to the Monge’s era, but at a new stage of development. The conclusion is drawn that, taking into account the realities and needs of modern production, it is necessary to modernize the descriptive geometry course for technical high educational institutions. The author's suggestions on course content changing are presented related to extension of one sections and reducing of another ones, taking into account the fact that in real design practice the designer solves geometric problems in three-dimensional space, rather than in a complex drawing. It is noted that in connection with the extensive use of CAD systems, the design stages and the composition of design documentation developed at each stage are changed. Such concepts as "electronic model" and "electronic document" have appeared and are widely used, that is confirmed by adoption of new USDD standards. In such a case the role and significance of some types of drawings may change in the near future, since modern CAD systems allow transfer to production not 2D drawings, but electronic models and product drawings.
{"title":"The Content of Descriptive Geometry Course For High Educational Institution’s In the Third Industrial Revolution’s Era","authors":"Юрий Поликарпов, Yu. V. Polikarpov","doi":"10.12737/ARTICLE_5BC453447DB654.91666264","DOIUrl":"https://doi.org/10.12737/ARTICLE_5BC453447DB654.91666264","url":null,"abstract":"A brief journey into the history of industrial revolutions has been presented. It is noted that our society has entered the third industrial revolution’s era. In this regard, the main consequences of the third industrial revolution have been noted. The stages of development for design methods and the basic science providing the design process have been considered. The historical necessity and significance of Gaspar Monge’s descriptive geometry appearance has been considered as well. Modern products design approaches using CAD systems are described. It is stated that design has again returned to three-dimensional space, in fact prior to the Monge’s era, but at a new stage of development. The conclusion is drawn that, taking into account the realities and needs of modern production, it is necessary to modernize the descriptive geometry course for technical high educational institutions. The author's suggestions on course content changing are presented related to extension of one sections and reducing of another ones, taking into account the fact that in real design practice the designer solves geometric problems in three-dimensional space, rather than in a complex drawing. It is noted that in connection with the extensive use of CAD systems, the design stages and the composition of design documentation developed at each stage are changed. Such concepts as \"electronic model\" and \"electronic document\" have appeared and are widely used, that is confirmed by adoption of new USDD standards. In such a case the role and significance of some types of drawings may change in the near future, since modern CAD systems allow transfer to production not 2D drawings, but electronic models and product drawings.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80340757","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 : 2018-08-21DOI: 10.12737/ARTICLE_5B55A695093294.45142608
Ю. Брылкин, Yuriy Brylkin
This paper is devoted to the problem of modeling a rough surface to ensure calculations for a flow around aircraft by high-enthalpy gas. The surface layer’s geometric characteristics along with the material’s chemical composition affect the surface’s optical indices and catalytic properties, and, consequently, on the measured heat flux. The problem of construction a geometric model for micro-surface has both fundamental and applied aspects. The fundamental nature stems from the fact that considered processes arising from the interaction of gas atoms and molecules with the surface are very complex ones. In such a case the correct interpretation for results of aircraft fragments’ ground experimental method is required. The work’s applied significance is determined by the need to optimize tools for flows diagnostic in high-enthalpy installations, in which simulation of thermal load affecting the aircraft in flight is taking place, as well as simulation of technological processes for heat-shielding materials and coatings development. Effective way for modeling of undifferentiated surfaces for gas dynamics problems solving is the use of fractal methods accounting the roughness at the micro- and nano-scale. They are based on the assertion that the natural surface’s structure has the same fractality at all levels. The development of this hypothesis has led to the emergence of a whole direction – material engineering – allowing most adequately describe self-organizing structures. Also, with the development of nanotechnologies, fractal geometry has found its own place in solving problems related to obtaining certain materials properties. As has been shown in the paper, fractal theory is a good mathematical tool for study of rigid bodies’ surface geometry and mechanisms influencing on the obtaining surface structure.
{"title":"Modeling of Surface’s Micro- And Nanostructures for Solving of Gas Dynamics, Heat And Mass Transfer Problems","authors":"Ю. Брылкин, Yuriy Brylkin","doi":"10.12737/ARTICLE_5B55A695093294.45142608","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B55A695093294.45142608","url":null,"abstract":"This paper is devoted to the problem of modeling a rough surface to ensure calculations for a flow around aircraft by high-enthalpy gas. The surface layer’s geometric characteristics along with the material’s chemical composition affect the surface’s optical indices and catalytic properties, and, consequently, on the measured heat flux. The problem of construction a geometric model for micro-surface has both fundamental and applied aspects. The fundamental nature stems from the fact that considered processes arising from the interaction of gas atoms and molecules with the surface are very complex ones. In such a case the correct interpretation for results of aircraft fragments’ ground experimental method is required. The work’s applied significance is determined by the need to optimize tools for flows diagnostic in high-enthalpy installations, in which simulation of thermal load affecting the aircraft in flight is taking place, as well as simulation of technological processes for heat-shielding materials and coatings development. Effective way for modeling of undifferentiated surfaces for gas dynamics problems solving is the use of fractal methods accounting the roughness at the micro- and nano-scale. They are based on the assertion that the natural surface’s structure has the same fractality at all levels. The development of this hypothesis has led to the emergence of a whole direction – material engineering – allowing most adequately describe self-organizing structures. Also, with the development of nanotechnologies, fractal geometry has found its own place in solving problems related to obtaining certain materials properties. As has been shown in the paper, fractal theory is a good mathematical tool for study of rigid bodies’ surface geometry and mechanisms influencing on the obtaining surface structure.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"158 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78067719","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 : 2018-08-21DOI: 10.12737/ARTICLE_5B55A35D683A33.30813949
Игорь Боровиков, I. Borovikov, Геннадий Юрьевич Иванов, G. Ivanov, Н. Суркова, N. Surkova
This publication is devoted to the application of transformations at descriptive geometry’s problems solution. Using parametric calculus lets rationally select the number of transformations in the drawing. In Cartesian coordinates, on condition that an identical coordinate plane exists, the difference between parameters of linear forms, given and converted ones, is equal to the number of transformations in the composition. In affine space under these conditions, this difference is equal to two. Based on parameters calculation the conclusion is confirmed that the method of rotation around the level line, as providing the transformation of the plane of general position to the level plane, is a composition of two transformations: replacement of projections planes and rotation around the projection line. In various geometries (affine, projective, algebraic ones, and topology) the types of corresponding transformations are studied. As a result of these transformations are obtained affine, projective, bi-rational and topologically equivalent figures respectively. Such transformations are widely used in solving of applied problems, for example, in the design of technical surfaces of dependent sections. At the same time, along with transformation invariants, the simplicity of the algorithm for constructing of corresponding figures should be taken into account, with the result that so-called stratified transformations are preferred. A sign of transformation’s stratification is a value of dimension for a set of corresponding points’ carriers. This fact explains the relative simplicity of the algorithm for constructing the corresponding points in such transformations. In this paper the use of stratified transformations when finding the points of intersection of a curve with a surface, as well as in the construction of surfaces with variable cross-section shape are considered. The given examples show stratification idea possibilities for solving the problems of descriptive geometry.
{"title":"On Application of Transformations at Descriptive Geometry’s Problems Solution","authors":"Игорь Боровиков, I. Borovikov, Геннадий Юрьевич Иванов, G. Ivanov, Н. Суркова, N. Surkova","doi":"10.12737/ARTICLE_5B55A35D683A33.30813949","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B55A35D683A33.30813949","url":null,"abstract":"This publication is devoted to the application of transformations at descriptive geometry’s problems solution. Using parametric calculus lets rationally select the number of transformations in the drawing. In Cartesian coordinates, on condition that an identical coordinate plane exists, the difference between parameters of linear forms, given and converted ones, is equal to the number of transformations in the composition. In affine space under these conditions, this difference is equal to two. Based on parameters calculation the conclusion is confirmed that the method of rotation around the level line, as providing the transformation of the plane of general position to the level plane, is a composition of two transformations: replacement of projections planes and rotation around the projection line. In various geometries (affine, projective, algebraic ones, and topology) the types of corresponding transformations are studied. As a result of these transformations are obtained affine, projective, bi-rational and topologically equivalent figures respectively. Such transformations are widely used in solving of applied problems, for example, in the design of technical surfaces of dependent sections. At the same time, along with transformation invariants, the simplicity of the algorithm for constructing of corresponding figures should be taken into account, with the result that so-called stratified transformations are preferred. A sign of transformation’s stratification is a value of dimension for a set of corresponding points’ carriers. This fact explains the relative simplicity of the algorithm for constructing the corresponding points in such transformations. In this paper the use of stratified transformations when finding the points of intersection of a curve with a surface, as well as in the construction of surfaces with variable cross-section shape are considered. The given examples show stratification idea possibilities for solving the problems of descriptive geometry.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83694180","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 : 2018-08-21DOI: 10.12737/ARTICLE_5B559F018F85A7.77112269
Д. Волошинов, D. Voloshinov
This paper is devoted to analysis of Dandelin spheres problem based on the constructive geometric approach. In the paper it has been demonstrated that the traditional approach used to this problem solving leads to obtaining for only a limited set of heterogeneous solutions. Consideration of the problem in the context of plane and space’s projective properties by structural geometry’s methods allows interpret this problem’s results in a new way. In the paper it has been demonstrated that the solved problem has a purely projective nature and can be solved by a unified method, which is impossible to achieve if conduct reasoning and construct proofs only on affine geometry’s positions. The research’s scientific novelty is the discovery and theoretical justification of a new classification feature allowing classify as Dandelin spheres the set of spheres pairs with imaginary tangents to the quadric, as well as pairs of imaginary spheres with a unified principle for constructive interrelation of images, along with real solutions. The work’s practical significance lies in the extension of application areas for geometric modeling’s constructive methods to the solution of problems, in the impro vement of geometric theory, in the development of system for geometric modeling Simplex’s functional capabilities for tasks of objects and processes design automation. The algorithms presented in the paper demonstrate the deep projective nature and interrelation of such problems as Apollonius circles and spheres one, Dandelin spheres one and others, as well as lay the groundwork for researches in the direction of these problems’ multidimensional interpretations. The problem solution can be useful for second-order curves’ blending function realization by means of circles with a view to improve the tools of CAD-systems’ design automation without use of mathematical numerical methods for these purposes.
{"title":"On the Peculiarities of the Constructive Solution For Dandelin Spheres Problem","authors":"Д. Волошинов, D. Voloshinov","doi":"10.12737/ARTICLE_5B559F018F85A7.77112269","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B559F018F85A7.77112269","url":null,"abstract":"This paper is devoted to analysis of Dandelin spheres problem based on the constructive geometric approach. In the paper it has been demonstrated that the traditional approach used to this problem solving leads to obtaining for only a limited set of heterogeneous solutions. Consideration of the problem in the context of plane and space’s projective properties by structural geometry’s methods allows interpret this problem’s results in a new way. In the paper it has been demonstrated that the solved problem has a purely projective nature and can be solved by a unified method, which is impossible to achieve if conduct reasoning and construct proofs only on affine geometry’s positions. The research’s scientific novelty is the discovery and theoretical justification of a new classification feature allowing classify as Dandelin spheres the set of spheres pairs with imaginary tangents to the quadric, as well as pairs of imaginary spheres with a unified principle for constructive interrelation of images, along with real solutions. The work’s practical significance lies in the extension of application areas for geometric modeling’s constructive methods to the solution of problems, in the impro vement of geometric theory, in the development of system for geometric modeling Simplex’s functional capabilities for tasks of objects and processes design automation. The algorithms presented in the paper demonstrate the deep projective nature and interrelation of such problems as Apollonius circles and spheres one, Dandelin spheres one and others, as well as lay the groundwork for researches in the direction of these problems’ multidimensional interpretations. The problem solution can be useful for second-order curves’ blending function realization by means of circles with a view to improve the tools of CAD-systems’ design automation without use of mathematical numerical methods for these purposes.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"90 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78452369","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 : 2018-08-21DOI: 10.12737/ARTICLE_5B55A5163FA053.07622109
Николай Сальков, N. Sal'kov
In technical inventions related to innovative developments, the role of one of the main components belongs to geometry. A follow hypothesis has been adopted: in technical inventions the geometrical component is the determining one. This hypothesis applied to technical inventions can be confirmed by any copyright certificate, any patent both in Russia and abroad. In proposed paper this statement is proved by examples developed based on geometry of following inventions. 1. Screen feeder for sticky masses. Screen feeder’s grates are made in pairs, and between grate pairs there are gaps for screening of material’s size-defined fractions. In the screen-feeder has been proposed such geometry of grates that grates of each pair could clean each other, thereby preventing sticking on the surface and destroying the gaps between the pairs, which transforms the usual screen-feeder with cylindrical grates at an ordinary feeder. 2. Double-screw mixer for paste-like masses. The mixer consists of two contiguous worms. Their surfaces are the helical ones, in cross-section consisting of two quarters of circles stacked at the ends. Such cross-section allows homogenize the mixed paste-like material in the best way, and also deliver it under higher pressure in an extrusion head, that improves a final product. 3. Machine for processing of multi-faceted surfaces. This invention serves for manufacture of worms with a cross-section composed of two, three, etc. pieces of circles of the same radius and angle. Worms, made with this machine, are designed for the above mentioned two-screw mixer. 4. Method of mechanical processing. This method is also intended to manufacture of worms for two-screw mixer.
{"title":"The Geometric Component Of Technical Innovations","authors":"Николай Сальков, N. Sal'kov","doi":"10.12737/ARTICLE_5B55A5163FA053.07622109","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B55A5163FA053.07622109","url":null,"abstract":"In technical inventions related to innovative developments, the role of one of the main components belongs to geometry. A follow hypothesis has been adopted: in technical inventions the geometrical component is the determining one. This hypothesis applied to technical inventions can be confirmed by any copyright certificate, any patent both in Russia and abroad. In proposed paper this statement is proved by examples developed based on geometry of following inventions. 1. Screen feeder for sticky masses. Screen feeder’s grates are made in pairs, and between grate pairs there are gaps for screening of material’s size-defined fractions. In the screen-feeder has been proposed such geometry of grates that grates of each pair could clean each other, thereby preventing sticking on the surface and destroying the gaps between the pairs, which transforms the usual screen-feeder with cylindrical grates at an ordinary feeder. 2. Double-screw mixer for paste-like masses. The mixer consists of two contiguous worms. Their surfaces are the helical ones, in cross-section consisting of two quarters of circles stacked at the ends. Such cross-section allows homogenize the mixed paste-like material in the best way, and also deliver it under higher pressure in an extrusion head, that improves a final product. 3. Machine for processing of multi-faceted surfaces. This invention serves for manufacture of worms with a cross-section composed of two, three, etc. pieces of circles of the same radius and angle. Worms, made with this machine, are designed for the above mentioned two-screw mixer. 4. Method of mechanical processing. This method is also intended to manufacture of worms for two-screw mixer.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"463 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79856302","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 : 2018-08-21DOI: 10.12737/ARTICLE_5B559A548FA209.41386317
Владимир Вышнепольский, Vladimir Vyshnyepolskiy, Нина Кадыкова, N. Kadykova, Николай Сальков, N. Sal'kov
In May 2018 the Engineering Graphics Chair celebrates 90 years from the date of its foundation. The Chair was organized in 1928. The paper tells the Chair’s history, its teachers and heads, as well as a brief description of its scientific work. In 1900 were established the Moscow Higher Feminine Courses (MHFCs). A year after the October revolution, in late 1918, MHFCs were transformed into the 2nd Moscow state University. In 1930 the 2nd MSU was reorganized into three independent institutes: medical, chemical-technological and pedagogical ones. In May 1928 was organized the Chair of Technical Drawing, this moment is the counting of Engineering Graphics Chair existence. The first head of the Chair was S.G. Borisov. Than the Chair was supervised by Associate Professor A.A. Sintsov (from September 1932 till January 1942), Associate Professor M.Ya. Khanyutin (in 1942–1952), Associate Professor N.I. Noskov (in 1954–1962), Associate Professor F.T. Karpechenko (in 1962–1972), Senior Lecturer N.A. Sevruk (in 1972–1982), Professor, Doctor of Engineering E.K. Voloshin-Chelpan (from January 1982 to August 2007), Associate Professor V.I. Vyshnepolsky (from August 2007 till present). Currently, on the Chair are carrying out researches in the following directions: Higher School’s Pedagogy; Academic Competitions of Regional and All-Russia’s Level; Loci; Geometry of Cyclic Surfaces; Theory of Kinetic Geometry; Geometries; Geometric Transformations; Theory of Fractals; Famous Geometers’ Biographies.
{"title":"90-Year Anniversary of Mitkht’s Engineering Graphics Chair","authors":"Владимир Вышнепольский, Vladimir Vyshnyepolskiy, Нина Кадыкова, N. Kadykova, Николай Сальков, N. Sal'kov","doi":"10.12737/ARTICLE_5B559A548FA209.41386317","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B559A548FA209.41386317","url":null,"abstract":"In May 2018 the Engineering Graphics Chair celebrates 90 years from the date of its foundation. The Chair was organized in 1928. The paper tells the Chair’s history, its teachers and heads, as well as a brief description of its scientific work. In 1900 were established the Moscow Higher Feminine Courses (MHFCs). A year after the October revolution, in late 1918, MHFCs were transformed into the 2nd Moscow state University. In 1930 the 2nd MSU was reorganized into three independent institutes: medical, chemical-technological and pedagogical ones. In May 1928 was organized the Chair of Technical Drawing, this moment is the counting of Engineering Graphics Chair existence. The first head of the Chair was S.G. Borisov. Than the Chair was supervised by Associate Professor A.A. Sintsov (from September 1932 till January 1942), Associate Professor M.Ya. Khanyutin (in 1942–1952), Associate Professor N.I. Noskov (in 1954–1962), Associate Professor F.T. Karpechenko (in 1962–1972), Senior Lecturer N.A. Sevruk (in 1972–1982), Professor, Doctor of Engineering E.K. Voloshin-Chelpan (from January 1982 to August 2007), Associate Professor V.I. Vyshnepolsky (from August 2007 till present). Currently, on the Chair are carrying out researches in the following directions: Higher School’s Pedagogy; Academic Competitions of Regional and All-Russia’s Level; Loci; Geometry of Cyclic Surfaces; Theory of Kinetic Geometry; Geometries; Geometric Transformations; Theory of Fractals; Famous Geometers’ Biographies.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79020572","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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