Pub Date : 2018-08-21DOI: 10.12737/ARTICLE_5B559DC3551F95.26045830
Д. Волошинов, D. Voloshinov
While using conventional tools for solving geometric problems, it is difficult to obtain and analyze results where imaginary geometric images appear. Despite the recognition of legitimacy and scientific value of imaginary solutions presenting in geometric constructions, the question on such solutions’ appropriateness and practical feasibility remains no completely clear up till now. That’s why, for most practitioners imaginary solutions are presented as something unattainable or unimportant. However, the introduction of imaginary geometric images into the practice of geometric modeling makes it possible to obtain solutions in an exhaustiveness, to develop unified algorithms for solving problems that were usually presented as either not solvable or reduced to solutions in partial settings. The use of computer technologies and the paradigm of constructive geometric modeling allow eliminate this problem’s acuteness, and direct efforts both at geometric theory’s improvement and introduction of scientific achievements in this area at the field of practical applications. Automation means for geometric experiment make it possible to find new regularities in seemingly well-known mathematical facts, to come to more general understanding of geometric concepts and images. This paper is devoted to analysis of some geometric schemes and to discussion of arising from it questions related to the theory of second-order curves creation by the methods of constructive synthesis. In the paper it has been demonstrated that the currently used definitions of second-order curves’ center and diameters contradict the principle of conics indistinguishability in projective geometry. The ways for eliminating of these contradictions have been proposed, and a unified algorithm for the second-order curves’ foci creation has been developed based on these ways.
{"title":"A Unified Constructive Algorithm For Second- Order Curves’ Foci Creation","authors":"Д. Волошинов, D. Voloshinov","doi":"10.12737/ARTICLE_5B559DC3551F95.26045830","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B559DC3551F95.26045830","url":null,"abstract":"While using conventional tools for solving geometric problems, it is difficult to obtain and analyze results where imaginary geometric images appear. Despite the recognition of legitimacy and scientific value of imaginary solutions presenting in geometric constructions, the question on such solutions’ appropriateness and practical feasibility remains no completely clear up till now. That’s why, for most practitioners imaginary solutions are presented as something unattainable or unimportant. However, the introduction of imaginary geometric images into the practice of geometric modeling makes it possible to obtain solutions in an exhaustiveness, to develop unified algorithms for solving problems that were usually presented as either not solvable or reduced to solutions in partial settings. The use of computer technologies and the paradigm of constructive geometric modeling allow eliminate this problem’s acuteness, and direct efforts both at geometric theory’s improvement and introduction of scientific achievements in this area at the field of practical applications. Automation means for geometric experiment make it possible to find new regularities in seemingly well-known mathematical facts, to come to more general understanding of geometric concepts and images. This paper is devoted to analysis of some geometric schemes and to discussion of arising from it questions related to the theory of second-order curves creation by the methods of constructive synthesis. In the paper it has been demonstrated that the currently used definitions of second-order curves’ center and diameters contradict the principle of conics indistinguishability in projective geometry. The ways for eliminating of these contradictions have been proposed, and a unified algorithm for the second-order curves’ foci creation has been developed based on these ways.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"458 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86682592","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_5B55A16B547678.01517798
О. Графский, O. Grafskiy, Ю. Пономарчук, Yu. Ponomarchuk, В. Суриц, V. Suric
When studying the theory of contour construction in “Affine and Projective Geometry” course on educational program specializations “Computer-Aided Design Systems” and “Applied Informatics in Design” a unit of computational and graphic task "Contour Construction" is carrying out in structural design. In this computational and graphic task the contour constructions are carrying out by second-order curves (a circle — by the radius and graphical method; a hyperbola, an ellipse, a parabola — by means of Pascal curves, taking into account positions of engineering discriminant). The constructions of an arc of ellipse, hyperbola, and parabola are carried out based on Pascal theorem: in any hexagon, which vertices belong to a second-order series, three points of the opposite sides’ intersection lie on one straight line — the Pascal line. However, in construction of a conic (a second-order curve), it is necessary to draw students’ attention to the fact that the points belonging to a second-order series (a second-order curve, or a conic) make a geometrical locus of intersection of Pascal hexagon’s adjacent opposite sides. By this method students successfully construct conjugate arcs of an ellipse and a hyperbola with other conics. The construction of a parabola arc, conjugated with other conics, is carried out by the method of engineering discriminant (it is more convenient to divide line segments in halves: a median and a triangle side, which is opposite to its vertex lying on a parabola arc). It should be noted that theoretical and practical material on this subject corresponds to the assimilation of Study Plan’s necessary competences (in accordance with each educational program), however, some aspects of this subject are accepted by students simply by trust. The aim of this paper is research of construction methods for parabola, applied to contour simulation.
{"title":"Properties Features of Parabola at Its Simulation","authors":"О. Графский, O. Grafskiy, Ю. Пономарчук, Yu. Ponomarchuk, В. Суриц, V. Suric","doi":"10.12737/ARTICLE_5B55A16B547678.01517798","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B55A16B547678.01517798","url":null,"abstract":"When studying the theory of contour construction in “Affine and Projective Geometry” course on educational program specializations “Computer-Aided Design Systems” and “Applied Informatics in Design” a unit of computational and graphic task \"Contour Construction\" is carrying out in structural design. In this computational and graphic task the contour constructions are carrying out by second-order curves (a circle — by the radius and graphical method; a hyperbola, an ellipse, a parabola — by means of Pascal curves, taking into account positions of engineering discriminant). The constructions of an arc of ellipse, hyperbola, and parabola are carried out based on Pascal theorem: in any hexagon, which vertices belong to a second-order series, three points of the opposite sides’ intersection lie on one straight line — the Pascal line. However, in construction of a conic (a second-order curve), it is necessary to draw students’ attention to the fact that the points belonging to a second-order series (a second-order curve, or a conic) make a geometrical locus of intersection of Pascal hexagon’s adjacent opposite sides. By this method students successfully construct conjugate arcs of an ellipse and a hyperbola with other conics. The construction of a parabola arc, conjugated with other conics, is carried out by the method of engineering discriminant (it is more convenient to divide line segments in halves: a median and a triangle side, which is opposite to its vertex lying on a parabola arc). It should be noted that theoretical and practical material on this subject corresponds to the assimilation of Study Plan’s necessary competences (in accordance with each educational program), however, some aspects of this subject are accepted by students simply by trust. The aim of this paper is research of construction methods for parabola, applied to contour simulation.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78276963","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_5B55A829CEE6C0.74112002
Виктор Короткий, V. Korotkiy, Екатерина Александровна Усманова, E. Usmanova
Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.
{"title":"Second Order Curves on Computer Screen","authors":"Виктор Короткий, V. Korotkiy, Екатерина Александровна Усманова, E. Usmanova","doi":"10.12737/ARTICLE_5B55A829CEE6C0.74112002","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B55A829CEE6C0.74112002","url":null,"abstract":"Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78452697","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_5B559C70BECF44.21848537
Д. Волошинов, D. Voloshinov
The Apollonius problem on construction of circles, tangent to three arbitrary given circles of a plane, is one of classical geometry’s well-studied problems. The presented paper’s materials are directed at development a unified theory for Apollonius problem solving, taking into account it’s not only real, but also invisible complex-valued images. In the paper it has been demonstrated, that fundamental geometric structures, on which Apollonius problem is based on, are applicable not only to real, but also to complex-valued data, that makes possible to eliminate many exceptions, currently existing in it. In this paper Apollonius problem’s fundamental nature and its strong correlation with projective and quadratic geometric transformations has been disclosed. It has been proved that Apollonius problem and its analogues have a single solution method, in contrast to the prevailing idea that these problems can be solved only by separate particular methods. A concept of geometric experiment proposed by the author has allowed find out many previously unknown and discussed in this paper common factors, due to the set of many computational tests in the system Simplex for visual design of geometric models. In this paper is considered an example for solving an analogue of Apollonian problem for three-dimensional space, but proposed algorithm’s operation is universal, and it can be equally applied to solving similar problems in spaces of arbitrary dimensions. Obtained results demonstrate capabilities of methods for constructive modeling and multidimensional descriptive geometry in application to solving of complex mathematical problems, and determine the trends in development for automation systems of constructive geometric modeling.
{"title":"Visual-Graphic Design of a Unitary Constructive Model to Solve Analogues For Apollonius Problem Taking into Account Imaginary Geometric Images","authors":"Д. Волошинов, D. Voloshinov","doi":"10.12737/ARTICLE_5B559C70BECF44.21848537","DOIUrl":"https://doi.org/10.12737/ARTICLE_5B559C70BECF44.21848537","url":null,"abstract":"The Apollonius problem on construction of circles, tangent to three arbitrary given circles of a plane, is one of classical geometry’s well-studied problems. The presented paper’s materials are directed at development a unified theory for Apollonius problem solving, taking into account it’s not only real, but also invisible complex-valued images. In the paper it has been demonstrated, that fundamental geometric structures, on which Apollonius problem is based on, are applicable not only to real, but also to complex-valued data, that makes possible to eliminate many exceptions, currently existing in it. In this paper Apollonius problem’s fundamental nature and its strong correlation with projective and quadratic geometric transformations has been disclosed. It has been proved that Apollonius problem and its analogues have a single solution method, in contrast to the prevailing idea that these problems can be solved only by separate particular methods. A concept of geometric experiment proposed by the author has allowed find out many previously unknown and discussed in this paper common factors, due to the set of many computational tests in the system Simplex for visual design of geometric models. In this paper is considered an example for solving an analogue of Apollonian problem for three-dimensional space, but proposed algorithm’s operation is universal, and it can be equally applied to solving similar problems in spaces of arbitrary dimensions. Obtained results demonstrate capabilities of methods for constructive modeling and multidimensional descriptive geometry in application to solving of complex mathematical problems, and determine the trends in development for automation systems of constructive geometric modeling.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76033555","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 : 2018-04-25DOI: 10.12737/ARTICLE_5AD0971A86AF78.65167837
Михаил Решетников, M. Reshetnikov, С. Рязанов, S. Ryazanov
{"title":"Estimation of Worm-Gear Drives’ Parameters Based on Computer Graphics 3D-Methods","authors":"Михаил Решетников, M. Reshetnikov, С. Рязанов, S. Ryazanov","doi":"10.12737/ARTICLE_5AD0971A86AF78.65167837","DOIUrl":"https://doi.org/10.12737/ARTICLE_5AD0971A86AF78.65167837","url":null,"abstract":"","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"124 1","pages":"34-38"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91101913","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 : 2018-04-25DOI: 10.12737/ARTICLE_5AD09A84CBD105.88047545
Р Н Булычев, R. Bulychev, Т. Аюшеев, T. Ayusheev
{"title":"Discription of Process for Sheet Material Deforming With Using of Parametric Solid-State Simulation","authors":"Р Н Булычев, R. Bulychev, Т. Аюшеев, T. Ayusheev","doi":"10.12737/ARTICLE_5AD09A84CBD105.88047545","DOIUrl":"https://doi.org/10.12737/ARTICLE_5AD09A84CBD105.88047545","url":null,"abstract":"","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"10 1","pages":"48-56"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77102483","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 : 2018-04-25DOI: 10.12737/ARTICLE_5AD085A6D75BB5.99078854
О. Графский, O. Grafskiy, Ю. Пономарчук, Yu. Ponomarchuk, А. Холодилов, A. Holodilov
{"title":"Geometry of Electrostatic Fields","authors":"О. Графский, O. Grafskiy, Ю. Пономарчук, Yu. Ponomarchuk, А. Холодилов, A. Holodilov","doi":"10.12737/ARTICLE_5AD085A6D75BB5.99078854","DOIUrl":"https://doi.org/10.12737/ARTICLE_5AD085A6D75BB5.99078854","url":null,"abstract":"","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"55 1","pages":"10-19"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79291012","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 : 2018-04-25DOI: 10.12737/ARTICLE_5AD09D62E8A792.47611365
Ю. Савельев, Yu. Savel'ev, Е. Бабич, E. Babich
{"title":"Computer Method for Learning of Descriptive Geometry. Technical Task","authors":"Ю. Савельев, Yu. Savel'ev, Е. Бабич, E. Babich","doi":"10.12737/ARTICLE_5AD09D62E8A792.47611365","DOIUrl":"https://doi.org/10.12737/ARTICLE_5AD09D62E8A792.47611365","url":null,"abstract":"","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"12 1","pages":"67-74"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76443964","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 : 2018-04-25DOI: 10.12737/ARTICLE_5AD07ED61BC114.52669586
Геннадий Юрьевич Иванов, G. Ivanov
{"title":"Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane","authors":"Геннадий Юрьевич Иванов, G. Ivanov","doi":"10.12737/ARTICLE_5AD07ED61BC114.52669586","DOIUrl":"https://doi.org/10.12737/ARTICLE_5AD07ED61BC114.52669586","url":null,"abstract":"","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"10 1","pages":"3-9"},"PeriodicalIF":0.0,"publicationDate":"2018-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73056350","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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