Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21F207BFD6E4.78537377
Владимир Вышнепольский, Vladimir Vyshnepol'skiy, К. Киршанов, K. Kirshanov, К.А. Егиазарян, K. Egiazaryan
The loci (L) equally spaced from a sphere and a straight line, and from a conic surface and a plane, are considered. The following options have been considered. The straight line passes through the center of the sphere (a = 0), at the same time completely at spheres’ positive radiuses a surface of rotation is obtained, forming which the parabola is, and a rotation axis – this straight line. The parabola’s top forms the biggest parallel on the site points of intersection of the parabola’s forming with the rotation axis. Let's call such paraboloid a perpendicular paraboloid of rotation. The straight line crosses the sphere, but does not pass through the center (0 < a < R/2) – a perpendicular paraboloid, at that the surface is also completely obtained at radiuses’ positive values. The straight line is tangent to the sphere (a = R/2) – a surface which projections are parabolas, lemniscates and circles, and a piece from a tangency point to the sphere center – at radiuses positive values; a beam from the sphere center, perpendicular to this straight line – at radiuses negative values, at that the beam and the piece belong to one straight line. The straight line lies out of the sphere (α > R/2) – two different surfaces, having the general properties with a hyperbolic paraboloid, are obtained, one of which is obtained at radius positive values, and another one – at radius negative values. It has been noticed that loci, equally spaced from a sphere and a straight line, and from a cylinder and a point, coincide at equal radiuses and distances from axes to points and straight lines if to take into account the surfaces obtained both at positive, and negative values of radiuses. Locus, equally spaced from the conic surface of rotation and the plane, are two elliptic conic surfaces which in case 7.4.1 degenerate in the conic surfaces of rotation. In cases 7.4.3 and 7.4.4 one elliptic conic surface degenerates in a plane and a parabolic cylinder respectively.
{"title":"Loci of Points Equally Spaced From Two Given Geometrical Figures. Part 3","authors":"Владимир Вышнепольский, Vladimir Vyshnepol'skiy, К. Киршанов, K. Kirshanov, К.А. Егиазарян, K. Egiazaryan","doi":"10.12737/ARTICLE_5C21F207BFD6E4.78537377","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21F207BFD6E4.78537377","url":null,"abstract":"The loci (L) equally spaced from a sphere and a straight line, and from a conic surface and a plane, are considered. The following options have been considered. The straight line passes through the center of the sphere (a = 0), at the same time completely at spheres’ positive radiuses a surface of rotation is obtained, forming which the parabola is, and a rotation axis – this straight line. The parabola’s top forms the biggest parallel on the site points of intersection of the parabola’s forming with the rotation axis. Let's call such paraboloid a perpendicular paraboloid of rotation. The straight line crosses the sphere, but does not pass through the center (0 < a < R/2) – a perpendicular paraboloid, at that the surface is also completely obtained at radiuses’ positive values. The straight line is tangent to the sphere (a = R/2) – a surface which projections are parabolas, lemniscates and circles, and a piece from a tangency point to the sphere center – at radiuses positive values; a beam from the sphere center, perpendicular to this straight line – at radiuses negative values, at that the beam and the piece belong to one straight line. The straight line lies out of the sphere (α > R/2) – two different surfaces, having the general properties with a hyperbolic paraboloid, are obtained, one of which is obtained at radius positive values, and another one – at radius negative values. It has been noticed that loci, equally spaced from a sphere and a straight line, and from a cylinder and a point, coincide at equal radiuses and distances from axes to points and straight lines if to take into account the surfaces obtained both at positive, and negative values of radiuses. Locus, equally spaced from the conic surface of rotation and the plane, are two elliptic conic surfaces which in case 7.4.1 degenerate in the conic surfaces of rotation. In cases 7.4.3 and 7.4.4 one elliptic conic surface degenerates in a plane and a parabolic cylinder respectively.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"1937 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87751955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21FA732F6B62.81431444
В.М. Панченко, V. M. Panchenko
When studying ''Engineering and Computer Graphics'' discipline, extramural students are faced with a number of difficulties. Age groups of these students differ from full-time students from behind a greater number of students related to more age categories. Also, unlike full-time students, the level of extramural students’ primary education is higher, but it has been acquired for a long time, and knowledge, in the vast majority of cases, leaves much to be desired. In addition to the described differences it is possible to report a lesser amount of free time, which an extramural student can use for his independent work because of his primary employment’s strained activity timetable. An important moment that plays a key role in discipline understanding is the complexity of "Engineering and Computer Graphics" subject itself, which requires drawing skills (in the school some students did not even have such a discipline) and the ability for spatial thinking. In the presented paper have been considered features on age groups, primary education and drawing skill level for two streams of extramural students learning on the "Railway Operations" specialty in the Russian University of Transport (MIIT). In view of students’ contingent peculiarities the use of modern teaching tools in the process of studying "Engineering and Computer Graphics" discipline has been suggested as a method for enhancement of effectiveness for understanding of educational material. As an illustration of obtained theoretical concepts has been presented a plan for carrying out a laboratory work on "Engineering and Computer Graphics" discipline using modern teaching techniques. In the process of performing the laboratory work, modern teaching tools are used, and after its completion the trainees receive a useful solid piece (a stand for a smartphone) made on a 3D printer, obtained with the help of a three-dimensional model prepared by students, that increases the efficiency of received material’s understanding.
{"title":"Modern Means of Graphic Disciplines Teaching for Extramural Students","authors":"В.М. Панченко, V. M. Panchenko","doi":"10.12737/ARTICLE_5C21FA732F6B62.81431444","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21FA732F6B62.81431444","url":null,"abstract":"When studying ''Engineering and Computer Graphics'' discipline, extramural students are faced with a number of difficulties. Age groups of these students differ from full-time students from behind a greater number of students related to more age categories. Also, unlike full-time students, the level of extramural students’ primary education is higher, but it has been acquired for a long time, and knowledge, in the vast majority of cases, leaves much to be desired. In addition to the described differences it is possible to report a lesser amount of free time, which an extramural student can use for his independent work because of his primary employment’s strained activity timetable. An important moment that plays a key role in discipline understanding is the complexity of \"Engineering and Computer Graphics\" subject itself, which requires drawing skills (in the school some students did not even have such a discipline) and the ability for spatial thinking. In the presented paper have been considered features on age groups, primary education and drawing skill level for two streams of extramural students learning on the \"Railway Operations\" specialty in the Russian University of Transport (MIIT). In view of students’ contingent peculiarities the use of modern teaching tools in the process of studying \"Engineering and Computer Graphics\" discipline has been suggested as a method for enhancement of effectiveness for understanding of educational material. As an illustration of obtained theoretical concepts has been presented a plan for carrying out a laboratory work on \"Engineering and Computer Graphics\" discipline using modern teaching techniques. In the process of performing the laboratory work, modern teaching tools are used, and after its completion the trainees receive a useful solid piece (a stand for a smartphone) made on a 3D printer, obtained with the help of a three-dimensional model prepared by students, that increases the efficiency of received material’s understanding.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74050136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21F80E2925C6.80568562
Ирина Дмитриева, I. Dmitrieva, Геннадий Юрьевич Иванов, G. Ivanov
Qualified presentation of the topic "Tangent Plane and Surface Normal" in terms of competence approach is possible with the proper level for students' attention focusing on both intra-subject and inter-subject relations of descriptive geometry. Intra-subject connections follow from the position that the contingence is a particular (limit) case of intersection. Therefore, the line of intersection of the tangent plane and the surface, or two touching surfaces, has a special point at the tangency point. It is known from differential geometry [1] that this point can be nodal, return, or isolated one. In turn, this point’s appearance depends on differential properties of the surface(s) in this point’s vicinity. That's why, for the competent solution of the considered positional problem account must be also taken of the inter-subject connections for descriptive and differential geometry. In the training courses of descriptive geometry tangent planes are built only to the simplest surfaces, containing, as a rule, the frames of straight lines and circles. Therefore, the tangent plane is defined by two tangents drawn at the tangency point to two such lines. In engineering practice, as such lines are used cross-sections a surface by planes parallel to any two coordinate planes. That is, from the standpoints for the course of higher mathematics, the problem is reduced to calculation for partial derivatives. Although this topic is studied after the course of descriptive geometry, it seems possible to give geometric explanation for computation of partial derivatives in a nutshell. It also seems that the study of this topic will be stimulated by a story about engineering problems, which solution is based on construction of the tangent plane and the normal to the technical surface. In this paper has been presented an example for the use of surface curvature lines for programming of milling processing for 3D-harness surfaces.
{"title":"Competence Approach in Teaching the Topic \"Tangent Plane and Normal\"","authors":"Ирина Дмитриева, I. Dmitrieva, Геннадий Юрьевич Иванов, G. Ivanov","doi":"10.12737/ARTICLE_5C21F80E2925C6.80568562","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21F80E2925C6.80568562","url":null,"abstract":"Qualified presentation of the topic \"Tangent Plane and Surface Normal\" in terms of competence approach is possible with the proper level for students' attention focusing on both intra-subject and inter-subject relations of descriptive geometry. Intra-subject connections follow from the position that the contingence is a particular (limit) case of intersection. Therefore, the line of intersection of the tangent plane and the surface, or two touching surfaces, has a special point at the tangency point. It is known from differential geometry [1] that this point can be nodal, return, or isolated one. In turn, this point’s appearance depends on differential properties of the surface(s) in this point’s vicinity. That's why, for the competent solution of the considered positional problem account must be also taken of the inter-subject connections for descriptive and differential geometry. In the training courses of descriptive geometry tangent planes are built only to the simplest surfaces, containing, as a rule, the frames of straight lines and circles. Therefore, the tangent plane is defined by two tangents drawn at the tangency point to two such lines. In engineering practice, as such lines are used cross-sections a surface by planes parallel to any two coordinate planes. That is, from the standpoints for the course of higher mathematics, the problem is reduced to calculation for partial derivatives. Although this topic is studied after the course of descriptive geometry, it seems possible to give geometric explanation for computation of partial derivatives in a nutshell. It also seems that the study of this topic will be stimulated by a story about engineering problems, which solution is based on construction of the tangent plane and the normal to the technical surface. In this paper has been presented an example for the use of surface curvature lines for programming of milling processing for 3D-harness surfaces.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85491957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21F96DCE5DE8.36096061
Алексей Бойков, A. Boykov
In this paper the visibility concept in the context of modeling of multidimensional spaces’ objects is clarified. It is concluded that such model’s visibility should be defined as unambiguity and completeness of information presented in this model and consistent with the student’s experience in the area of modeling a space of higher dimension (3D) by elements of spaces of lower dimension (2D). Such possibilities are presented by the generalized complex drawing. Examples for objects 4D-modeling using two 3D or three 2D flat projections are presented, some properties of the 4D generalized drawing are listed. The solution of problems with 4D-objects is considered on the example of 4D-pyramid section construction, and deploying its lateral surface. It is shown that to simplify the solution of these problems is required a system allowing automatically perform repetitive sequences of constructions. A list of elementary constructions is presented, and a method for recording of composite constructions and based on them algorithms for problems solving is shown. It is demonstrated that a 3D-scan of 4D-pyramid’s lateral surface, constructed with 2D drawing, can be imported into CAD as a 3D-model. The deploying of the 4D-cone’s lateral surface is considered. The resulting scan’s surface 3D-model imported into CAD is shown. Cases are indicated when a multidimensional space’s object 3D-model may be more visual than a flat one. As an example, 2D-models for imaginary continuations of lines and circles of the complex plane (simulated by Euclidean 4D-space) are presented. Two 3D-projections for imaginary continuations of a circle with a real radius as 3D-space surfaces are shown. It is noted that in order to combine in an educational course the multidimensional space’s objects modeling and work in CAD the tasks on designing of complex technical surfaces by means of output in multidimensional space are suitable. A brief review of sources is given, in which theoretical foundations and the use of key geometrical methods for surfaces construction are considered; an example of a surface constructed by a progressive key method and imported into CAD is shown. The concept of a product’s electronic model (PEM) is described, in which the modeled object’s 3D-simulator as its visual representation is combined with numerous 2D-layers, which elements automatically perform geometrical and graphical calculations in spaces of any dimensions, and control 3D-model’s dimensions and shape through constructive and parametric links. Conclusions are drawn about the possibility of visual multidimensional modeling in the educational process, the advantages of using a complex drawing for solving of problems with multidimensional objects, the need to use special systems of constructive geometric modeling that automate repetitive sequences of constructions. It is also concluded that multidimensional objects’ 2D-models can and should be directly involved in the PEM formation.
{"title":"About Building of Models for Objects in Space of Four and More Dimensions in Educational Process","authors":"Алексей Бойков, A. Boykov","doi":"10.12737/ARTICLE_5C21F96DCE5DE8.36096061","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21F96DCE5DE8.36096061","url":null,"abstract":"In this paper the visibility concept in the context of modeling of multidimensional spaces’ objects is clarified. It is concluded that such model’s visibility should be defined as unambiguity and completeness of information presented in this model and consistent with the student’s experience in the area of modeling a space of higher dimension (3D) by elements of spaces of lower dimension (2D). Such possibilities are presented by the generalized complex drawing. Examples for objects 4D-modeling using two 3D or three 2D flat projections are presented, some properties of the 4D generalized drawing are listed. The solution of problems with 4D-objects is considered on the example of 4D-pyramid section construction, and deploying its lateral surface. It is shown that to simplify the solution of these problems is required a system allowing automatically perform repetitive sequences of constructions. A list of elementary constructions is presented, and a method for recording of composite constructions and based on them algorithms for problems solving is shown. It is demonstrated that a 3D-scan of 4D-pyramid’s lateral surface, constructed with 2D drawing, can be imported into CAD as a 3D-model. The deploying of the 4D-cone’s lateral surface is considered. The resulting scan’s surface 3D-model imported into CAD is shown. Cases are indicated when a multidimensional space’s object 3D-model may be more visual than a flat one. As an example, 2D-models for imaginary continuations of lines and circles of the complex plane (simulated by Euclidean 4D-space) are presented. Two 3D-projections for imaginary continuations of a circle with a real radius as 3D-space surfaces are shown. It is noted that in order to combine in an educational course the multidimensional space’s objects modeling and work in CAD the tasks on designing of complex technical surfaces by means of output in multidimensional space are suitable. A brief review of sources is given, in which theoretical foundations and the use of key geometrical methods for surfaces construction are considered; an example of a surface constructed by a progressive key method and imported into CAD is shown. The concept of a product’s electronic model (PEM) is described, in which the modeled object’s 3D-simulator as its visual representation is combined with numerous 2D-layers, which elements automatically perform geometrical and graphical calculations in spaces of any dimensions, and control 3D-model’s dimensions and shape through constructive and parametric links. Conclusions are drawn about the possibility of visual multidimensional modeling in the educational process, the advantages of using a complex drawing for solving of problems with multidimensional objects, the need to use special systems of constructive geometric modeling that automate repetitive sequences of constructions. It is also concluded that multidimensional objects’ 2D-models can and should be directly involved in the PEM formation.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87474421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21F6E832B4D2.25216268
И. Беглов, I. Beglov, Вячеслав Рустамян, V. Rustamyan, И. Антонова, I. Antonova
Previously, the method of rotating of flat geometric objects around curvilinear axes was described by us. The next step in the path of our research should be the development of methods for the automated creation of surfaces digital models obtained by the described rotation method. We have created models of surfaces, the axis and the forming curve of which are circles lying in the same plane. Several cases of mutual disposition for such circles were analyzed. Modeling was carried out using constructive techniques. Surfaces were created using the “surface by section” operation. The centers of such circular sections belong to the axis of rotation, if it is a circle. Using the special tools incorporated in the KOMPAS-3D program, we have cut the surfaces modeled in this way by planes, and obtained a number of flat sections. Taking into account the difficulties occurring during the study of such complex geometric objects by means of flat graphic constructions, as well as graphic computer modeling, we have realized the need to create a mathematical apparatus describing these objects’ shape. The required mechanism should be applicable to any pair of second-order curves interconnected as “axis — generatix”. We have considered an elementary example – the rotation of a point around a curve elliptical axis. In this paper a solution for the problem of finding a system of equations describing a set of point positions, which it will successively take when rotating around the elliptic axis, is presented. It is possible to apply a similar mathematical apparatus to axes having the form of other quadrics, for example, hyperbolas or parabolas, as well as to generatices consisting of more than one point, that is, to forming curves.
{"title":"Mathematical Interpretation for a Method of Rotation of a Point Around a Second Order Curved Axis","authors":"И. Беглов, I. Beglov, Вячеслав Рустамян, V. Rustamyan, И. Антонова, I. Antonova","doi":"10.12737/ARTICLE_5C21F6E832B4D2.25216268","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21F6E832B4D2.25216268","url":null,"abstract":"Previously, the method of rotating of flat geometric objects around curvilinear axes was described by us. The next step in the path of our research should be the development of methods for the automated creation of surfaces digital models obtained by the described rotation method. We have created models of surfaces, the axis and the forming curve of which are circles lying in the same plane. Several cases of mutual disposition for such circles were analyzed. Modeling was carried out using constructive techniques. Surfaces were created using the “surface by section” operation. The centers of such circular sections belong to the axis of rotation, if it is a circle. Using the special tools incorporated in the KOMPAS-3D program, we have cut the surfaces modeled in this way by planes, and obtained a number of flat sections. Taking into account the difficulties occurring during the study of such complex geometric objects by means of flat graphic constructions, as well as graphic computer modeling, we have realized the need to create a mathematical apparatus describing these objects’ shape. The required mechanism should be applicable to any pair of second-order curves interconnected as “axis — generatix”. We have considered an elementary example – the rotation of a point around a curve elliptical axis. In this paper a solution for the problem of finding a system of equations describing a set of point positions, which it will successively take when rotating around the elliptic axis, is presented. It is possible to apply a similar mathematical apparatus to axes having the form of other quadrics, for example, hyperbolas or parabolas, as well as to generatices consisting of more than one point, that is, to forming curves.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82473571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21F593838774.44754853
Тагир Пшуков, Tagir Pshukov, Мурат Османович Мамчуев, M. Mamchuev
It is known that squaring the circle (the problem consisting in construction of a square with the same area as a given circle), together with duplication of cube and angle trisection, is one of the most famous unsolv able problems of constructive geometry for construction with compass and straightedge. The solution of squaring the circle problem is reduced to the straightening of the circle, that is, to the construction of a segment equal in length to the circle, and its insolvability is connected with the pi character transcendence. In this paper, the limiting case of one of Christian Huygens theorems, which establishes an estimate for length of circumference of a circle through perimeters of regular polygons inscribed in circle and circumscribed about it, is proved. On this basis has been proposed and justified an approximate method for squaring the circle problem solving, which allows consistently construct arbitrarily exact solutions of the problem. We will approximate an arc of a circle whose radius is a multiple of the given circle’s radius, with the help of a segment which is parallel to a shrinking it chord, and then will increase or decrease this segment in the required number of times, so that the resulting segment’s length would be approximately equal to half of the given circle’s circumference. The approximation accuracy will be the higher the smaller arc of the circle we will consider. But possibilities of real tools are limited, and not suitable for both too small and too large drawing scales. In order to cope with this problem, an algorithm for scaled approximation has been proposed, in which it is sufficient to increase (or reduce) the drawing fragment, so that all the time sta y within the sheet of the same size. Perhaps this approach will be useful for other constructions, including the exact ones, where it is necessary to come to very large or vice versa very small drawings’ dimensions.
{"title":"Approximate Solution for Squaring the Circle Problem","authors":"Тагир Пшуков, Tagir Pshukov, Мурат Османович Мамчуев, M. Mamchuev","doi":"10.12737/ARTICLE_5C21F593838774.44754853","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21F593838774.44754853","url":null,"abstract":"It is known that squaring the circle (the problem consisting in construction of a square with the same area as a given circle), together with duplication of cube and angle trisection, is one of the most famous unsolv able problems of constructive geometry for construction with compass and straightedge. The solution of squaring the circle problem is reduced to the straightening of the circle, that is, to the construction of a segment equal in length to the circle, and its insolvability is connected with the pi character transcendence. In this paper, the limiting case of one of Christian Huygens theorems, which establishes an estimate for length of circumference of a circle through perimeters of regular polygons inscribed in circle and circumscribed about it, is proved. On this basis has been proposed and justified an approximate method for squaring the circle problem solving, which allows consistently construct arbitrarily exact solutions of the problem. We will approximate an arc of a circle whose radius is a multiple of the given circle’s radius, with the help of a segment which is parallel to a shrinking it chord, and then will increase or decrease this segment in the required number of times, so that the resulting segment’s length would be approximately equal to half of the given circle’s circumference. The approximation accuracy will be the higher the smaller arc of the circle we will consider. But possibilities of real tools are limited, and not suitable for both too small and too large drawing scales. In order to cope with this problem, an algorithm for scaled approximation has been proposed, in which it is sufficient to increase (or reduce) the drawing fragment, so that all the time sta y within the sheet of the same size. Perhaps this approach will be useful for other constructions, including the exact ones, where it is necessary to come to very large or vice versa very small drawings’ dimensions.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85343681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21FBA3F26C35.85693389
О. Ю. Филимонова, O. Filimonova
In this paper features for creation of educational process in a military higher education institution when studying “Engineering and Computer Graphics” discipline are revealed. Military education is a part of the Russian Federation’s education system. In conditions of the Armed Forces modernization and development of new methods and ways for conduct of operations the young officers’ perfection acquires a big significance. Requirements applicable to military specialists reflect the concept of educational activity in general – possession of strong theoretical knowledge and formed practical skills at the tasks solution. The big part in the system of development for military engineering education is assigned to practical orientation of training. Future officer has to understand the processes for design, production and operation of cars and mechanisms with varying complexity, therefore be able to work with design documentation of any kind. In the course of “Engineering and Computer Graphics” discipline studying cadets are learned to read and carry out drawings, to develop their technical support, and also to design and model both two, and three-dimensional objects on a plane and in space. The efficiency of graphic training in a greater degree depends on educational activity’s organization. Application of education traditional forms in combination with innovative practice and methods, development of the system of didactic tools focused on increase in educational process’s intensity is the most optimal one for achievement of training maximum results. During realization of the tasks set by the state for training of competent military specialists, the educational process based on principles of personally focused training with developing orientation has been organized by “Engineering and Computer Graphics” discipline teachers of Military Academy of Troops Air Defense of Russian Federation Armed Forces. The developed system of didactic tools enhances the intensity and productivity of cadets’ educational activity, helps to cultivate professional qualities of future military specialists.
{"title":"“Engineering and Computer Graphics” Discipline in the System of Higher Military Education","authors":"О. Ю. Филимонова, O. Filimonova","doi":"10.12737/ARTICLE_5C21FBA3F26C35.85693389","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21FBA3F26C35.85693389","url":null,"abstract":"In this paper features for creation of educational process in a military higher education institution when studying “Engineering and Computer Graphics” discipline are revealed. Military education is a part of the Russian Federation’s education system. In conditions of the Armed Forces modernization and development of new methods and ways for conduct of operations the young officers’ perfection acquires a big significance. Requirements applicable to military specialists reflect the concept of educational activity in general – possession of strong theoretical knowledge and formed practical skills at the tasks solution. The big part in the system of development for military engineering education is assigned to practical orientation of training. Future officer has to understand the processes for design, production and operation of cars and mechanisms with varying complexity, therefore be able to work with design documentation of any kind. In the course of “Engineering and Computer Graphics” discipline studying cadets are learned to read and carry out drawings, to develop their technical support, and also to design and model both two, and three-dimensional objects on a plane and in space. The efficiency of graphic training in a greater degree depends on educational activity’s organization. Application of education traditional forms in combination with innovative practice and methods, development of the system of didactic tools focused on increase in educational process’s intensity is the most optimal one for achievement of training maximum results. During realization of the tasks set by the state for training of competent military specialists, the educational process based on principles of personally focused training with developing orientation has been organized by “Engineering and Computer Graphics” discipline teachers of Military Academy of Troops Air Defense of Russian Federation Armed Forces. The developed system of didactic tools enhances the intensity and productivity of cadets’ educational activity, helps to cultivate professional qualities of future military specialists.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90673848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-29DOI: 10.12737/ARTICLE_5C21F4A06DBB74.56415078
Николай Сальков, N. Sal'kov
Probably, it is impossible to find such industry where the ruled surfaces would not be used. They are used in agriculture, in the heavy and light industries, in construction, in aircraft manufacturing, and in military art. Ruled surfaces are used in the design of wings, tail and partially fuselage of aircraft, car bodies, in the project engineering of slopes and embankments of auto-roads, abutments of bridge supports, transitions from a vertical quay to inclined walls of embankments, various hydraulic structures, towers, masts, cooling towers, vaults and arches, overlaps of pavilions, circuses, stadiums and other building structures, as well as in the calculation of solar exposure. This paper deals with the formation of ruled surfaces in a single method of their definition. A number of examples for definition of ruled surfaces have been presented. These examples show that in general for definition of ruled surfaces it is required to have three guides and three geometric conditions characterizing the position of a rectilinear generator with respect to each of the guides. Both surfaces and lines can act as guides. The plane is selected separately from other surfaces. The geometric conditions are the intersection with the guide line and the tangent or intersection at a certain sharp angle with the guide surface. The table of 19 variants for guides has been given. An attempt to classify surfaces does not even consider in this paper since it is impossible to classify ruled surfaces, even within its class, due to the lack of a criterion showing their belonging to one or another species. It can be concluded that the classification of surfaces may be used only for educational purposes and in cases where the surface name is obvious.
{"title":"General Principles for Formation of Ruled Surfaces. Part 1","authors":"Николай Сальков, N. Sal'kov","doi":"10.12737/ARTICLE_5C21F4A06DBB74.56415078","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C21F4A06DBB74.56415078","url":null,"abstract":"Probably, it is impossible to find such industry where the ruled surfaces would not be used. They are used in agriculture, in the heavy and light industries, in construction, in aircraft manufacturing, and in military art. Ruled surfaces are used in the design of wings, tail and partially fuselage of aircraft, car bodies, in the project engineering of slopes and embankments of auto-roads, abutments of bridge supports, transitions from a vertical quay to inclined walls of embankments, various hydraulic structures, towers, masts, cooling towers, vaults and arches, overlaps of pavilions, circuses, stadiums and other building structures, as well as in the calculation of solar exposure. This paper deals with the formation of ruled surfaces in a single method of their definition. A number of examples for definition of ruled surfaces have been presented. These examples show that in general for definition of ruled surfaces it is required to have three guides and three geometric conditions characterizing the position of a rectilinear generator with respect to each of the guides. Both surfaces and lines can act as guides. The plane is selected separately from other surfaces. The geometric conditions are the intersection with the guide line and the tangent or intersection at a certain sharp angle with the guide surface. The table of 19 variants for guides has been given. An attempt to classify surfaces does not even consider in this paper since it is impossible to classify ruled surfaces, even within its class, due to the lack of a criterion showing their belonging to one or another species. It can be concluded that the classification of surfaces may be used only for educational purposes and in cases where the surface name is obvious.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89320523","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_5BC45ADD9A2B21.45929543
Алексей Бойков, A. Boykov, А.Ю. Сидоров, A. Sidorov, А. Федотов, A. Fedotov
In this paper the urgent problem of the formal approach to the teaching of descriptive geometry (DG) has been formulated. The authors consider the algorithm concept and approaches to formal description of methods (algorithms) for tasks solving. It is emphasized that the known methods for creating and presenting of algorithms for DG tasks solving do not reflect all possibilities of algorithmization as it is. In the third section the authors, in examples, emphasize the complexity of DG tasks solutions algorithmization. The diversity of solutions for one or another DG task is noted depending on location of initial figures that requires a suitable context analysis in solving, and, as a consequence, the algorithm choice. It is pointed out that the reason for this is different ways for expressing of figures’ geometric properties by means of drawing. General algorithms for applying the method of loci and geometric transformations to tasks solving are considered. From the loci position have been considered two basic tasks of DG: plotting a point drawing in the coordinates, and a perpendicular to the plane. The method of loci importance is emphasized in view of algorithms compilation simplicity and wide possibilities for tasks solving. The authors note that algorithmization does not reduce the importance of geometry knowledge or understanding of the tasks geometric content and used methods, but emphasizes the importance of the first stage for tasks solving — the stage of analysis at which basic decisions are made and its method is chosen. In conclusion it is emphasized that in the practice related to solving of DG educational tasks it is optimal to apply the algorithmization in point, as it enables to structure the course, operate with compact algorithms, and introduce automated technologies of constructive geometric modeling.
{"title":"On the Procedure For Algorithms Using In Solving Descriptive Geometry Tasks","authors":"Алексей Бойков, A. Boykov, А.Ю. Сидоров, A. Sidorov, А. Федотов, A. Fedotov","doi":"10.12737/ARTICLE_5BC45ADD9A2B21.45929543","DOIUrl":"https://doi.org/10.12737/ARTICLE_5BC45ADD9A2B21.45929543","url":null,"abstract":"In this paper the urgent problem of the formal approach to the teaching of descriptive geometry (DG) has been formulated. The authors consider the algorithm concept and approaches to formal description of methods (algorithms) for tasks solving. It is emphasized that the known methods for creating and presenting of algorithms for DG tasks solving do not reflect all possibilities of algorithmization as it is. In the third section the authors, in examples, emphasize the complexity of DG tasks solutions algorithmization. The diversity of solutions for one or another DG task is noted depending on location of initial figures that requires a suitable context analysis in solving, and, as a consequence, the algorithm choice. It is pointed out that the reason for this is different ways for expressing of figures’ geometric properties by means of drawing. General algorithms for applying the method of loci and geometric transformations to tasks solving are considered. From the loci position have been considered two basic tasks of DG: plotting a point drawing in the coordinates, and a perpendicular to the plane. The method of loci importance is emphasized in view of algorithms compilation simplicity and wide possibilities for tasks solving. The authors note that algorithmization does not reduce the importance of geometry knowledge or understanding of the tasks geometric content and used methods, but emphasizes the importance of the first stage for tasks solving — the stage of analysis at which basic decisions are made and its method is chosen. In conclusion it is emphasized that in the practice related to solving of DG educational tasks it is optimal to apply the algorithmization in point, as it enables to structure the course, operate with compact algorithms, and introduce automated technologies of constructive geometric modeling.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73183023","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_5BC4563CCF6884.11983902
Евгения Павловна Денисова, E. Denisova, Тимур Хуснетдинов, Timur Husnetdinov, Марианна Воронина, M. Voronina
This paper’s purpose is investigation of non-traditional projection systems and their projecting surfaces, the choice of such congruence parameters for conical helical lines, which allow cover the whole complex of requirements to the surface, obtained by projecting of an arbitrary flat or spatial line with congruence beams, as well as the use of computer graphics in surface visualization. In the paper has been presented an example of analytical interpretation for an image of curvilinear projection by conical helical lines with constant pitch, and a congruence example for conical helical lines located on coaxial cones with a common vertex and a variable angle of generatrix inclination to an axis. Have been investigated properties and defined parameters of the congruence helical line passing through a space arbitrary point which is not belonging to an axis. An approach for construction of spiral surfaces, which frame consists of beams projecting an arbitrary line. A form generation of surfaces by analytical methods and their visualization by means of computer graphics is one of applied geometry’s urgent problems in connection with the use of such methods in automated systems for scientific research, design, and manufacture on equipment with computer numerical control. The leading research method for this problem is the general analytical theory for surfaces’ applied form generation developed by Professor I.A. Skidan and formed a unique apparatus, based on mathematical support of computing technologies for design and creation of objects with complex forms. On examples of visualization for projecting surfaces by means of computer graphics it is possible to show applicability of analytical models in computer technologies for scientific researches, design and manufacturing.
{"title":"Projection by Conical Helical Lines With Constant Pitch","authors":"Евгения Павловна Денисова, E. Denisova, Тимур Хуснетдинов, Timur Husnetdinov, Марианна Воронина, M. Voronina","doi":"10.12737/ARTICLE_5BC4563CCF6884.11983902","DOIUrl":"https://doi.org/10.12737/ARTICLE_5BC4563CCF6884.11983902","url":null,"abstract":"This paper’s purpose is investigation of non-traditional projection systems and their projecting surfaces, the choice of such congruence parameters for conical helical lines, which allow cover the whole complex of requirements to the surface, obtained by projecting of an arbitrary flat or spatial line with congruence beams, as well as the use of computer graphics in surface visualization. In the paper has been presented an example of analytical interpretation for an image of curvilinear projection by conical helical lines with constant pitch, and a congruence example for conical helical lines located on coaxial cones with a common vertex and a variable angle of generatrix inclination to an axis. Have been investigated properties and defined parameters of the congruence helical line passing through a space arbitrary point which is not belonging to an axis. An approach for construction of spiral surfaces, which frame consists of beams projecting an arbitrary line. A form generation of surfaces by analytical methods and their visualization by means of computer graphics is one of applied geometry’s urgent problems in connection with the use of such methods in automated systems for scientific research, design, and manufacture on equipment with computer numerical control. The leading research method for this problem is the general analytical theory for surfaces’ applied form generation developed by Professor I.A. Skidan and formed a unique apparatus, based on mathematical support of computing technologies for design and creation of objects with complex forms. On examples of visualization for projecting surfaces by means of computer graphics it is possible to show applicability of analytical models in computer technologies for scientific researches, design and manufacturing.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"245 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73590278","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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