Geometry & Graphics最新文献

{"title":"Golden Section and Golden Rectangles When Building Icosahedron, Dodecahedron and Archimedean Solids Based On Them","authors":"В. Васильева, V. Vasil’eva","doi":"10.12737/ARTICLE_5D2C1CEB9F91B1.21353054","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C1CEB9F91B1.21353054","url":null,"abstract":"A brief history of the development of the regular polyhedrons theory is given. The work introduces the reader to modelling of the two most complex regular polyhedrons – Platonic solids: icosahedron and dodecahedron, in AutoCAD package. It is suggested to apply the method of the icosahedron and dodecahedron building using rectangles with their sides’ ratio like in the golden section, having taken the icosahedron’s golden rectangles as a basis. This method is well-known-of and is used for icosahedron, but is extremely rarely applied to dodecahedron, as in the available literature it is suggested to build the latter one as a figure dual to icosahedron. The work provides information on the first mentioning of this building method by an Italian mathematician L. Pacioli in his Divine Proportion book. In 1937, a Soviet mathematician D.I. Perepelkin published a paper On One Building Case of the Regular Icosahedron and Regular Dodecahedron, where he noted that this “method is not very well known of” and provided a building based “solely on dividing an intercept in the golden section ratio”. Taking into account the simplicity and good visualization of the building based on golden rectangles, a computer modeling of icosahedron and dodecahedron inscribed in a regular hexahedron is performed in the article. Given that, if we think in terms of the golden section concepts, the bigger side of the rectangle equals a whole intercept – side of the regular hexahedron, and the smaller sides of the icosahedron and dodecahedron rectangles are calculated as parts of the golden section ratio (of the bigger part and the smaller one, respectively). It is demonstrated how, using the scheme of a wireframe image of the dual connection of these polyhedrons as a basis, to calculate the sides of the rectangles in the golden section ratio in order to build an “infinite” cascade of these dual figures, as well as to build the icosahedron and dodecahedron circumscribed about the regular hexahedron. The method based on using the golden-section rectangles is also applied to building semiregular polyhedrons – Archimedean solids: a truncated icosahedron, truncated dodecahedron, icosidodecahedron, rhombicosidodecahedron, and rhombitruncated icosidodecahedron, which are based on icosahedron and dodecahedron.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73638651","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}

{"title":"Modeling Approximating the 16-Point Compartment the Response Surface With Respect To the Solution of the Inhomogeneous Heat Equation","authors":"Евгений Конопацкий, E. Konopatskiy","doi":"10.12737/ARTICLE_5D2C1A551A22C5.12136357","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C1A551A22C5.12136357","url":null,"abstract":"The paper proposes a computational method for solving differential equations of mathematical physics by approximating the desired solution using geometric objects of multidimensional space passing through predetermined points. The essence of the method is to simulate an approximating geometric object of a multidimensional affine space constructed on a regular multidimensional network of points. In this case, the response function values satisfying the solution of the original differential equation are calculated at the nodal points of the network. Modeling of approximating geometric object is carried out by means the arcs of algebraic curves passing through predetermined points. It should be noted that taking into account the boundary conditions does not require changes in the geometric algorithm or point equations. It is sufficient to use the necessary coordinates of the nodal boundary points corresponding to the boundary conditions of the solution of the differential equation. To achieve the required accuracy of the solution of differential equations, it is sufficient to compact the reference network of points. Under such conditions, it is possible to use as a single geometric object to approximate the solution of the differential equation, and composite, based on the simulation of multidimensional contours on a regular network of points of multidimensional space. A geometric classification of differential equations depending on the number of parameters determining the approximating geometric object in multidimensional space is proposed. An example of solving the inhomogeneous heat equation by means of an approximating response surface passing through 16 predetermined points is given. In this case, the required approximating compartment of the response surface passes through 3 straight lines that correspond to the boundary conditions and satisfies the solution of the original differential equation at the nodal points of the 16-point network. A comparison of the results of solving the inhomogeneous heat equation approximated by a 16-point compartment of the response surface with the reference compartment of the surface obtained by the method of separating variables is also presented.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74416160","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}

{"title":"Analytical Dependences of the Kinematic Forming Primary Surfaces of the Worm Gear","authors":"С. Рязанов, S. Ryazanov, Михаил Решетников, M. Reshetnikov","doi":"10.12737/ARTICLE_5D2C2DDA42FDA7.79858292","DOIUrl":"https://doi.org/10.12737/ARTICLE_5D2C2DDA42FDA7.79858292","url":null,"abstract":"The run-in method for obtaining the screw surface of a worm is based on the use of the worm gearing principle. In this case, the shaping surface (cutting tool) and the workpiece constitute a gear pair [4; 7]. The use of geometric modeling methods [8; 9] to simulate the process of shaping the working surface is based on the relative movement of intersecting objects in the form of a “workpiece-tool” system. This allows to obtain the necessary geometrical model that accurately reproduces the geometric configuration of the surfaces of the teeth of spatial gears [14; 15], where the producing surface of the tool moves in the selected reference system and its position at an arbitrary time is determined by a certain parameter, the motion parameter. The position of the cutting tool at the beginning and at the end of each pass is calculated using parametric equations, which make it possible to calculate the tool path for accurate processing of spatially complex surfaces [16–19]. In the process of mechanical action of a tool on a solid (workpiece), shaping occurs, which consists in the movement of the tool relative to the workpiece [30; 31]. The use of modern methods of three-dimensional computer graphics allows us to improve and accelerate the process of designing technological operations of tooth profiling, providing the final forms of the surfaces of the teeth in the form of visual and accurate computer-based solid-state models [39; 40]. The method is based on a virtual representation of the process of shaping in the form of intersection of solid-state 3D models of two objects (tools and workpieces), which generally perform a screw relative motion. As a result, the working surfaces of the teeth are formed as the envelopes of the tool producing surface [32–34]. For the formation of fission surfaces, mathematical dependences were obtained, which allow one to describe the mutual motion of a worm, a worm gear and a disk cutter [35–37]. These analytical dependences make it possible to simulate the virtual process of forming the side surfaces of the worm gearing elements [1–3; 5; 6]","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84908235","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}

{"title":"Training Procedure in Graphic Disciplines for Students of Technical High Educational Institutions","authors":"Марина Федосеева, M. Fedoseeva","doi":"10.12737/ARTICLE_5C91FED8650BB7.79232969","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C91FED8650BB7.79232969","url":null,"abstract":"In connection with transition of the higher education system to the bachelor's degree, a number of difficulties has arisen e in the student’s educational schedule. The question is especially acute for disciplines of general education, and a rather large reduction in classroom time has happened. Taking into account the rather low level of students training in the area of drawing, and sometimes its complete absence, due to the abolition of the subject of drawing from the mandatory school program, teachers are faced with more and more tasks on the formation of educational and methodical complexes [3; 6; 15–17; 23]. In this paper has been considered the issue related to complex training in the area of graphic disciplines for students of technical high educational institutions. These disciplines, as is known, are the basis of many special engineering disciplines, such as machine parts, theory of mechanisms and machines and so on. The fundamental component is the design documentation, the possession of which is necessary for a future engineer. Studying the general course of descriptive geometry and engineering graphics is not enough, training should be carried out taking into account the professional orientation and relevant competencies. However, despite the global automation in all areas, it is not necessary to completely abandon the traditional methods of training, for example, in descriptive geometry’s section. This course allows develop students' spatial reasoning [12–14; 19]. Work on descriptive geometry’s tasks gives for a student the opportunity to more clearly understand the projection principles, methods of drawings transformation, formation of complex surfaces, obtaining of visual images by constructing of axonometric projections or performing of a technical drawing. Another question in our opinion is the order for studying the section of engineering graphics, which is more appropriate to study using modern graphics programs [10; 11].","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"64 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87107533","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}

{"title":"General Principles for Formation of Ruled Surfaces. Part 2","authors":"Николай Сальков, N. Sal'kov","doi":"10.12737/ARTICLE_5C9201EB1C5F06.47425839","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C9201EB1C5F06.47425839","url":null,"abstract":"In this paper the consideration related to formation of ruled surfaces with a single method for their set up that was proposed in the first part of the work, is continued. In the first part of the work were considered more than twenty variants for set up of ruled surfaces, including four set ups by guide lines, overall known in the literature, for example, in the books of S.A. Frolov, A.V. Bubennikov, M.Ya. Gromov. However, the set up of ruled surfaces with the help of guide lines was proposed in a new aspect – as a constituent of a single method for setting absolutely all ruled surfaces, taking place in science and industry, and with additional variants. Besides, have been proposed equation systems, which solution leads to generation of equation for the given ruled surface. New set ups of ruled surfaces have included eighteen examples, which is the main material of the work’s first part. Also was proposed a table in which have been put all possible variants for set up of geometric figures’ guiding lines to obtain ruled surfaces. Of course, the proposed variants of guiding lines’ combination were presented in the enlarged form. In the proposed paper have been considered new, not presented before, variants for set up of ruled surfaces. Have been presented 19 examples, including the ones with one or two guiding planes, as well as when the guiding line belongs to one of guiding surfaces. Such surfaces can be considered as ruled surfaces of smooth transition. As in the first part of the work, the equation systems leading to the equation of the set ruled surface are proposed.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85177904","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}

{"title":"Visualization of Regular Polyhedrons during Their Formation","authors":"В.В. Романова, Viktoriia Borysivna Romanova","doi":"10.12737/ARTICLE_5C91FFD0916D52.90296375","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C91FFD0916D52.90296375","url":null,"abstract":"In this work the automated formation of surfaces correct to convex polyhedrons of Platon and two regular not convex star-shaped polyhedrons of Kepler-Poinsot by the kinematic method. Researches on realization of a goal were carried out in the environment of AutoCAD with use of the programs developed in the functional Autolisp programming language which is built in AutoCAD. The AutoLisp language and the AutoCAD environment are chosen for achievement of a goal as they allow showing bodies in the movement. The technique of formation of electronic models of the polyhedrons necessary for performance of visualization of polyhedrons is stated. The model is a set of compartments of a surface, issued in the form of the block. The user function in the AutoLisp language which identifier is team in the environment of AutoCAD is developed for each model. Each compartment was placed in the drawing layer which is taken away for it. When developing the user functions were taken into account to a possibility of the AutoCAD environment – the available teams for formation of surfaces. The user functions in the AutoLisp language for formation of the studied surfaces in the environment of AutoCAD are made by the defrosting method of the block containing surface compartments. In the course of \"defrosting\" of layers with compartments on the screen of the monitor process of formation of a surface is shown – drawings of compartments of a surface appear one by one. The last drawing is an image of a surface. The user functions in the AutoLisp language for formation of the studied surfaces in the environment of AutoCAD are made. The fragment of the program by training of one side of a tetrahedron is given Drawings of elements of surfaces of all regular polyhedrons of Platon and star-shaped polyhedrons of Kepler-Poinsot are provided in initial situation and in the course of stage-by-stage formation of these surfaces, the programs received in the environment of AutoCAD with use in the AutoLisp language. Drawings of elements of surfaces of all regular polyhedrons of Platon and star-shaped polyhedrons of Kepler-Poinsot are provided in initial situation and in the course of stage-by-stage formation of these surfaces, the programs received in the environment of AutoCAD with use in the AutoLisp language. The possibility of formation of surfaces of regular polyhedrons is shown by a kinematic method: the movement rectilinear forming on the directing lines as which edges of polyhedrons are used.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89391889","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}

{"title":"Reflection from Curved Mirrors in a Plane","authors":"Л. Жихарев, L. Zhikharev","doi":"10.12737/ARTICLE_5C9203ADB22641.01479568","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C9203ADB22641.01479568","url":null,"abstract":"Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79678464","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}

{"title":"Geometric Model for Generation of Contour- Parallel Lines’ Family for Cutting Tool’s Path Automated Computation","authors":"К. Панчук, K. Panchuk, Т. Мясоедова, T. Myasoedova, И В Крысова, I. Krysova","doi":"10.12737/ARTICLE_5C92012C51BBA1.17153893","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C92012C51BBA1.17153893","url":null,"abstract":"In this paper has been proposed a geometric model for forming problem of contour-parallel lines (equidistant lines) for a flat contour with an island, and has been obtained the problem’s analytical solution, which is relevant for computer-aided design of cutting tools processing pocket surfaces on CNC machines. The proposed geometric model is based on cyclograph mapping of space on a plane. Beyond the analytical solution the geometric model differs from the known algebraic models and their solutions for considered forming problem also by the fact that it allows obtain a more complete and evident representation on the relationship and interaction for all its geometric components at the stages of 3D computer visualization. A 3D geometric model based on a cyclograph mapping of space has been proposed for obtaining the families of equidistant lines for connected and multiply connected regions with closed contours taken as a basis for pocket surfaces modeling. An algorithm for the analytical solution of the problem related to equidistant families generation is getting from the geometric model. All stages of the analytical solution are accompanied by a figurative representation of geometric objects and their relations in the geometric model’s virtual electronic space. The proposed in this paper algorithm for the case of a doubly connected polygonal region can be used as a basis for generation of equidistant families for multiply connected polygonal regions. The presence of the analytical solution for the problem related to equidistant families generation simplifies greatly the automated calculation of the tool path and preparation of control programs for pocket surfaces manufacturing on CNC machines. Have been presented an example and algorithm providing support for working capacity of the proposed geometric model for considered forming problem.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"416 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75760071","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}

{"title":"Discrete Geometric Models for Estimating the Degree of Shading in Solar Energy","authors":"Л. Маркин, L. Markin","doi":"10.12737/ARTICLE_5C9202D8D821B0.81468033","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C9202D8D821B0.81468033","url":null,"abstract":"Geometric simulation and its software for estimating the efficiency of deployment of solar panels on spacecraft and solar concentrators on the ground are considered in this work. Both the physical and mathematical set up of the problem for estimating the energy efficiency of solar panels, taking into account their shading both by each other and by other elements of a space station has been described in this paper. It has been shown that the known methods for mechanization and automation of such calculations are focused on objects of relatively simple geometric shapes (such as buildings), and are inefficient for objects of complex and diverse geometric shape, characteristic both for spacecraft themselves and their solar panels. Therefore, to solve this problem, a receptor (voxel) geometric model digitizing the computational space has been chosen. The receptor model’s uniqueness is that comparing the values of receptor codes allows easy determine the intersection of objects. Has been described a developed receptor geometric model for estimating the effective area of solar panels, taking into account their shading when the object (for example, a spacecraft) is illuminated by a flow of solar energy from a given direction. The essential difference between the developed receptor geometric model and the classical one is that the former is multiform, i.e. uses not the 2-digit code (0 and 1), but the 4-digit one (0, 1, 2 and 3). Has been demonstrated a software implementation of the described geometric model in C#, and a graphical shell developed for this problem, allowing see the obtained results’ numerical values. Have been provided examples of its implementation in solving of practical problems. The results of verification for the described receptor geometric model have been demonstrated. All this allows speak about efficiency of using receptor geometric models both in singular computation calculations and for creating the appropriate algorithmic, mathematical support and software for the corresponding CAD systems.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75040368","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}

{"title":"Modern Approaches to Products Design in the Process of Students Teaching in Computer Graphics","authors":"Татьяна Усатая, T. Usataya, Любовь Дерябина, L. Deryabina, Елена Решетникова, E. Reshetnikova","doi":"10.12737/ARTICLE_5C91FD2BDE0FF7.07282102","DOIUrl":"https://doi.org/10.12737/ARTICLE_5C91FD2BDE0FF7.07282102","url":null,"abstract":"Using the computer graphics tools in the design allows improve the design quality and speed, as well as provide the qualitative front end engineering design. In this paper the problem related to improvement of quality in engineering training for students of technical high educational institutions, that today is considered as one of the main tasks for the system of higher professional education. A method proposed by authors allows solve this problem in the frame of students training in disciplines of graphic cycle, and directed to introducing the computer technologies in the teaching process of students. This method provides the development of students’ professional skills in the area of products and electronic devices design, and in the front end engineering design. In such a case, design is regarded from the standpoint of project-process approach. Project-process approach is a combination of interrelated projects implemented in the frame of process. A process is considered as a group of projects aimed at achievement of a planned result – a design object model. Design objects models in the area of mechanical engineering, electronics and electric power engineering, are presented as drawings, schemes and 3D models. That is why the emphasis is upon models and drawings building by means of CAD software (Autodesk AutoCAD, Autodesk Inventor, Kompas-Grafik, Kompas 3D). A high level of students’ competences can be achieved by modernization the educational content so that from their first teaching year the students could see the relation of learned graphical disciplines with their future professional occupation and prospects of production development and project activities. State-oftheart software for devices design provides for students an opportunity to extent their possibilities in the learning of graphical disciplines’ courses. The project-process approach is necessary for identifying and studying the relationship between the design as a process and the reformative human activity in general.","PeriodicalId":12604,"journal":{"name":"Geometry & Graphics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87732071","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}