scholarly journals The Algorithm for Crossing the N-dimensional Hyperquadric with N-1-dimensional Hyperspace

2021 ◽  
Author(s):  
Elena Lesnova ◽  
Denis Voloshinov
Keyword(s):  
Geometric Modeling ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Descriptive Geometry ◽  
Multidimensional Space ◽  
Geometric Constructions ◽  
Special Cases ◽  
Modeling Software ◽  
Automation Tools ◽  
Surface Curve

In descriptive geometry, the problem of finding a surface curve section with a plane is common. One such surface curve is a quadric. Due to the increased demand for tasks related to quadric, the synthetic modeling method becomes relevant. In recent years, geometric constructions of dimensions of more than three began to be studied more and more often. Multidimensional geometric shapes in multidimensional space are typically constructed using geometric modeling software. However, without additional building automation tools, software does not sufficiently facilitate human labor. The larger the dimension of the constructions, the more cumbersome and time consuming the drawing process becomes. The increasing complexity of constructions requires automation of constructions that can be traditimatized. Geometric constructions made using automation tools make us rethink the process of structural geometric modeling in descriptive geometry. Within the framework of the article, the algorithm for crossing the N-dimensional hyperquadric with N-1-dimensional hyperspace is presented. Special cases of this geometric construction are also considered: intersection of a three- dimensional quadric with a plane and intersection of a four-dimensional hyperquadric with a three-dimensional space. The implementation of the developed algorithm is carried out using the Simplex system and the built-in interpreter of the prolog logical programming language.

scholarly journals Integrating Urban Building Space and Transportation Space to Solve Traffic Congestion

2014 ◽  
Vol 3 (1) ◽  
pp. 7 ◽  
Author(s):  
Hong Zhang
Keyword(s):  
Urban Space ◽  
Traffic Congestion ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Urban Morphology ◽  
Multidimensional Space ◽  
Urban Sustainable Development ◽  
Urban Function ◽  
Space Separation ◽  
Three Dimensional Space

<p>Today, with urban function system increasingly complicated, there exist problems which are seriously hindering urban sustainable development in most cities such as traffic jams, constructive destruction, building space separation with traffic space, poor urban space resource utilization and so on. So the article makes a number of integration methods of urban building space and transportation space from the perspective of urban morphology integration. It tries to integrate urban environment with techniques of multidimensional space interludes, cascading, infiltration between building space and traffic space in three-dimensional space coordinates, to achieve the objectives   of proper division, solving traffic congestion problems and the establishment of a new dynamic three-dimensional transport system.</p>

Features of Distance Learning in Geometric and Graphic Disciplines Using Methods of Constructive Geometric Modeling

2021 ◽  
pp. 46-56
Author(s):  
E. Boyashova
Keyword(s):  
Distance Learning ◽  
Geometric Modeling ◽  
Information Technologies ◽  
Graphical Representation ◽  
Teaching Experience ◽  
Educational Process ◽  
Descriptive Geometry ◽  
Geometric Constructions ◽  
Professional Activities

The article is devoted to the peculiarities of teaching the discipline "Descriptive geometry" in the conditions of distance learning, it examines the application of information technologies in the educational process in geometric and graphic disciplines. Increasing the speed of information processes, reducing the number of hours for mastering the discipline. the conditions of distance learning set new tasks for teachers and dictate their requirements for teaching graphic disciplines and the use of teaching experience in a new reality; there is a need to introduce and develop new forms of education without losing the quality of education. Geometric-graphic disciplines occupy one of the important places in technical education, the complexity of the study of which lies in the development of a graphical representation of phenomena, objects and processes by methods of constructive geometric modeling. The knowledge and skills acquired by students contribute to the development of spatial, imaginative and rational thinking, which is necessary for future professional activities. Descriptive geometry is a discipline that is not easy to master on your own without a conscious understanding of the logic and sequence of geometric constructions, without deep knowledge of theoretical foundations and constant, repeated implementation of practical tasks. The acquisition of practical skills in mastering the methods of discipline has become more difficult in the current epidemiological situation. In modern conditions of distance learning, the use of the Simplex geometric modeling system made it possible to develop and propose a new concept of geometric-graphic interaction, which significantly reduced the time for completing and checking educational tasks in real time. The proposed technology reveals the deep informational essence of the studied discipline "Descriptive Geometry" and becomes a powerful research tool for students. The integration of traditional teaching methods in the graphic preparation of students with computer and communication facilities increases the possibilities of communication and improves the quality of teaching.

An Executable Notation, With Illustrations From Elementary Crystallography

1994 ◽  
Author(s):  
Donald B. Mclntyre
Keyword(s):  
Plate Tectonics ◽  
Periodic Structures ◽  
Dimensional Space ◽  
Reciprocal Lattice ◽  
Three Dimensional ◽  
Cubic Symmetry ◽  
Multiple Data ◽  
Special Cases ◽  
Unit Cells ◽  
Crystallographic Axes

Elementary crystallography is an ideal context for introducing students to mathematical geology. Students meet crystallography early because rocks are made of crystalline minerals. Moreover, morphological crystallography is largely the study of lines and planes in real three-dimensional space, and visualizing the relationships is excellent training for other aspects of geology; many algorithms learned in crystallography (e.g., rotation of arrays) apply also to structural geology and plate tectonics. Sets of lines and planes should be treated as entities, and crystallography is an ideal environment for introducing what Sylvester (1884) called "Universal Algebra or the Algebra of multiple quantity." In modern terminology, we need SIMD (Single Instruction, Multiple Data) or even MIMD. This approach, initiated by W.H. Bond in 1946, dispels the mysticism unnecessarily associated with Miller indices and the reciprocal lattice; edges and face-normals are vectors in the same space. The growth of mathematical notation has been haphazard, new symbols often being introduced before the full significance of the functions they represent had been understood (Cajori, 1951; Mclntyre, 1991b). Iverson introduced a consistent notation in 1960 (e.g., Iverson 1960, 1962, 1980). His language, greatly extended in the executable form called J (Iverson, 1993), is used here. For information on its availability as shareware, see the Appendix. Publications suitable as tutorials in , J are available (e.g., Iverson. 1991; Mclntyre, 1991 a, b; 1992a,b,c; 1993). Crystals are periodic structures consisting of unit cells (parallelepipeds) repeated by translation along axes parallel to the cell edges. These edges define the crystallographic axes. In a crystal of cubic symmetry they are orthogonal and equal in length (Cartesian). Those of a triclinic crystal, on the other hand, are unequal in length and not at right angles. The triclinic system is the general case; others are special cases. The formal description of a crystal gives prominent place to the lengths of the axes (a, b, and c) and the interaxial angles ( α, β, and γ). A canonical form groups these values into a 2 x 3 table (matrix), the first row being the lengths and the second the angles.

scholarly journals Integrating Urban Building Space and Transportation Space to Solve Traffic Congestion

2014 ◽  
Vol 3 ◽  
pp. 7
Author(s):  
Hong Zhang
Keyword(s):  
Urban Space ◽  
Traffic Congestion ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Urban Morphology ◽  
Multidimensional Space ◽  
Urban Sustainable Development ◽  
Urban Function ◽  
Space Separation ◽  
Three Dimensional Space

<p>Today, with urban function system increasingly complicated, there exist problems which are seriously hindering urban sustainable development in most cities such as traffic jams, constructive destruction, building space separation with traffic space, poor urban space resource utilization and so on. So the article makes a number of integration methods of urban building space and transportation space from the perspective of urban morphology integration. It tries to integrate urban environment with techniques of multidimensional space interludes, cascading, infiltration between building space and traffic space in three-dimensional space coordinates, to achieve the objectives   of proper division, solving traffic congestion problems and the establishment of a new dynamic three-dimensional transport system.</p>

scholarly journals Methodological Foundations for Teaching Computer Graphics for Students in IT Areas

2020 ◽  
pp. short28-1-short28-8
Author(s):  
Vitaly Karabchevsky
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Information Technologies ◽  
Three Dimensional ◽  
Descriptive Geometry ◽  
Modeling Tools ◽  
Geometric Shapes ◽  
Basic Set ◽  
The Creation ◽  
Education Of Students

Computer technologies of graphic education of students studying programming and information technologies are considered. Particular attention is paid to the joint use of descriptive geometry methods and three-dimensional geometric modeling tools in the creation and study of models of geometric shapes. A basic set of competencies has been identified, allowing to solve the main types of computer graphics tasks, methods for achieving these competencies are considered.

Fractals In Three-Dimensional Space. I-Fractals

2017 ◽  
Vol 5 (3) ◽  
pp. 51-66 ◽  
Author(s):  
Л. Жихарев ◽  
L. Zhikharev
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Georg Cantor ◽  
Fractal Sets ◽  
Three Dimensional Modeling ◽  
Mathematical Programs ◽  
Dimensional Modeling ◽  
Cutting Out ◽  
Modeling Software ◽  
Three Dimensional Models

It has long been known that there are fractals, which construction resolve into cutting out of elements from lines, curves or geometric shapes according to a certain law. If the fractal is completely self-similar, its dimensionality is reduced relative to the original object and usually becomes fractional. The whole fractal is often decomposing into a set of separate elements, organized in the space of corresponding dimension. German mathematician Georg Cantor was among the first to propose such fractal set in the late 19th century. Later in the early 20th century polish mathematician Vaclav Sierpinski described the Sierpinski carpet – one of the variants for the Cantor set generalization onto a two-dimensional space. At a later date the Austrian Karl Menger created a three-dimensional analogue of the Sierpinski fractal. Similar sets differ in a number of parameters from other fractals, and therefore must be considered separately. In this paper it has been proposed to call these fractals as i-fractals (from the Latin interfican – cut). The emphasis is on the three-dimensional i-fractals, created based on the Cantor and Sierpinski principles and other fractal dependencies. Mathematics of spatial fractal sets is very difficult to understand, therefore, were used computer models developed in the three-dimensional modeling software SolidWorks and COMPASS, the obtained data were processing using mathematical programs. Using fractal principles it is possible to create a large number of i-fractals’ three dimensional models therefore important research objectives include such objects’ classification development. In addition, were analyzed i-fractals’ geometry features, and proposed general principles for their creation.

The Content of Descriptive Geometry Course For High Educational Institution’s In the Third Industrial Revolution’s Era

2018 ◽  
Vol 6 (3) ◽  
pp. 49-55 ◽  
Author(s):  
Юрий Поликарпов ◽  
Yuriy Polikarpov
Keyword(s):  
Industrial Revolution ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Descriptive Geometry ◽  
Cad Systems ◽  
Course Content ◽  
The Third ◽  
Stage Of Development ◽  
High Educational ◽  
Three Dimensional Space

A brief journey into the history of industrial revolutions has been presented. It is noted that our society has entered the third industrial revolution’s era. In this regard, the main consequences of the third industrial revolution have been noted. The stages of development for design methods and the basic science providing the design process have been considered. The historical necessity and significance of Gaspar Monge’s descriptive geometry appearance has been considered as well. Modern products design approaches using CAD systems are described. It is stated that design has again returned to three-dimensional space, in fact prior to the Monge’s era, but at a new stage of development. The conclusion is drawn that, taking into account the realities and needs of modern production, it is necessary to modernize the descriptive geometry course for technical high educational institutions. The author's suggestions on course content changing are presented related to extension of one sections and reducing of another ones, taking into account the fact that in real design practice the designer solves geometric problems in three-dimensional space, rather than in a complex drawing. It is noted that in connection with the extensive use of CAD systems, the design stages and the composition of design documentation developed at each stage are changed. Such concepts as "electronic model" and "electronic document" have appeared and are widely used, that is confirmed by adoption of new USDD standards. In such a case the role and significance of some types of drawings may change in the near future, since modern CAD systems allow transfer to production not 2D drawings, but electronic models and product drawings.

On the Educational-Methodical Complex of Discipline «Engineering and Computer Graphics» in Terms of Modernization of the Course of Descriptive Geometry

2018 ◽  
Vol 6 (5) ◽  
pp. 34-40 ◽  
Author(s):  
Юрий Поликарпов ◽  
Yuriy Polikarpov ◽  
М. Семашко ◽  
M. Semashko ◽  
Л. Худякова ◽  
...  
Keyword(s):  
Computer Graphics ◽  
Computer Aided Design ◽  
Dimensional Space ◽  
New Products ◽  
Three Dimensional ◽  
Machine Building ◽  
Educational Process ◽  
Descriptive Geometry ◽  
Aided Design ◽  
Three Dimensional Space

In connection with the use of machine-building enterprises to create new products of computer-aided design, which solve the problem in three-dimensional space, the problem of modernization of the course of descriptive geometry becomes relevant. The article describes the experience of the Department of descriptive geometry and drawing of the Ufa state aviation technical University for the modernization of the course of descriptive geometry. The questions of development of educational and methodical complex of the modernized discipline “engineering and computer graphics”, about its components which are prepared by Department and are used in educational process at training of bachelors in the directions which are included in the enlarged group of 150000 «Mechanical engineering» are in detail considered.

scholarly journals Geometrical Model of the Manufacturing Surface of the Equivalent Working Surface of the Fine Tooth Dolbyak

2019 ◽  
Author(s):  
С. Рязанов ◽  
S. Ryazanov ◽  
Михаил Решетников ◽  
Mihail Reshetnikov
Keyword(s):  
Computer Modeling ◽  
Geometric Modeling ◽  
Dimensional Space ◽  
Geometric Model ◽  
Three Dimensional ◽  
Geometrical Model ◽  
Analytic Description ◽  
Computer Algorithms ◽  
Gear Cutting ◽  
Working Surface

Existing mathematical models for calculating gearing are quite complex and do not always make it possible to quickly and accurately obtain the desired result. A simpler way to find a suitable gear option that satisfies the task is to use computer modeling and computer graphics methods, and in particular solid-state modeling algorithms. The use of geometric modeling techniques to simulate the process of shaping the working surface of gearing is based on the relative movement of intersecting objects in the form of a “workpiece-tool” system. This allows you to obtain the necessary geometric model that accurately reproduces the geometric configuration of the surfaces of the teeth of spatial gears, taking into account the technological features of their production on gear cutting machines. This information allows you to perform on the computer imitation control the movement of the cutting tool. Ultimately, this boils down to the problem of analytic description and computer representation of curves and surfaces in three-dimensional space. As the gear cutting tools, the most widely used are disk and worm modular mills (shaver), gear cutting heads, dolbyaki and lath tools. At the moment there are no computer algorithms for obtaining the “dolbyak” producing surfaces, which are obtained by a tool with a modified producing surface. A change in the geometric shape of the tool producing surface will lead to a change in its working surfaces, which may lead to an improvement in their contact. This article shows the application of the developed methods and algorithms of geometric and computer modeling, which are intended for shaping the working surfaces of the Dolbyak tool. Their application will speed up the process of calculating intermediate adjustments of machines used for cutting gears, bypassing complex mathematical calculations that, under conditions of aging of the gear-cutting machines, their wear and the inevitable reduction in the accuracy of their kinematic chains.

scholarly journals Distance Between Two Circles In Any Number Of Dimensions Is A Vector Ellipse

2021 ◽  
Author(s):  
Asli Pinar Tan
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Mathematical Basis ◽  
Multiple Dimensions ◽  
Special Cases ◽  
Position Data ◽  
Gravitational Pull ◽  
Elliptical Motion ◽  
The Universe ◽  
Moving Bodies

Based on measured astronomical position data of heavenly objects in the Solar System and other planetary systems, all bodies in space seem to move in some kind of elliptical motion with respect to each other, whereas objects follow parabolic escape orbits while moving away from Earth and bodies asserting a gravitational pull, and some comets move in near-hyperbolic orbits when they approach the Sun. In this article, it is first mathematically proven that the &ldquo;distance between points on any two different circles in three-dimensional space&rdquo; is equivalent to the &ldquo;distance of points on a vector ellipse from another fixed or moving point, as in two-dimensional space.&rdquo; Then, it is further mathematically demonstrated that &ldquo;distance between points on any two different circles in any number of multiple dimensions&rdquo; is equivalent to &ldquo;distance of points on a vector ellipse from another fixed or moving point&rdquo;. Finally, two special cases when the &ldquo;distance between points on two different circles in multi-dimensional space&rdquo; become mathematically equivalent to distances in &ldquo;parabolic&rdquo; or &ldquo;near-hyperbolic&rdquo; trajectories are investigated. Concepts of &ldquo;vector ellipse&rdquo;, &ldquo;vector hyperbola&rdquo;, and &ldquo;vector parabola&rdquo; are also mathematically defined. The mathematical basis derived in this Article is utilized in the book &ldquo;Everyhing Is A Circle: A New Model For Orbits Of Bodies In The Universe&rdquo; in asserting a new Circular Orbital Model for moving bodies in the Universe, leading to further insights in Astrophysics.

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