The Problem of Emplacement of Triangular Geometric Net on the Sphere With Nodes on the Same Level

Abstract

Мордовский государственный университет им. Н.П. Огарева, г. Саранск, РОССИЯ Аннотация: Исследован один из методов образования треугольных сетей на сфере. Поставлены условия задачи размещения треугольной сети на сфере. Критерием оценки эффективности решения задачи явля-ется минимальное число типоразмеров панелей купола, возможность укрупнительной сборки и предва-рительного напряжения. Приведено решение одного варианта задачи размещения треугольной сети в сов-местимом сферическом треугольнике и, соответственно, на сфере. Размещение на сфере правильных и неправильных шестиугольников, вписанных в окружности, т.е. фигур плоских или составленных в свою очередь из сферических треугольников с минимальными размерами ребер, имеет эффективное решение в виде сети, образованной на основе окружностей минимальных радиусов, т.е. окружностей на сфере, полу-ченных при касании двух смежных окружностей, центры которых находятся на наименьшем расстоянии друг от друга. Задача выравнивания опор на одном уровне может быть решена размещением в системе правильных шестиугольников и пятиугольников неправильных шестиугольников, вписанных в окружно-сти минимальных размеров.Ключевые слова: сборная сферическая оболочка, треугольная геометрическая сеть, описанная окружность, правильный шестиугольник, разрезка, купол, оптимизация THE PROBLEM OF EMPLACEMENT OF TRIANGULAR GEOMETRIC NET ON THE SPHERE WITH NODES ON THE SAME LEVEL Vasilij D. Antoshkin Mordovian State University, Saransk, RussiaAbstract: One of the methods of formation of triangular networks in the field is investigated. Conditions of the problem of locating a triangular network in the area are delivered. The criterion for assessing the effectiveness of the solution of the problem is the minimum number of sizes of the dome panels, the possibility of pre-assembly and pre-stressing. The solution of the problem of one embodiment of a triangular network of accommodation in a compatible spherical triangle and, accordingly, on the sphere. Placing on the area of regular and irregular hexagon inscribed in a circle, ie, flat figures or composed in turn of spherical triangles with minimum dimensions of the ribs, is an effective solution in the form of a network formed by circles of minimum radii, ie, circles on a sphere obtained at the touch of three adjacent circles whose centers are at the shortest distance from each other. Task align the supports at one level can be resolved by placement in the regular hexagons and irregular pentagons hexagons inscribed in a circle of minimum size..

The task 3 of forming a network on the sphere from the circles of the same radius

“…As a result of this work, geodesic domes are offered with a specific cuts, which is performed by splitting the faces with large circles along the midpoints of the edges and the centers of the faces of a regular spherical polyhedron, for example, a cube or an octahedron, etc. [1][2][3][4][5][6][7][8][9]. The first cut of the spherical shell framework is obtained by splitting into spherical elements and shapes, for example, arched arcs, polygons, triangles, etc., paired circles of the same radius parallel to the specified large circles passing through the middle of adjacent ribs.…”

Network from the circles of the same radius for sustainable energy-efficient roof structures

“…The diagrams in figures 1 and 2 show the placement of paired circles of the same radius [1][2][3][4][5][6][7][8][9][10][11][12][13] on the basis of a spherical icosahedron. On the known location conditions of the centers of the circles in the centers of opposite planes, the problem of forming a geometric network on a sphere with the centers of unites located on the circles parallel to the equator circle (i.e., at one level) is reduced to solving a system of equations for spherical triangles shown in figure 1.…”

“…Independent variants of arrangement of paired circles can be combined in pairs, or all three independent systems can be combined into a single frame of paired arches [1][2][3][4][5][6][7][8][9][10][11][12][13]. To do this, we estimate the possibility of forming the second variant of such geometric network on the sphere ( figure 2), where the vertices of the faces O, the correct spherical twenty-triangle (icosahedron) should be chosen as the poles for the construction.…”

“…Using the known Napier's expressions [14] for the sides and angles of right-angled spherical triangles, respectively, we obtain again a system of equations (1). Ie, in the end, we can use the unchanged equations (1)(2)(3)(4)(5)…”

“…In a geodesic dome, on the basis of a regular spherical polyhedron (icosahedron), there is a possibility of such placement in the form of a network with the minimum sizes of arch segments and with pair arches unites comprised on the basis of circles of the same radii formed on the ground of the correct spherical polyhedra, with preservation of the minimum number of standard sizes which will provide an effective arrangement of support unites much lower or higher than the equator and at one, quite certain level [1][2][3][4][5][6][7][8][9][10][11][12][13]. In the investigated regular spherical polyhedron ( Fig.…”