Geometry & Graphics Geometry & Graphics Геометрия и графика 2308-4898 10832 10.12737/18054 Научные проблемы геометрии Scientific problems of geometry Научные проблемы геометрии Roof Skeletons and Graph Theory Trees Каркасы крыш и деревья теории графов Козневски Э. Kozniewski E. 17 03 2016 17 03 2016 4 1 12 20 https://naukaru.ru/en/nauka/article/10832/view

Проблеме конструирования крыш посвящено много работ [1; 6; 7; 9 и др.]. При этом в некоторых работах предлагается использовать компьютер и специальные программы [10; 13; 16 и др.]. Геометрия эффективного проектирования крыш является актуальным научным направлением, поэтому в образовательный процесс архитектурных и строительных специальностей вносятся соответственные инновации [2; 8; 14; 15; 17–20 и др.]. Крыши, рассмотренные в данной статье, определяются как специальные геометрические многогранные поверхности, создание которых основано на двух положениях: 1) все карнизы крыши образуют планарный (односвязный или K-связный) многоугольник, называемый базой; 2) каждый скат вальмовой крыши имеет один и тот же угол наклона с (горизонтальной) плоскостью, содержащей базу. При этом все вершины и ребра такой крыши, образующие каркас крыши, определяют граф. Ортогональные проекции каркаса крыши на основную плоскость предлагается рассматривать в качестве планарного графа. На основе положений, изложенных в научных работах [11] и [12], и в продолжение своих исследований для обычных крыш нам удалось сформулировать свойства, которые позволяют изучать формы крыш, распределенных по односвязному V-угольнику для произвольного целого числа v (v ≥ 3). Также мы предложили внедрить новые виды операций — расщепление и прививание графов — и сформулировали некоторые их свойства. С помощью данных операций возможно создание циклического графа при использовании двух или более деревьев. В частности, операция прививания крыши позволяет проводить экспериментальное моделирование крыш.

The problem of constructing roofs of many papers [1; 6; 7; 9; etc.]. In this case, some studies suggested to use a computer and special programs [10; 13; 16; etc.]. The geometry of the efficient design of roofs is actual scientific direction, so in the educational process of architectural and building trades made corresponding innovations [2; 8; 14; 15; 17–20; etc.]. The roofs considered in this article are defined as special geometric polyhedral surfaces, on the basis of two assumptions: (1) all eaves of a roof form a planar (simply-connected or k-connected) polygon called the base, (2) every hipped roof end makes the same slope angle with the (horizontal) plane which contains the base. All the vertices and edges of such a roof, forming the roof skeleton, determine a graph. The orthogonal projection of a roof skeleton onto the base plane leads to a planar graph. On the basis of [11] and [12] and continuing the investigations for regular roofs, we formulate further properties which enable studying the shapes of the roofs spread over simply connected v-gons for an arbitrary integer v (v ≥ 3). We also introduce new operations: splitting and grafting of graphs, and formulate some properties of these operations. By means of such operations, the creation of a graph with cycles using two or more trees is possible. In particular, the operation of the grafting of a roof allows modeling an atrial roof.

 геометрия крыши обобщенный многоугольник регулярные графы элементарное дерево расщепление дерева прививание дерева основные крыши примитив крыши классификация форм крыши. geometry of roofs generalized polygon regular graphs elementary tree splitting of trees grafting of trees basic roofs primitive of roof classification of shapes roofs.

1. IntroductionThis article is the third part of work which deals with the geometrical properties of the roofs of buildings, considered as a special class of polyhedral surfaces from the view point of Graph Theory. In [11] and [12] we formulated and proved the Euler formula for regular roofs and some useful properties of roofs. We classified the shapes of regular roofs over simply connected v-gons for v ≤ 8. In this paper we continue the geometrical characterization of roofs. We introduce some new concepts and definitions concerning trees: elementary tree, splitting and grafting of trees, and suitable concepts for roofs: elementary roof (roof primitive), decomposition and joining of roofs. We formulate and prove some properties of these objects, which are the key to carrying out a description of the shapes of regular roofs spread over simply connected v-gons for an arbitrary integer v (v ≥ 3). In particular, we prove that for every tree TR of degree at most 3 we can construct a roof such that the graph (T, R') of the line of disappearing ridges of this roof is isomorphic with TR. We follow the notation and terminology of [11] and [12].

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