Roof Skeletons and Graph Theory Trees

E. Kozniewski

doi:10.12737/18054

ROOF SKELETONS AND GRAPH THEORY TREES Roof Skeletons and Graph Theory Trees

АА Аа

Published in Geometry & Graphics · Pages 12–20 · Rubric: Scientific problems of geometry

DOI: https://doi.org/10.12737/18054

Received: 02.03.2016 Accepted: 17.03.2016 Published: 17.03.2016 Language of publication: ENG

Authors

E. Kozniewski

ABSTRACT

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.

KEYWORDS

geometry of roofs, generalized polygon, regular graphs, elementary tree, splitting of trees, grafting of trees, basic roofs, primitive of roof, classification of shapes roofs.

Text

1. Introduction

This 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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JATS XML

Zamestitel' glavnogo redaktora, chlen redsoveta

Vladimir Vyshnyepolskiy

Glavnyy redaktor

A. A. Ishchenko

Vypuskayuschiy redaktor

Dar'ya Sklyankina

501@infra-m.ru

Deputy editor-in-chief, member of the editorial Board

N. Sal'kov

nikolaysalkov@mail.ru

ROOF SKELETONS AND GRAPH THEORY TREES Roof Skeletons and Graph Theory Trees

1. Introduction

Confirmation