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.
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. 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].
1. Adigamova Z.S., Lihnenko E.V. Innovacionniy podhod k proektirovaniyu krysh pri razrabotke energoeffektivnyh zdaniy [Innovative Approach to Design of Roofs when Developing Power Effective Residential Buildings]. Integraciya, partnerstvo i innovacii v stroitelnoy nauke i obrazovanii. 2015, p. 128-131. (in Russian).
2. Aleksandrova E.P., Kochurova L.V., Nosov K.G., Stolbova I.D. Intensifikaciya graficheskoy podgotovki studentov na osnove geometricheskogo modelirovaniya [Intensification of Graphics Training of Students on the Basis of the Geometric Simulation]. Problemy kachestva graficheskoy podgotovki studentov v tehnicheskom vuze: tradicii i innovacii. 2015, vol. 1, p. 213-223. (in Russian).
3. Buckley R., Lewinter M. Introductory Graph Theory with Applications. Waveland Press, Inc., 2003, Reprinted 2013.
4. Deo N. Graph Theory with Application to Engineering and Computer Science. PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1974.
5. Diestel R. Graph Theory. 4th edition, 2010, Springer-Verlag, Berlin.
6. Domaski T. The Impact of Loads on Fire Safety of Timber Roofs in Mountain Region in Poland. Bezpieczestwo i technika pozarnicza, 2015, Volume 37, r. 87-96. (in Polish) DOI:https://doi.org/10.12845/bitp.37.1.2015.7.
7. Elovikova A.V., Demeneva N.V. Reshenie zadachi optimizacii rashoda krovelnogo materiala dlya chetyryohskatnoy kryshi [Solving Problems Optimization Flow Hipped Roofing Material for Roof]. Molodyojnaya nauka 2015: tehnologii, innovacii. 2015, p. 91-94. (in Russian).
8. Grochowski B. Descriptive Geometry from an Applied Perspective. Wydawnictwo Naukowe PWN, Warszawa 1995 (in Polish).
9. Istomin B.S., Turkina E.A. Arhitekturniy potencial prostranstva krysh mnogoetajnih jilyh zdaniy [Architectural Potential Roof Space High-rise Housing]. Jilichnoe stroitelstvo, 2013, I. 10, pp. 28-31. (in Russian).
10. Korotkiy V.A., Khmarova L.I. Nachertatelnaya geometriya na ekrane kompyutera [Descriptive geometry on computer screen]. Geometriya i grafika, 2013, V. 1, I. 1, pp. 32-34. DOI:https://doi.org/10.12737/469. (in Russian.)
11. Koniewski E. Geometry of Roofs from the View Point of Graph Theory. Journal for Geometry and Graphics, Vol. 8 (2004), I. 1, pr. 41-58.
12. Koniewski E. On the Existence of Shapes of Roofs. Journal for Geometry and Graphics, Volume 8 (2004), I. 2, pr. 185-198.
13. Petukhova A.V. Ingenerno-graficheskaya podgotovka studentov ctroitelnyh specialnostey s vspolzovaniem sovremennyh programmnyh kompleksov [Engineering Graphics Course Using Modern Software Systems for Students of Civil Engineering University]. Geometriya i grafika, 2015, V. 3, I. 1, p. 47-58. DOI:https://doi.org/10.12737/10458. (in Russian).
14. Panchuk K.L., Lyashkov A.A., Kaygorodtseva N.V., Leonova L.M. Geometricheskoe modelirovanie v injenernoy i kompyuternoy grafike: uchebnoe posobie [Geometric modeling in engineering and computer graphics: a tutorial]. Omsk, OmSTU Publ., 2015, p. 460. (in Russian).
15. Seregin V.I., Ivanov G.S., Borovikov I.F., Senchenkova L.S. Geometricheskie preobrazovaniya v nachertatelnoy geometrii i ingenernoy grafike [Geometric Transformations in Descriptive Geometry and Engineering Graphics]. Geometriya i grafika, 2015, V. 3, I. 2, pp. 23-28. DOI:https://doi.org/10.12737/12165. (in Russian).
16. Volkov V.Y., Kaygorodtseva N.V., Panchuk K.L. Sovremennye napravleniya i perspektivy razvitiya nauchnyh issledovaniy po geometrii i grafike: obzor dokladov na Mejdunarodnoy koferencii ICGG 2014 [Modern Direction and Prospects for Development of Scientific Research on the Geometry and Graphics: A Review of Reports of the International Conference ICGG 2014]. Problemy kachestva graficheskoy podgotovki studentov v tehnicheskom vuze: tradicii i innovacii, 2015, V. 1, pp. 99-110. (in Russian).
17. Volkov V.Y., Panchuk K.L., Kaygorodtseva N.V. O vozmojnom napravlenii razvitiya kafedr geometro-graficheskoy podgotovki [About the Possible Direction of Development Departments of Geometric-Graphics Preparation]. Problemy kachestva graficheskoy podgotovki studentov v tehnicheskom vuze: tradicii i innovacii, 2014, V. 1, pp. 161-165. (in Russian).
18. Volkov V.Y., Yurkov V.Y., Panchuk K.L., Kaygorodtseva N.V. Elementy matematizacii teoreticheskih osnov nachertatelnoy geometrii [Matematization Elements of Theoretical Fundamentals of Descriptive Geometry]. Geometriya i grafika, 2015, V. 3, I. 1, pp. 3-15. DOI:https://doi.org/10.12737/10453. (in Russian).
19. Voloshinov D.V., Solomonov R.N. Konstruktivnoe geometricheakoe modelirovanie kak perspektiva prepodavaniya graficheskih disciplin [Constructive geometric modeling as graphic disciplines’ teaching prospect]. Geometriya i grafika, 2013, V. 1, I. 2, pp. 10-13. DOI:https://doi.org/10.12737/778. (in Russian).
20. Vyshnepolskiy V.I., Salkov N.A. Celi i metody obucheniya graficheskim disciplinam [The aims and methods of teaching drawing]. Geometriya i grafika, 2013, V. 1, I. 2, pp. 8-9. DOI:https://doi.org/10.12737/777. (in Russian).
