Isavnin A.G.Branch of Kazan (Volga) Federal University in Naberezhnye Chelny, Naberezhnye Chelny, Republic of Tatarstan, Russian Federation isavnin@mail.ru
Sharipov R.Sh.Branch of Kazan (Volga) Federal University in Naberezhnye Chelny, Naberezhnye Chelny, Republic of Tatarstan, Russian Federation radik@sharipov.com
Subject The article addresses the problem of finding Steiner points in the spherical coordinate system and using them in road construction, taking into account the topographic features that influence the cost of construction. Objectives The aim of the article is to analyze the possibility of applying the Steiner problem in spherical coordinates given the topographic features of the surface in road building. Methods The study was conducted in the spherical coordinate system using the algorithm for solving the Steiner problem with convergence points. Results We considered a possibility to find a solution to the Steiner problem using spherical coordinates for three and six points, calculated the cost of road construction given the relief, on the case of connecting three settlements of the Moscow oblast. Conclusions and Relevance Applying the Steiner problem for points with geocentric coordinates enables to take into account topographic features in the design and reconstruction of roads. Using the Steiner problem, it is possible to significantly reduce the scope of construction, shorten the length of the path between the object, and save money. It also enables to calculate more accurately the amount of required investment and estimate return on invested funds.
Keywords: Steiner Tree Problem, Steiner point, spherical coordinate system, road design, relief
References:
Bern M.W., Graham R.L. The Shortest-Network Problem. Scientific American, 1989, no. 3, pp. 64–70.
Courant R., Robbins H. Chto takoe matematika? Elementarnyi ocherk idei i metodov [What is Mathematics? An Elementary Approach to Ideas and Methods]. Moscow, MTsNMO Publ., 2001, 568 p.
Lysov A.V., Shiganov A.S. Geodezicheskie raboty pri zemleustroistve [Geodetic works in land development]. Saratov, Saratov State Agrarian University named after N.I. Vavilov Publ., 2007, 147 p.
Orlov N.N. [Building the optimal connections]. Izvestiya YuFU. Tekhnicheskie nauki = Izvestiya SFedU. Engineering Sciences, 2009, vol. 93, no. 4, pp. 88–93. (In Russ.)
Orlov N.N. [Solution to the Euclidean Steiner Tree Problem]. Izvestiya TRTU = Bulletin of Taganrog State University of Radio-Engineering, 2005, no. 3, pp. 81–86. (In Russ.)
Gordeev E.N., Tarastsov O.G. [The Steiner Tree Problem: a survey]. Diskretnaya matematika = Discrete Mathematics and Applications, 1993, vol. 5, no. 2, pp. 3–28. (In Russ.)
Grigoreva D.R., Faizullina A.G., Sharipov R.Sh., Basyrov R.R. Use of Steiner Problem in Solving Practical Problems of Road Construction. Modern Applied Science, 2015, vol. 9, no. 4, pp. 294–302.
Lisin A.V., Faizullin R.T. [Heuristic algorithm for finding an approximate solution to the Steiner problem based on physical analogies]. Komp'yuternaya optika = Computer Optics, 2013, vol. 37, no. 4, pp. 503–510. (In Russ.)
Lotarev D.T. [A solution to the Steiner three point problem on the plane using MatLab tools]. Trudy ISA RAN = Proceedings of ISA RAS, 2008, vol. 32, pp. 159–165. (In Russ.)
Lotarev D.T., Uzdemir A.P. [Locating transport networks in non-homogenous territories]. Avtomatika i telemekhanika = Automation and Remote Control, 2002, no. 7, pp. 114–124. (In Russ.)
Lotarev D.T., Uzdemir A.P. [Transformation of the Steiner problem on the Euclidean plane to the Steiner problem on the graph]. Avtomatika i telemekhanika = Automation and Remote Control, 2005, no. 10, pp. 82–92. (In Russ.)
