Flexibility of curves on a single-sheet hyperboloid
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Flexibility of curves on a single-sheet hyperboloid Miroslav D. Maksimovi´c
Received: 16 September 2019 / Accepted: 21 April 2020 © Springer Nature B.V. 2020
Abstract Hyperbolic towers are towers in the shape of a single-sheet hyperboloid, and they are interesting in architecture. In this paper, we deal with the infinitesimal bending of a curve on a hyperboloid of one sheet; that is, we study the flexibility of the net-like structures used to make a hyperbolic tower. Visualization of infinitesimal bending has been carried out using Mathematica, and some examples are presented and discussed. Keywords Curve · Hyperbolic tower · Hyperboloid · Infinitesimal bending · Shukhov tower
1 Introduction The Russian engineer Vladimir Shukhov designed many towers using net-like structures in the shape of hyperboloids of a single sheet (see [1–3]). The first hyperbolic tower was Shukhov Tower in Polibino. With his work, Shukhov induced many architects to design similar hyperboloid structures. As a result, there are a number of modern hyperboloid shape towers in the world: The Canton Tower, The Kobe Port Tower, The Sydney Tower, etc. Hyperbolic towers are used for radio stations, power stations, as a cooling towers, skyscrapers, etc. The hyperboloid is the design standard for all nuclear cooling towers and some coal-fired power plants [4]. The demand for economical, tall and lightweight cooling towers has driven engineers towards designing hyperbolic structures, especially in regions with high-seismic ground motions [5]. A hyperboloid structure as a mechanical model of the carbon bond was studied in [6]. A hyperboloid is a doubly ruled surface so it can be built using straight steel beams, producing a strong structure more cheaply than other methods [7]. However, despite the prevalence and importance of the hyperbolic structures, there are surprisingly few papers dealing with single-sheet hyperboloids, in the context of the flexibility of its net-like structure. Infinitesimal bending of surfaces and curves is a part of the infinitesimal bending theory—one of the main parts of global Differential geometry. Infinitesimal bending of surfaces and curves was studied in [8–13]. The theory of infinitesimal bending has applications in various sciences: physics, biology [14,15], architecture [16], etc. In this The author was supported by the research project 174025 of the Serbian Ministry of Science. M. D. Maksimovi´c (B) Department of Mathematics, Faculty of Sciences, University of Priština - Kosovska Mitrovica, Kosovska Mitrovica, Serbia e-mail: [email protected]
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M. D. Maksimovi´c
article, we will study flexibility of net-like structures of hyperbolic towers by examining infinitesimal bending of u- and v-parameter curves on the single-sheet hyperboloid.
2 Preliminaries In this section, basic facts of the theory of infinitesimal bending of curves are given according to Efimov [9] and Velimirovi´c [11,12]. Definition 1 Let us consider a continuous regular curve C ⊂ R3 C : r = r(t), t ∈ I ⊆ R,
(1)
that is included i
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