A Pseudopolynomial Algorithm for Alexandrov's Theorem

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes; an abbreviated v2 was at WADS 200

On the total mean curvature of non-rigid surfaces

Using Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.Comment: 4 page

Многочлены объема для некоторых многогранников в пространствах постоянной кривизны

It is known that for each simplicial polyhedron P in 3-space there exists a monic polynomial Q depending on the combinatorial structure of P and the lengths of its edges only such that the volume of the polyhedron P as well as one of any polyhedron isometric to P and with the same combinatorial structure are roots of the polynomial Q. But this polynomial contains many millions of terms and it cannot be presented in an explicit form. In this work we indicate some special classes of polyhedra for which these polynomials can be found by a sufficiently effective algorithm which also works in spaces of constsnt curvature of any dimension.Известно, что для каждого симплициального многогранника P в 3-пространстве существует многочлен Q, зависящий только от комбинаторного строения многогранника и длин его ребер, такой, что объемы многогранника P и любого другого изометричного P многогранника с таким же комбинаторным строением являются корнями многочлена Q. Но этот многочлен содержит много миллионов слагаемых, и его нельзя выписать в явном виде. В работе мы указываем один класс многогранников, для которых эти многочлены можно выписать в компактной форме, верной также в пространствах постоянной кривизны любой размерности

Smarandache theorem in hyperbolic geometry

In the paper a hyperbolic version of the Smarandache pedal polygon theorem is considered. © A.V. Kostin and I.Kh. Sabitov, 2014

Гиперболический тетраэдр: вычисление объема с применением к доказательству формулы Шлефли

We propose a new approach to the problem of calculations of volumes in the Lobachevsky space, and we apply this method to tetrahedra. Using some integral formulas, we present an explicit formula for the volume of a tetrahedron in the function of the coordinates of its vertices as well as in the function of its edge lengths. Finally, we give a direct analitic proof of the famous Schläfli formula for tetrahedra.Мы предлагаем один новый подход к проблеме вычисления объемов тел в пространстве Лобачевского и применяем его к тетраэдру. Используя некоторые интегральные соотношения, мы даем явные формулы для объема тетраэдра в функции координат его вершин, а также длин его ребер. Наконец, мы даем в случае тетраэдра прямое аналитическое доказательство знаменитой формулы Шлефли

Volumes of polytopes in spaces of constant curvature

We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a solution for Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find a few versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.Comment: 22 pages, 9 figures, 58 reference

Deflected mode of junction of pipes of different diameters in the constructions of contact-line supports of electrical transport

© Research India Publications 2015. Rapid pace of development of energy, communications, telecommunications and other industries of economy stimulate fabrication of structural steel with application of tubular rods (round pipe, polyhedral bent studding, profile of closed section and so on), processing a number of constructional qualities which provide decline in demand for steel, decrease intensity of wind loading, increase corrosion resistance [1-3]. Such constructions can be referred to the transmission towers, supports for wind-generated installations, towers of cellular communications, supports of urban illumination, supports of contact-line networks of electrical transport, supports for advertizing structures, supports for lighting (traffic signal installations) and the others. In designing constructions from the pipes one of the most important tasks is support of bearing capacity of node points of junction. It is substantiated by experimental data indicating that in many cases a carrying capacity of the whole construction is determined by the strength of junction bond of its elements. The designs from pipes are performed from separate shafts or in the form of flat, or space grid systems. Urge towards decline in demand for steel in these constructions leads naturally to the use of tubular rods of different diameter. On the whole, the effectiveness of tubular constructions is determined to large extent by constructive design of node points of connection of tubular rods. In practical building it is applied different types junction of tube bars, including assemblies from pipes of various diameter. It has been first developed numerical methods of analysis of determination of deflected mode of the pipes of different diameters by push fit of one into the other. For ECM, using the language FORTRAN it had been coded «AutoRSS. 01», which allows to DM components of telescopic joint of pipes being different in diameter. It has been carried out comparative assessment of the results of calculation of DM joint units according to the suggested program «AutoRSS. 01» the known programs, realizing the method of finite-elements method (FEM). It has been performed the analysis of the results of calculation according to the suggested program «AutoRSS. 01» and determined an optimal push of one pipe into the other, which is within the limits of 2÷2. 3d, where d-diameter of smaller pipe

Development of the method of dynamic tests support of air transmission lines

© Published under licence by IOP Publishing Ltd. Known methods of testing supports create longitudinal and transverse static load applied to the support. In real conditions the majority of the damage of the supports is connected with the influence on them of dynamic loads, which can exceed the static. The proposed method allows to conduct experimental studies of cascade processes of destruction of anchor site due to a wire breakage, when the potential energy of the tension wires is converted into a dynamic influence on the design of supports