200 research outputs found
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
ΠΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Ρ ΠΎΠ±ΡΠ΅ΠΌΠ° Π΄Π»Ρ Π½Π΅ΠΊΠΎΡΠΎΡΡΡ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠΎΠ² Π² ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π°Ρ ΠΏΠΎΡΡΠΎΡΠ½Π½ΠΎΠΉ ΠΊΡΠΈΠ²ΠΈΠ·Π½Ρ
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.ΠΠ·Π²Π΅ΡΡΠ½ΠΎ, ΡΡΠΎ Π΄Π»Ρ ΠΊΠ°ΠΆΠ΄ΠΎΠ³ΠΎ ΡΠΈΠΌΠΏΠ»ΠΈΡΠΈΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠ° P Π² 3-ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π΅ ΡΡΡΠ΅ΡΡΠ²ΡΠ΅Ρ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½ Q, Π·Π°Π²ΠΈΡΡΡΠΈΠΉ ΡΠΎΠ»ΡΠΊΠΎ ΠΎΡ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΎΡΠ½ΠΎΠ³ΠΎ ΡΡΡΠΎΠ΅Π½ΠΈΡ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠ° ΠΈ Π΄Π»ΠΈΠ½ Π΅Π³ΠΎ ΡΠ΅Π±Π΅Ρ, ΡΠ°ΠΊΠΎΠΉ, ΡΡΠΎ ΠΎΠ±ΡΠ΅ΠΌΡ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠ° P ΠΈ Π»ΡΠ±ΠΎΠ³ΠΎ Π΄ΡΡΠ³ΠΎΠ³ΠΎ ΠΈΠ·ΠΎΠΌΠ΅ΡΡΠΈΡΠ½ΠΎΠ³ΠΎ P ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠ° Ρ ΡΠ°ΠΊΠΈΠΌ ΠΆΠ΅ ΠΊΠΎΠΌΠ±ΠΈΠ½Π°ΡΠΎΡΠ½ΡΠΌ ΡΡΡΠΎΠ΅Π½ΠΈΠ΅ΠΌ ΡΠ²Π»ΡΡΡΡΡ ΠΊΠΎΡΠ½ΡΠΌΠΈ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Π° Q. ΠΠΎ ΡΡΠΎΡ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½ ΡΠΎΠ΄Π΅ΡΠΆΠΈΡ ΠΌΠ½ΠΎΠ³ΠΎ ΠΌΠΈΠ»Π»ΠΈΠΎΠ½ΠΎΠ² ΡΠ»Π°Π³Π°Π΅ΠΌΡΡ
, ΠΈ Π΅Π³ΠΎ Π½Π΅Π»ΡΠ·Ρ Π²ΡΠΏΠΈΡΠ°ΡΡ Π² ΡΠ²Π½ΠΎΠΌ Π²ΠΈΠ΄Π΅. Π ΡΠ°Π±ΠΎΡΠ΅ ΠΌΡ ΡΠΊΠ°Π·ΡΠ²Π°Π΅ΠΌ ΠΎΠ΄ΠΈΠ½ ΠΊΠ»Π°ΡΡ ΠΌΠ½ΠΎΠ³ΠΎΠ³ΡΠ°Π½Π½ΠΈΠΊΠΎΠ², Π΄Π»Ρ ΠΊΠΎΡΠΎΡΡΡ
ΡΡΠΈ ΠΌΠ½ΠΎΠ³ΠΎΡΠ»Π΅Π½Ρ ΠΌΠΎΠΆΠ½ΠΎ Π²ΡΠΏΠΈΡΠ°ΡΡ Π² ΠΊΠΎΠΌΠΏΠ°ΠΊΡΠ½ΠΎΠΉ ΡΠΎΡΠΌΠ΅, Π²Π΅ΡΠ½ΠΎΠΉ ΡΠ°ΠΊΠΆΠ΅ Π² ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π°Ρ
ΠΏΠΎΡΡΠΎΡΠ½Π½ΠΎΠΉ ΠΊΡΠΈΠ²ΠΈΠ·Π½Ρ Π»ΡΠ±ΠΎΠΉ ΡΠ°Π·ΠΌΠ΅ΡΠ½ΠΎΡΡΠΈ
Smarandache theorem in hyperbolic geometry
In the paper a hyperbolic version of the Smarandache pedal polygon theorem is considered. Β© A.V. Kostin and I.Kh. Sabitov, 2014
ΠΠΈΠΏΠ΅ΡΠ±ΠΎΠ»ΠΈΡΠ΅ΡΠΊΠΈΠΉ ΡΠ΅ΡΡΠ°ΡΠ΄Ρ: Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠ΅ ΠΎΠ±ΡΠ΅ΠΌΠ° Ρ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ΠΌ ΠΊ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²Ρ ΡΠΎΡΠΌΡΠ»Ρ Π¨Π»Π΅ΡΠ»ΠΈ
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 SchlΓ€fli formula for tetrahedra.ΠΡ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΠΌ ΠΎΠ΄ΠΈΠ½ Π½ΠΎΠ²ΡΠΉ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ ΠΊ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ΅ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ ΠΎΠ±ΡΠ΅ΠΌΠΎΠ² ΡΠ΅Π» Π² ΠΏΡΠΎΡΡΡΠ°Π½ΡΡΠ²Π΅ ΠΠΎΠ±Π°ΡΠ΅Π²ΡΠΊΠΎΠ³ΠΎ ΠΈ ΠΏΡΠΈΠΌΠ΅Π½ΡΠ΅ΠΌ Π΅Π³ΠΎ ΠΊ ΡΠ΅ΡΡΠ°ΡΠ΄ΡΡ. ΠΡΠΏΠΎΠ»ΡΠ·ΡΡ Π½Π΅ΠΊΠΎΡΠΎΡΡΠ΅ ΠΈΠ½ΡΠ΅Π³ΡΠ°Π»ΡΠ½ΡΠ΅ ΡΠΎΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ, ΠΌΡ Π΄Π°Π΅ΠΌ ΡΠ²Π½ΡΠ΅ ΡΠΎΡΠΌΡΠ»Ρ Π΄Π»Ρ ΠΎΠ±ΡΠ΅ΠΌΠ° ΡΠ΅ΡΡΠ°ΡΠ΄ΡΠ° Π² ΡΡΠ½ΠΊΡΠΈΠΈ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°Ρ Π΅Π³ΠΎ Π²Π΅ΡΡΠΈΠ½, Π° ΡΠ°ΠΊΠΆΠ΅ Π΄Π»ΠΈΠ½ Π΅Π³ΠΎ ΡΠ΅Π±Π΅Ρ. ΠΠ°ΠΊΠΎΠ½Π΅Ρ, ΠΌΡ Π΄Π°Π΅ΠΌ Π² ΡΠ»ΡΡΠ°Π΅ ΡΠ΅ΡΡΠ°ΡΠ΄ΡΠ° ΠΏΡΡΠΌΠΎΠ΅ Π°Π½Π°Π»ΠΈΡΠΈΡΠ΅ΡΠΊΠΎΠ΅ Π΄ΠΎΠΊΠ°Π·Π°ΡΠ΅Π»ΡΡΡΠ²ΠΎ Π·Π½Π°ΠΌΠ΅Π½ΠΈΡΠΎΠΉ ΡΠΎΡΠΌΡΠ»Ρ Π¨Π»Π΅ΡΠ»ΠΈ
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 and . 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
Β© 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 Β«AutoRSS. 01Β», 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 Β«AutoRSS. 01Β» 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 Β«AutoRSS. 01Β» and determined an optimal push of one pipe into the other, which is within the limits of 2Γ·2. 3d, where d-diameter of smaller pipe
Development of the method of dynamic tests support of air transmission lines
Β© 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
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