40 research outputs found
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
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
An infinitesimally nonrigid polyhedron with nonstationary volume in the Lobachevsky 3-space
We give an example of an infinitesimally nonrigid polyhedron in the
Lobachevsky 3-space and construct an infinitesimal flex of that polyhedron such
that the volume of the polyhedron isn't stationary under the flex.Comment: 10 pages, 2 Postscript figure
Volume formula for a -symmetric spherical tetrahedron through its edge lengths
The present paper considers volume formulae, as well as trigonometric
identities, that hold for a tetrahedron in 3-dimensional spherical space of
constant sectional curvature +1. The tetrahedron possesses a certain symmetry:
namely rotation through angle in the middle points of a certain pair of
its skew edges.Comment: 27 pages, 2 figures; enhanced and improved exposition, typos
corrected; Arkiv foer Matematik, 201
Volume Polynomials for Some Polyhedra in Spaces of Constant Curvature
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