共查询到20条相似文献,搜索用时 15 毫秒
1.
Eric Berberich Efi Fogel Dan Halperin Michael Kerber Ophir Setter 《Mathematics in Computer Science》2010,4(1):67-91
We describe the algorithms and implementation details involved in the concretizations of a generic framework that enables
exact construction, maintenance, and manipulation of arrangements embedded on certain two-dimensional orientable parametric
surfaces in three-dimensional space. The fundamentals of the framework are described in a companion paper. Our work covers
arrangements embedded on elliptic quadrics and cyclides induced by intersections with other algebraic surfaces, and a specialized
case of arrangements induced by arcs of great circles embedded on the sphere. We also demonstrate how such arrangements can
be used to accomplish various geometric tasks efficiently, such as computing the Minkowski sums of polytopes, the envelope
of surfaces, and Voronoi diagrams embedded on parametric surfaces. We do not assume general position. Namely, we handle degenerate
input, and produce exact results in all cases. Our implementation is realized using Cgal and, in particular, the package that provides the underlying framework. We have conducted experiments on various data sets,
and documented the practical efficiency of our approach. 相似文献
2.
Rolf Walter 《manuscripta mathematica》1989,63(3):343-363
A previous result in Euclidean geometry [7] on H-tori with plane and spherical curvature lines is extended here to the two noneuclidean geometries. There result families of H-tori with only spherical curvature lines, which are explicitly representable by elliptic and theta functions (or ordinary integrals of elementary functions). Among the geometric properties, it is shown that the midpoints of the generating spheres vary on geodesics. The hyperbolic case is more similar to the Euclidean situation than the elliptic one. In elliptic geometry the constructed surfaces depend on one additional rational parameter and, as a limiting case, there are even countably many minimal tori of this type. 相似文献
3.
Representations of solutions of equations describing the diffusion and quantum dynamics of particles in a Riemannian manifold are discussed under the assumption that the mass of particles is anisotropic and depends on both time and position. These equations are evolution differential equations with secondorder elliptic operators, in which the coefficients depend on time and position. The Riemannian manifold is assumed to be isometrically embedded into Euclidean space, and the solutions are represented by Feynman formulas; the representation of a solution depends on the embedding. 相似文献
4.
In this paper, it is proved that the surfaces of positive curvature with no umbilical points in 3-dimensional Euclidean space whose characteristic curves form a Tchebychef net are translation surfaces and that the characteristic curves are represented on the unit sphere by a rhombic net. The determination of these surfaces depends on two elliptic integrals of the first kind. Furthermore, the case where these elliptic integrals reduce to elementary integrals is studied and it is shown that the surfaces corresponding to this case belong to one of the following two classes: (a) Translation surfaces of positive curvature with plane characteristic curves as generators lying in two planes intersecting each other under a constant angle. The special case where these planes are perpendicular gives an analogue of the Scherk's minimal surfaces of translation. (b) Translation surfaces of revolution of positive curvature with characteristic curves as generators which are circular helices. 相似文献
5.
Simon Blatt 《Journal of Evolution Equations》2010,10(1):21-27
We show that for a large class of geometric evolution equations of immersed surfaces in the Euclidean space, there are compact
embedded surfaces that lose their embeddedness and compact strictly convex surfaces that lose their convexity under these
evolution equations. 相似文献
6.
We study the integrability problem for evolution systems on phase spaces with a nonflat metric. We show that if the phase space is a sphere, the Hamiltonian systems are generated by the action of the Hamiltonian operators on the variations of the phase-space geodesics and the integrability problem for the evolution systems reduces to the integrability problem for the equations of motion for the frames on the phase space. We relate the bi-Hamiltonian representation of the evolution systems to the differential-geometric properties of the phase space. 相似文献
7.
First, we review the authors' recent results on translating solutions to mean curvature flows in Euclidean space as well as in Minkowski space, emphasizing on the asymptotic expansion of rotationally symmetric solutions. Then we study the sufficient condition for which the translating solution is rotationally symmetric. We will use a moving plane method to show that this condition is optimal for the symmetry of solutions to fully nonlinear elliptic equations without ground state condition. 相似文献
8.
B. G. Konopelchenko 《Studies in Applied Mathematics》1996,96(1):9-51
A method is considered to induce surfaces in three-dimensional (pseudo) Euclidean space via the solutions to two-dimensional linear problems (20 LPs) and their integrable dynamics (deformations) via the 2 + 1-dimensional nonlinear integrable equations associated with these 2D LPs. Coordinates Xi of the induced surfaces are defined as integrals over certain bilinear combinations of the wave functions ψ of these 20 LPs. General formulation as well as three concrete examples are considered. Some properties and features of such induction are discussed. Three-dimensional Riemann spaces associated with 2 + 1-dimensional nonlinear integrable equations are considered also. 相似文献
9.
S. A. Nazarov 《Journal of Mathematical Sciences》2009,162(3):373-392
We study optimal patterns of a patch made of an elastic anisotropic homogeneous material for covering a hole in a two-dimensional
body possessing different physical characteristics. In addition to the optimization problem for inclusions in two-dimensional
and three-dimensional elastic and piezoelectric bodies, we also consider similar problems for an arbitrary formally selfadjoint
elliptic system of differential equations in multidimensional domains. A condition for the stationarity of the energy functional
is obtained; for a free parameter the matrix of orthogonal transformations of the Euclidean space is taken; the result is
based on an algebraic fact about small increments of orthogonal and unitary matrices. Bibliography: 23 titles. Illustrations:
1 figure. 相似文献
10.
M. I. Shtogrin 《Proceedings of the Steklov Institute of Mathematics》2006,252(1):264-271
Two-dimensional polyhedra homeomorphic to closed two-dimensional surfaces are considered in the three-dimensional Euclidean space. While studying the structure of an arbitrary face of a polyhedron, an interesting particular case is revealed when the magnitude of only one plane angle determines the sign of the curvature of the polyhedron at the vertex of this angle. Due to this observation, the following main theorem of the paper is obtained: If a two-dimensional polyhedron in the three-dimensional Euclidean space is isometric to the surface of a closed convex three-dimensional polyhedron, then all faces of the polyhedron are convex polygons. 相似文献
11.
The work deals with the Dirichlet problem for elliptic equations with nonhomogeneous anisotropic degeneracy in a possibly unbounded domain of multidimensional Euclidean space. The existence of weak solutions is proved. Some conditions are established connecting the character of nonlinearity of the equation and the geometric characteristics of the domain which guarantee the one-dimensional localization (vanishing) of weak solutions. The equation with anisotropic degeneracy is shown to admit localized solutions even in the absence of absorption. 相似文献
12.
Dr. Johann Lang 《Journal of Geometry》1989,34(1-2):114-126
In [8] the author gave a report on some properties of flag space motions, especially of the composition of screw motions or rotations in flag space. Planet motions more generally are motions which can be composed by two one-parameter-groups. These motions are investigated with respect to their orbits, their multiple ways of construction and the tubular surfaces they can determine. Some of them yield tubular screw surfaces, some others move every sphere the way that it again envellopes a sphere. These motions, which have no Euclidean counterpart, determine non-trivial, kinematically generated LIE-transformations in flag space. 相似文献
13.
Hiroyuki Chihara 《Journal of Fourier Analysis and Applications》2008,14(2):301-325
We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them
to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose
we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the
symbols of multipliers.
The author was supported by the JSPS Grant-in-Aid for Scientific Research #17540140. 相似文献
14.
Ioannis M. Roussos 《数学学报(英文版)》1999,15(2):269-276
We study the real Bonnet surfaces which accept one unique nontrivial isometry that preserves the mean curvature, in the three-dimensional
Euclidean space. We give a general criterion for these surfaces and use it to determine the tangential developable surfaces
of this kind. They are determined implicitly by elliptic integrals of the third kind. Only the tangential developable surfaces
of circular helices are explicit examples for which we completely determine the above unique nontrivial isometry.
Dedication Dedicated to Siuping Ho for all her invaluable support and encouragement. 相似文献
15.
16.
We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a by-product we find another sphere covering inequality that can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and discuss their applications to elliptic equations with exponential nonlinearities in dimension 2 . The approach in this paper extends, improves, and unifies several inequalities about solutions of elliptic equations with exponential nonlinearities. © 2020 Wiley Periodicals LLC 相似文献
17.
《Journal of Applied Mathematics and Mechanics》1998,62(2):193-200
A dynamical system is constructed in the multiplicative group of the quarternion algebra H that serves as the configuration space. A homomorphism H → SO(3) is used such that the unit sphere S3 ⊂ H, invariant under the system, is transformed into the rotation group SO(3). The homomorphic image of the system is identical with the dynamics of rotational motion of a rigid body. The equations of motion are completely integrated in the Euler case. To this end Weierstrass' elliptic functions are used. The following goals are achieved within the framework of the method: (a) when representing the algorithms for modelling the dynamics it suffices to use only one chart from the atlas of the phase space manifold, (b) the point in the configuration space of the actual motion lies on the unit sphere, which ensures the best accuracy in numerical procedures, and (c) in the majority of applications the right-hand sides of the equations of perturbed motion depend polynomially on the phase variables, which simplifies the use of computer algebra in analytic theories. 相似文献
18.
In pseudo-Riemannian geometry the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. Furthermore, the space of all geodesics has a structure of a Jacobi manifold. We describe the geometry of these structures and their generalizations. We also introduce and study pseudo-Euclidean billiards, emphasizing their distinction from Euclidean ones. We present a pseudo-Euclidean version of the Clairaut theorem on geodesics on surfaces of revolution. We prove pseudo-Euclidean analogs of the Jacobi–Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space. 相似文献
19.
È. I. Semenov 《Siberian Mathematical Journal》2008,49(1):166-174
We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form. 相似文献
20.
Roman Samulyak 《数学物理学报(B辑英文版)》2010,30(2):499-521
The embedded boundary method for solving elliptic and parabolic problems in geometrically complex domains using Cartesian meshes by Johansen and Colella (1998, J. Comput. Phys. 147, 60) has been extended for elliptic and parabolic problems with interior boundaries or interfaces of discontinuities of material properties or solutions. Second order accuracy is achieved in space and time for both stationary and moving interface problems. The method is conservative for elliptic and parabolic problems with fixed interfaces. Based on this method, a front tracking algorithm for the Stefan problem has been developed. The accuracy of the method is measured through comparison with exact solution to a two-dimensional Stefan problem. The algorithm has been used for the study of melting and solidification problems. 相似文献