Search

In this paper, we study the deformation of triangle groups of type ( 3 , n , ∞ ) determined by three R -circles R 0 , R 1 , R 2 with only rotational symmetry, which generalizes the problem studied by Falbel and Parker in [4] .

In this work ternary non-stationary subdivision schemes, based on hyperbolic B-spline basis functions, have been presented. The proposed hyperbolic, ternary three point and four point subdivision schemes, give pleasing as well as consistent curves with the control polygons as compared to the existing non-stationary subdivision schemes obtained from trigonometric B-splines. The main speciality of the hyperbolic schemes is that, they can reproduce hyperbolas and parabolas quite efficiently which has been demonstrated with the help of curvature plots.

The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle., Comment: 14 pages, 7 figures

We study the Dirichlet–Cauchy problem for nonlinear hyperbolic equation of second order in a domain with edges. The aim of this paper is to prove the regularity of solution in weighted Sobolev spaces.

In this paper we investigate the group-invariant solutions of the hyperbolic mean curvature flow with a constant forcing term by the application of Lie group in differential equations. Based on the associated vector of the obtained symmetry, we construct the group-invariant optimal system of the hyperbolic Monge–Ampere equation.

For any closed connected orientable 3-manifold M , we present a method for constructing infinitely many hyperbolic spatial embeddings of a given finite graph with no vertex of degree less than two from hyperbolic spatial graphs in S 3 via the Heegaard splitting theory. These spatial embeddings are adjustable so as to take cycle subgraphs into specified homotopy classes of loops in M .

In this paper we present a new characterization of Mobius transformations by use of hyperbolic regular polygons.

We investigate further the concept of asymptotic connectivity as defined previously by the first author. In particular, we prove the existence of, and compute an upper bound for, the asymptotic connectivity of almost any regular hyperbolic tiling of the plane. Our results indicate fundamental differences between hyperbolic (in the sense of Gromov) and non-hyperbolic graphs.

The main aim of this note is to show that the inequality h D 2 ( x , y ) ≥ h D 2 ( x , z ) + h D 2 ( y , z ) − 2 h D ( x , z ) h D ( y , z ) cos ∠ h ( y , z , x ) holds for any hyperbolic domain D ⊂ R 2 and distinct points x , y , z ∈ D , where h D denotes the hyperbolic metric in D and ∠ h ( y , z , x ) the angle formed by the hyperbolic segments γ h [ z , x ] and γ h [ z , y ] . This shows that the answer to an open problem recently raised by Klen (2009) in [10] is positive.

We present an effective method to automatically construct IFS which can be used to generate symmetric fractal in a hyperbolic plane [ p , q ] + . First, the basic IFS, which was composed of i contraction affine transforms (where i takes one of {2, 3, 4 and 5}), was randomly constructed according to three inequalities. Then the symmetric IFS was constructed by applying the rotational symmetry group Z n to the basic IFS. By the Q contraction or Q 1 contraction in the central lattice of a hyperbolic plane [ p , q ] + (where Q or Q 1 is a vertex or a middle point of a border of the central lattice, respectively), the n -fold symmetry fractals from the symmetric IFS were limited near the central lattice of the hyperbolic plane. The fractal was randomly transformed into the unit circle by isometries of hyperbolic geometry. All the work was automatically done. Many patterns with hyperbolic symmetry have been produced by this method.

In this survey paper, we present several results linking quantifier-free axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions.

The Gromov-hyperbolic δ or “fatness” of a hyperbolic geodesic triangle, defined to be the infimum of the perimeters of all inscribed triangles, is given an explicit analytical expression in term of the angle data of the triangle. By a hyperbolic extension of Fermat’s principle, the optimum inscribed triangle is easily constructed as the orthic triangle, that is, the triangle with its vertices at the feet of the altitudes of the original triangle. From the analytical expression of the optimum perimeter δ , a Tarski–Seidenberg computer algebra argument demonstrates that the δ , scaled by the diameter of the triangle, never exceeds 3/2 in a Riemannian manifold of constant nonpositive curvature. As probably the most important corollary, a finite metric geodesic space in which the ratio δ /diam is (strictly) bounded from above by 3/2 for all geodesic triangles exhibits the same metric properties as a negatively curved Riemannian manifold. The specific applications targeted here are those involving such very large but finite graphs as the Internet and the Protein Interaction Network. It is indeed argued that negative curvature is the precise mathematical formulation of their visually intuitive core concentric property.

Following a brief review of the history of the link between Einstein’s velocity addition law of special relativity and the hyperbolic geometry of Bolyai and Lobachevski, we employ the binary operation of Einstein’s velocity addition to introduce into hyperbolic geometry the concepts of vectors, angles and trigonometry. In full analogy with Euclidean geometry, we show in this article that the introduction of these concepts into hyperbolic geometry leads to hyperbolic vector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry.

In this paper we investigate the mean curvature H of a radial graph in hyperbolic space Hn+1. We obtain an integral inequality for H, and find that the lower limit of H at infinity is less than or equal to 1 and the upper limit of H at infinity is more than or equal to −1. As a byproduct we get a relation between the n-dimensional volume of a bounded domain in an n-dimensional hyperbolic space and the (n−1)-dimensional volume of its boundary. We also sharpen the main result of a paper by P.-A. Nitsche dealing with the existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space.

Using results from integral geometry, we find inequalities involving mean curvature integrals of convex hypersurfaces in hyperbolic space. Such inequalities generalize the Minkowski formulas for euclidean convex sets.

Let M3 be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M3. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.

We construct examples of surfaces in hyperbolic space which do not satisfy the Chern–Lashof inequality (which holds for immersed surfaces in Euclidean space). To cite this article: R. Langevin, G. Solanes, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

New pointwise inversion formulae are obtained for the d-dimensional totally geodesic Radon transform on the n-dimensional real hyperbolic space, 1⩽d⩽n−1, in terms of polynomials of the Laplace–Beltrami operator and intertwining fractional integrals. Similar results are established for hyperbolic cosine and sine transforms.

International audience; We show that the Teichmüller space of the triangle groups of type (p,q,∞) in the automorphism group of the two-dimensional complex hyperbolic space contains open sets of 0, 1 and two real dimensions. In particular, we identify the Teichmüller space near embeddings of the modular group preserving a complex geodesic.

Images are created using probabilistic iterated function systems that involve both affine transformations of the plane and isometries of hyperbolic geometry. Figures of attractors with striking hyperbolic symmetry are the result.

Hyperbolic geometry is a fundamental aspect of modern physics. We explore in this paper the use of Einstein's velocity addition as a model of vector addition in hyperbolic geometry. Guided by analogies with ordinary vector addition, we develop hyperbolic vector spaces, called gyrovector spaces, which provide the setting for hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. The resulting gyrovector spaces enable Euclidean trigonometry to be extended to hyperbolic trigonometry. In particular, we present the hyperbolic law of cosines and sines and the Hyperbolic Pythagorean Theorem emerges when the common vector addition is replaced by the Einstein velocity addition.

The oriented configuration space X+6 of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X+6 is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R10 whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center.

We give here a pair of characterizations for a euclidean disk D which are concerned with the hyperbolic geometry in D and in domains which contain D.

We present a theorem on the Wigner angle and its relation with the defect of a triangle in hyperbolic geometry.

It is shown that, for site percolation on the dual Dirichlet tiling graph of a co-compact Fuchsian group of genus ≥ 2, infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = P{site is open}. In such cases, the set of limit points at ∞ of an infinite cluster is shown to be a perfect, nowhere dense set of Lebesgue measure 0. These results are also shown to hold for a class of hyperbolic triangle groups.

Explicit S-matrices are constructed for scattering on p-adic hyperbolic planes. Combining these with the known S-matrix on the real hyperbolic plane, an adelic S-matrix is obtained. It has poles at the nontrivial zeros of the Riemann zeta-function, and is closely related to scattering on the modular domain of the real hyperbolic plane. Generalizations of this work and their possible arithmetic relevance are outlined.

In the article [S. Yang, A. Fang, A new characteristic of Mobius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2) (2006) 660–664], we gave a new characterization of Mobius transformations on B2 = {x ∈ R2: |x| 1} by using Lambert quadrilaterals, which are defined to be hyperbolic quadrilaterals with angles π2 , π 2 , π 2 , φ. Precisely, we proved that for a continuous bijection f : B2 → B2, f is Mobius if and only if it preserves Lambert quadrilaterals in B2. Our proof is based on a geometric approach. Among the proof of Lemma 3 in [1], it is stated that “Let O ABC be a Lambert quadrilateral with AOC < π2 and d(O , A) = d(O ,C), and let P be the intersection of the unit sphere with the geodesic OC . Choose F on [C, P ] such that the angles C F B and C BF both equal π4 . This is possible since C F B goes from π2 to 0 as F goes from C to P , and C BF goes from 0 to π2 at the same time”. (Here O is the origin of B 2.) However, such F does not exist since the area of the hyperbolic triangle C BF equals zero if both C F B and C BF are equal to π4 . We would like to thank Oguzhan Demirel for pointing out this error. In this note, we will give a new proof of the main theorem in [1].

Article history: Received 1 February 2008 Available online 28 May 2008 Submitted by R. Gornet With the aim of revealing their purely geometric nature, we rephrase two theorems of S. Yang and A. Fang (S. Yang, A. Fang, A new characteristic of Mobius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006) 660-664) characterizing Mobius transformations as definability results in elementary plane hyperbolic geometry. We show not only that elementary plane hyperbolic geometry can be axiomatized in terms of the quaternary predicates λ or σ ,w ithλ(abcd) to be read as 'abcd is a Lambert quadrilateral' and σ (abcd) to be read as 'abcd is a Saccheri quadrilateral', but also that all elementary notions of hyperbolic geometry can be positively defined (i.e. by using only quantifiers (∀ and ∃) and the connectives ∨ and ∧ in the definiens) in terms of λ or σ .

Suppose x and y are two points in the upper half-plane H+, and suppose Γ is a discontinuous group of conformal automorphisms of H+ having compact fundamental domain S. Denote by NT(x, y) the number of points of the form γy (γ ϵ Γ) in the closed disc of hyperbolic radius T centered about x, and set QT(x, y) = NT(x, y) − V(T)A, where V(T) is the hyperbolic area of the disc, and A is the hyperbolic area of S. The asymptotic behavior of the quantity ⊢LxL(QT(x,y))2 is estimated in terms of small eigenvalues of the Laplacian on functions automorphic under Γ.

The Transformation of waves of different kinds at some interior points of nonhomogeneous media is studied for waves defined by linear hyperbolic variational systems. A formal normal form is given for the light hypersurface (i.e. for the dispersion relation) at its generic singular points in terms of the contact geometry. This normal form describes the behaviour of the rays and of the wavefronts at the singular points corresponding to the multiple eigenvalues of the principal symbol of a generic hyperbolic variational problem.