An Inversion Invariant of a Pair of Circles

An inversion invariant of two oriented circles is introduced. It is close to Coxeter's inversive distance between two non-intersecting circles, but it is defined for any pair of oriented circles (straight lines). Two topics are discussed as applications: the problem of C¹-conjunction of circles and properties of plane curves with monotone curvature. Bibliography: 10 titles.     

Binary relations as single primitive notions for hyperbolic three-space and the inversive plane

By interpreting J.A. Lester's [9] result on inversive-distance-preserving mappings as an axiomatizability statement, and by using the Liebmann isomorphism between the inversive plane and hyperbolic three-space, we point out that hyperbolic three-spaces (and inversive geometry) coordinatized by Euclidean fields can be axiomatized with planes (or circles) as variables, by using only the plane-orthogonality (or circle-orthogonality) predicate _p (or _c), or by using only the predicate δ′ (or δ), where δ′(p,p′) (or δ(A, B)) is interpreted as ‘the distance between the planes p and p′ is equal to the length of the segment s whose angle of parallelism is (i. e. II(s) = )’ (or as ‘the numerical distance between the disjoint circles A and B has the value , which corresponds to s via Liebmann's isomorphism’).     

A note on the finite convergence of alternating projections

We establish sufficient conditions for finite convergence of the alternating projections method for two non-intersecting and potentially nonconvex sets. Our results are based on a generalization of the concept of intrinsic transversality, which until now has been restricted to sets with nonempty intersection. In the special case of a polyhedron and closed half space, our sufficient conditions define the minimum distance between the two sets that is required for alternating projections to converge in a single iteration.     

Non-rigidity of Spherical Inversive Distance Circle Packings

We give a counterexample of Bowers–Stephenson’s conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.     

REMARKS ON THE ZIG-ZAG THEOREM

The classical Zig-zag Theorem [1] says that if an equilateral closed 2m-gon shuttles between two given circles of the Euclidean 3-space, then the vertices of the polygon can be moved smoothly along the circles without changing the lengths of the sides of the polygon. First we prove that the Zig-zag Theorem holds also in the hyperbolic, Euclidean and spherical n-spaces, and in fact the circles can be replaced by straight lines or any kind of cycles. In the second part of the paper we restrict our attention to planar zig-zag configurations. With the help of an alternative formulation of the Zig-zag Theorem, we establish two duality theorems for periodic zig-zags between two circles. This revised version was published online in August 2006 with corrections to the Cover Date.     

Diffeomorphisms preserving R‐circles in three dimensional CR manifolds

R ‐circles in (non‐degenerate) three dimensional CR manifolds are the analogues to traces of Lagrangian totally geodesic planes on S³ viewed as the boundary of two dimensional complex hyperbolic space. They form a family of certain Legendrian curves on the manifold. We prove that a diffeomorphism between three dimensional CR manifolds which preserve circles is either a CR diffeomorphism or a conjugate CR diffeomorphism.     

Coxeter polytopes with a unique pair of non-intersecting facets

We consider compact hyperbolic Coxeter polytopes whose Coxeter diagram contains a unique dotted edge. We prove that such a polytope in d-dimensional hyperbolic space has at most d+3 facets. In view of results by Kaplinskaja [I.M. Kaplinskaya, Discrete groups generated by reflections in the faces of simplicial prisms in Lobachevskian spaces, Math. Notes 15 (1974) 88-91] and the second author [P. Tumarkin, Compact hyperbolic Coxeter n-polytopes with n+3 facets, Electron. J. Combin. 14 (2007), R69, 36 pp.], this implies that compact hyperbolic Coxeter polytopes with a unique pair of non-intersecting facets are completely classified. They do exist only up to dimension 6 and in dimension 8.     

On blocking sets of inversive planes

Let S be a blocking set in an inversive plane of order q. It was shown by Bruen and Rothschild 1 that |S| ≥ 2q for q ≥ 9. We prove that if q is sufficiently large, C is a fixed natural number and |S = 2q + C, then roughly 2/3 of the circles of the plane meet S in one point and 1/3 of the circles of the plane meet S in four points. The complete classification of minimal blocking sets in inversive planes of order q ≤ 5 and the sizes of some examples of minimal blocking sets in planes of order q ≤ 37 are given. Geometric properties of some of these blocking sets are also studied. © 2004 Wiley Periodicals, Inc.     

Inversive localization at semiprlme goldie ideals

P. M. Cohn [6] introduced a method of localizing at a semiprime ideal of a noncommutative Noetherian ring by inverting certain matrices. This paper continues the study of the technique of inversive localization, in a more general setting. The inversive localization is characterized by its structure modulo its Jacobson radical. This is in marked contrast to the torsion theoretic localization, and the two constructions coincide only when the localization can actually be obtained by inverting elements rather than matrices. The inversive localization is computed for the class of left Artinian rings, and it is then shown that at a minimal prime ideal of an order in a left Artinian ring the inversive localization must be left Artinian. On the other hand, the inversive localization at a semiprime ideal of a left Noetherian ring need not be left Noetherian.     

Eine axiomatik ovoidaler

In the first paragraph of this paper we start with a geometry consisting of points, circles, and an equivalencerelation on the set of points. There is just one circle through any three pairwise non-parallel points and any four non-concyclic points generate a Laguerre-plane or an inversive plane. If dimension d>2, we show that we must have the geometry induced on a cone of a projective space by plane-cuts. In the following part this result is used to give a characterization of the geometry of paraboloids in affine spaces.     

Construction of spirals with prescribed boundary conditions

Spirality, regarded as monotonicity of curvature, is preserved under inversions. This property is used for constructing a spiral transition curve with predefined curvature elements at the endpoints. These boundary conditions define two invariant values: Coxeter’s inversive distance and the width of the lens. In order to solve the problem, it suffices to realize the corresponding values on two curvature elements of any known spiral. The rest is achieved by inversion. In particular, any boundary conditions compatible with spirality can be satisfied by inverting an arc of the logarithmic spiral. Bibliography: 9 titles.     

连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性

本文研究任意有限维空间中连接两个具有一维不稳定流形的双曲鞍点异宿环的稳定性.借助适当的线性变换和坐标变换,将局部稳定流形和不稳定流形拉直,利用奇异流映射和正则流映射构造了Poincaré映射.通过技巧性地估计向量的模,给出了在横截面上Poincaré映射的初始点与首次回归点离异宿轨道与横截面交点的距离之比,得到了高维空间中连接两个带有一维不稳定流形的异宿环的非常简洁的稳定性判据.     

On embedding a cycle in a plane graph

Consider a planar drawing Γ of a planar graph G such that the vertices are drawn as small circles and the edges are drawn as thin stripes. Consider a non-simple cycle c of G. Is it possible to draw c as a non-intersecting closed curve inside Γ, following the circles that correspond in Γ to the vertices of c and the stripes that connect them? We show that this test can be done in polynomial time and study this problem in the framework of clustered planarity for highly non-connected clustered graphs.     

Conics in the hyperbolic plane intrinsic to the collineation group

In the manner of Steiner??s interpretation of conics in the projective plane we consider a conic in a planar incidence geometry to be a pair consisting of a point and a collineation that does not fix that point. We say these loci are intrinsic to the collineation group because their construction does not depend on an imbedding into a larger space. Using an inversive model we classify the intrinsic conics in the hyperbolic plane in terms of invariants of the collineations that afford them and provide metric characterizations for each congruence class. By contrast, classifications that catalogue all projective conics intersecting a specified hyperbolic domain necessarily include curves which cannot be afforded by a hyperbolic collineation in the above sense. The metric properties we derive will distinguish the intrinsic classes in relation to these larger projective categories. Our classification emphasizes a natural duality among congruence classes induced by an involution based on complementary angles of parallelism relative to the focal axis of each conic, which we refer to as split inversion (Definition 5.3).     

Harmonische Involutionen und Kreisspiegelungen in Möbiusebenen

An involutory automorphism of an inversive plane whose set of fixed points consists of exactly two points resp. of a circle is called a harmonic involution resp. an inversion. In this paper we study inversive planes with sufficiently many such involutions. Herrn Walter Benz zum 75. Geburtstag gewidmet     

I piani di inversione come modelli di una classe di spazi metrici

Summary The purpose of this paper is to represent a class of metric spaces on the elliptic and hyperbolic inversive planes by inversions and their transformations.

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Lavoro eseguito nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche (sez. n. 4) del C.N.R., presso l'Istituto di Matematica Applicata della Facoltà di Ingegneria, Roma.

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Entrata in Redazione il 24 maggio 1972.     

A tale of two conformally invariant metrics

The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic metric in the disk. We prove that the Harnack distance is never greater than the hyperbolic distance and if the two distances agree for one pair of distinct points, then either the domain is simply connected or it is conformally equivalent to the punctured disk.     

Inversive planes with circles determined by tangents

In an inversive planeJ, one derived affine plane is such that every circle is uniquely determined by straight line tangents. The validity of 4-point Pascal conditions on one of the circles then plays a role in coordinatization ofJ.     

Construction of inversive congruential pseudorandom number generators with maximal period length

The inversive congruential method for generating uniform pseudorandom numbers is a particularly attractive alternative to linear congruential generators with their well-known inherent deficiencies like the unfavourable coarse lattice structure in higher dimensions. In the present paper the modulus in the inversive congruential method is chosen as a power of an arbitrary odd prime. The existence of inversive congruential generators with maximal period length is proved by a new constructive characterization of these generators.     

Sulla rappresentazione gruppale dei piani metrici sui piani di inversione

Summary The purpose of this paper is to give a group representation of the metric planes over the real numbers (exactly the elliptic plane, the hyperbolic plane, the DeSitter plane, the euclidean plane and the Minkowski plane) on the elliptic and hyperbolic inversive planes by inversions and their transformations.

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Dedicato al Prof. BeniaminoSegre per il 70^mo compleanno.

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Lavoro eseguito nell'ambito del Gruppo Naz. per le Strutture Algebriche e Geometriche (sez. n. 4) del C.N.R. presso l'Istituto di Matematica Applicata della Facoltà di Ingegneria dell'Università di Roma.

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Entrata in Redazione il 12 febbraio 1973.