The regular p-gonal prism tilings and their optimal hyperball packings in the hyperbolic 3-space

Summary We investigate the regular p-gonal prism tilings (mosaics) in the hyperbolic 3-space that were classified by I. Vermes in<span lang=EN-US style='font-size:10.0pt; mso-ansi-language:EN-US'>[12]and [13]. The optimal hyperball packings of these tilings are generated by the ``inscribed hyperspheres' whose metric data can be calculated by our method -- based on the projective interpretation of the hyperbolic geometry -- by the volume formulas of J. Bolyai and R. Kellerhals, respectively. We summarize in some tables the data and the densities of the optimal hyperball packings to each prism tiling in the hyperbolic space H³.     

Popular distances in 3-space

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn^1/3 log log n m(n) n^3/5 β(n), where c> 0 is a constant and β(n) is an extremely slowly growing function, related to the inverse of the Ackermann function.     

Critical sets in 3-space

Given a non-empty compact set C ?R ³, is C the set of critical points for some smooth proper functionf :R ³ →R ₊? In this paper we prove that the answer is “yes” for Antoine’s Necklace and most but not all tame links.     

Incenter iterations in 3-space

Consider a 3-dimensional point set which contains the incenters of all the nondegenerate tetrahedra with vertices from . In this paper we prove that then is dense in its convex hull. This settles the last unsolved variation in a sequence of similar questions initiated by D. Ismailescu, where he required to include other simplex centers, e.g. the orthocenters or the circumcenters. Our method allows us to generalize the planar incenter problem, showing that the denseness follows from a much weaker assumption for planar point sets. Supported by OTKA Grant K68398.     

A characterization of Weingarten surfaces in hyperbolic 3-space

We study 2-dimensional submanifolds of the space \({\mathbb{L}}({\mathbb{H}}^{3})\) of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. Such a surface is Lagrangian iff there exists a surface in ?³ orthogonal to the geodesics of Σ.We prove that the induced metric on a Lagrangian surface in \({\mathbb{L}}({\mathbb{H}}^{3})\) has zero Gauss curvature iff the orthogonal surfaces in ?³ are Weingarten: the eigenvalues of the second fundamental form are functionally related. We then classify the totally null surfaces in \({\mathbb{L}}({\mathbb{H}}^{3})\) and recover the well-known holomorphic constructions of flat and CMC 1 surfaces in ?³.     

A characterization of compact convex polyhedra in hyperbolic 3-space

Summary In this paper we study the extrinsic geometry of convex polyhedral surfaces in three-dimensional hyperbolic spaceH ³. We obtain a number of new uniqueness results, and also obtain a characterization of the shapes of convex polyhedra inH ³ in terms of a generalized Gauss map. This characterization greatly generalizes Andre'ev's Theorem.Oblatum 12-XI-1991 & 29-V-1992     

Finding stabbing lines in 3-space

A line intersecting all polyhedra in a setℬ is called a “stabber” for the setℬ. This paper addresses some combinatorial and algorithmic questions about the setℒ(ℬ) of all lines stabbingℬ. We prove that the combinatorial complexity ofℒ(ℬ) has an upper bound, wheren is the total number of facets inℬ, andc is a suitable constant. This bound is almost tight. Within the same time bound it is possible to determine if a stabbing line exists and to find one. The research of M. Pellegrini was partially supported by Eni and Enidata within the AXL project, and by NSF Grant CCR-8901484. A preliminary version appeared in theProceedings of the Second ACM-SIAM Symposium on Discrete Algorithms, pp. 24–31.     

Flexible polyhedra in Minkowski 3-space

 We prove that (non-immersed) flexible polyhedra do exist in the Minkowski 3-space and each of them preserves the (generalized) volume and the (total) mean curvature during a flex. To prove the latter result, we introduce the notion of the angle between two arbitrary non-null nonzero vectors in the Minkowski plane. Received: 16 August 2001 Published online: 19 May 2003 Mathematics Subject Classification (2000): 52C25, 51B20, 52B70, 52B11, 51M25     

Mannheim partner curves in 3-space

In this paper, we study Mannheim partner curves in three dimensional space. We obtain the necessary and sufficient conditions for the Mannheim partner curves in Euclidean space and Minkowski space , respectively. Some examples are also given. Supported by NSFC, No.10371013; Joint Research of NSFC and KOSEF, NEU     

A circular embedding of a graph in Euclidean 3-space

A spatial embedding of a graph G is an embedding of G into the 3-dimensional Euclidean space . J.H. Conway and C.McA. Gordon proved that every spatial embedding of the complete graph on 7 vertices contains a nontrivial knot. A linear spatial embedding of a graph is an embedding which maps each edge to a single straight line segment. In this paper, we construct a linear spatial embedding of the complete graph on 2n−1 (or 2n) vertices which contains the torus knot T(2n−5,2) (n4). A circular spatial embedding of a graph is an embedding which maps each edge to a round arc. We define the circular number of a knot as the minimal number of round arcs in among such embeddings of the knot. We show that a knot has circular number 3 if and only if the knot is a trefoil knot, and the figure-eight knot has circular number 4.     

Surface-dependent growth kinematics in euclidean 3-space

In this paper, we investigate the surface-dependent growth model in Euclidean 3-space. The surface-dependent model is developed for modeling the kinematics of surface growth for objects that can be generated by the curves on the surface, such as parasites and plants. This paper includes two main purposes for this model. The first is to parameterize this model using quaternions and homothetic motions, while expressing matrix representations of the surface-dependent growth model. The second one is to construct the surface-dependent growth model by using the growth velocity components related to the Darboux frame at each point of the generating curve. Moreover, to support the theory studied in the paper, various examples are illustrated.     

三维Minkowski空间中的极小曲面

给出了三维Minkowski空间中类空曲面活动标价的四元素表示,并利用四元素既适合于旋转结构的侵入又适合于2×2矩阵处理极小曲面的分析特性,经研究得到了R12中的极小曲面Weierstrass表示和曲面的可积条件.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Ruled Weingarten surfaces in Minkowski 3-space

We characterize all ruled surfaces in Minkowski 3-space with a relation between the Gauss and mean curvature (Weingarten surfaces). It turns out that, except if the rulings are in a null direction, these are given by Lorentzian screw motions of straight lines. However, if the rulings are always in a null direction, then every ruled surface is Weingarten. Received: 9 February 1998 / Revised version: 20 December 1998     

Isothermic constrained Willmore tori in 3-space

Annals of Global Analysis and Geometry - We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore...