We present a method for solving the following problem: Given a set of data points scattered in three dimensions and an initial triangular mesh M0, produce a mesh M, of the same topological type as M0, that fits the data well and has a small number of vertices. Our approach is to minimize an energy function that explicitly models the competing desires of conciseness of representation and fidelity to the data. We show that mesh optimization can be effectively used in at least two applications: surface reconstruction from unorganized points, and mesh simplification (the reduction of the number of vertices in an initially dense mesh of triangles).

We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al.

We study finite and infiite entangled graphs in the bond percolation process in three dimensions with density p. After a discussion of the relevant defi®nitions, the entanglement critical probabilities are defined. The size of the maximal entangled graph at the origin is studied for small p, and it is shown that this graph has radius whose tail decays at least as fast as exp … an = log n†; indeed, the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant a. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation.

Abstract. This paper describes a linear-time algorithm for computing the intersection of two convex polyhedra in 3-space. Applications of this result to computing intersections, convex hulls, and Voronoi diagrams are also given. Key words, computational geometry, convex polyhedra AMS(MOS) subject classifications. 68Q25, 68H05 1. Introduction. Giventwo

and precomputation of solid textures. The stretch caused by a given parameterization determines the sampling rate on the surface. In this article, we present an automatic parameterization method for segmenting a surface into patches that are then flattened with little stretch. Many objects consist of regions of relatively simple shapes, each of which has a natural parameterization. Based on this observation, we describe a three-stage feature-based patch creation method for manifold surfaces. The first two stages, genus reduction and feature identification, are performed with the help of distance-based surface functions. In the last stage, we create one or two patches for each feature region based on a covariance matrix of the feature’s surface points. To reduce stretch during patch unfolding, we notice that stretch is a 2 × 2 tensor, which in ideal situations is the identity. Therefore, we use the Green-Lagrange tensor to measure and to guide the optimization process. Furthermore, we allow the boundary vertices of a patch to be optimized by adding scaffold triangles. We demonstrate our feature-based patch creation and patch unfolding methods for several textured models. Finally, to evaluate the quality of a given parameterization, we describe an image-based error measure that takes into account stretch, seams, smoothness, packing efficiency, and surface visibility.

Abstract. Thurston’s Ending Lamination Conjecture states that a hyperbolic 3-manifold with finitely generated fundamental group is uniquely determined by its topological type and its end invariants. In this paper we prove this conjecture for Kleinian surface groups. The main ingredient is the establishment of a uniformly bilipschitz model for a Kleinian surface group. The first half of the proof appeared in [47], and a subsequent paper [15] will establish the Ending Lamination Conjecture in general. Contents 1. The ending lamination conjecture 1 2. Background and statements 8 3. Knotting and partial order of subsurfaces 25

While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin ...

) Elizabeth Borowsky (borowsky@hpl.hp.com) Hewlett-Packard Laboratories Palo-Alto, CA 94303 U.S.A. Eli Gafni (eli@cs.ucla.edu) Computer Science Department University of California, Los Angeles Los Angeles, CA 90024 U.S.A. July 1, 1996 Abstract In a sequence of two pioneering papers Herlihy and Shavit characterized waitfree shared-memory computations. The derivation of the characterization involves homology for the necessary conditions, and complex geometry arguments for the sufficiency. This paper gives an alternative proof of the conditions using familiar algorithmic arguments. Our only reliance on geometry is the use of a corollary to the simplicial approximation. Furthermore, this paper is the first to present another consequence of the relation between distributed algorithms and topology: that certain theorems in topology are naturally proven by distributed algorithms interpretations. Our techniques can be extended to characterize models that are more complex than the wait-free...

There is a positive constant c1 such that for any diagram D representing the unknot, there is a sequence of at most 2 c1n Reidemeister moves that will convert it to a trivial knot diagram, where n is the number of crossings in D. A similar result holds for elementary moves on a polygonal knot K embedded in the 1-skeleton of the interior of a compact, orientable, triangulated PL 3-manifold M. There is a positive constant c2 such that for each t ≥ 1, if M consists of t tetrahedra, and K is unknotted, then there is a sequence of at most 2 c2t elementary moves in M which transforms K to a triangle contained inside one tetrahedron of M. We obtain explicit values for c1 and c2.

. Topological and metric aspects of the multiresolution representation of geographic maps are considered. The combinatorial structure of maps is mathematically modelled through abstract cell complexes, and maps at different detail are related through continuous functions over such complexes. Metric aspects of multiresolution are controlled through the concept of homotopy. Two alternative multiresolution models are proposed, which are implicitly defined by a sequence of map simplifications that fulfil both topological and metric consistency rules. 1 Introduction The representation of spatial data at different resolution in the context of a unified model is a topic of relevant interest in spatial information theory. Indeed, multiresolution modelling offers interesting capabilities for spatial representation and reasoning: from support to map generalisation and automated cartography [15], to efficient browsing over large GISs, to structured solutions in wayfinding and planning [25]. Curr...