Triangulated n-manifolds are determined by their [n2] + 1-skeletons

We will reconstruct compact, triangulated n-manifolds—without-boundaries from just their $\text{[}\text{n}\text{2}\text{] + 1}$ skeletons. Therefore, two such manifolds are isomorphic if they have simplicially isomorphic $\text{[}\text{n}\text{2}\text{+ 1}$-skeletons. Furthermore, when n is even, and the $\text{n}\text{2}$-homology group is zero, then the $\text{n}\text{2}$-skeleton is sufficient.     

运动目标空间位置坐标激光跟踪测量的数学模型

运动目标空间位置激光跟踪测量是国际计量测试领域的前沿课题，该测量系统集激光干涉测距、精密机械、计算机及控制系统和现代数值计算于一体，对空间运动目标进行跟踪并实时测量其坐标和姿态。它的测量范围为１～５ｍ，测量精度为５０μｍ／ｍ。它的测量原理主要是干涉法和三角法。在干涉法中，根据跟踪测量机构数量的多少又可分为一站法和多站法。一站法采用球坐标测量法，多站法通过解最小二乘方程组计算出被测点的坐标。文中对各种测量方法的数学理论模型进行了较为详细地论述。     

Delaunay triangulation and 3D adaptive mesh generation

Delaunay triangulation and its complementary structure the Voronoi polyhedra form two of the most fundamental constructs of computational geometry. Delaunay triangulation offers an efficient method for generating high-quality triangulations. However, the generation of Delaunay triangulations in 3D with Watson's algorithm, leads to the appearance of silver tetrahedra, in a relatively large percentage. A different method for generating high-quality tetrahedralizations, based on Delaunay triangulation and not presenting the problem of sliver tetrahedra, is presented. The method consists in a tetrahedra division procedure and an efficient method for optimizing tetrahedral meshes, based on the application of a set of topological Delaunay transformations for tetrahedra and a technique for node repositioning. The method is robust and can be applied to arbitrary unstructured tetrahedral meshes, having as a result the generation of high-quality adaptive meshes with varying density, totally eliminating the appearance of sliver elements. In this way it offers a convenient and highly flexible algorithm for implementation in a general purpose 3D adaptive finite element analysis system. Applications to various engineering problems are presented     

Reverse Monte Carlo and Voronoi-Delaunay simulation and structure analysis for liquid metals

This paper discusses the applicability of the integrated reverse Monte Carlo and Voronoi-Delaunay approach to local structure analysis and medium-scale structure determination of liquid metals. The advantages of this approach are shown: generation of 3D structural models using experimental diffraction data and subsequent structure determination by atomic coordinate analysis.Original Russian Text Copyright © 2004 by A. S. Roik, V. P. Kazimirov, and V. E. SokolskiiTranslated from Zhurnal Strukturnoi Khimii, Vol. 45, No. 4, pp. 683–691, July–August, 2004.This revised version was published online in April 2005 with a corrected cover date.     

A flexible triangulation method to describe the solvent-accessible surface of biopolymers

A relatively simple protein solvent-accessible surface triangulation method for continuum electrostatics applications employing the boundary element method is presented. First, the protein is placed onto a three-dimensional lattice with a specified lattice spacing. To each lattice point, a box is assigned. Boxes located in the solvent region and in the interior of the protein are removed from the set. Improper connections between boxes and the possible existence of cavities in the interior of the protein which would destroy the proper connectivity of the triangulated surface are taken care of. The remaining set of boxes define the outer contour of the protein. Each free face exposed to the solvent of the remaining set of boxes is triangulated after the surface defined by the free faces has been smoothed to follow the shape of the macromolecule more accurately. The final step consists of a mapping of triangle vertices onto a set of surface points which define the solvent-accessible surface. Normal vectors at triangle vertices are obtained also from the free faces which define the orientation of the surface. The algorithm was tested for six molecules including four proteins; a dipeptide, a helical peptide consisting of 25 residues, calbindin, lysozyme, calmodulin and cutinase. For each molecule, total areas have been calculated and compared with the result computed from a dotted solvent-accessible surface. Since the boundary element method requires a low number of vertices and triangles to reduce the number of unknowns for reasons of efficiency, the number of triangles should not be too high. Nevertheless, credible results are obtained for the total area with relative errors not exceeding 12% for a large lattice spacing (0.30 nm) while close to zero for a smaller lattice spacing (down to 0.16 nm). The output of the triangulation computer program (written in C++) is rather simple so that it can be easily converted to any format acceptable for any molecular graphics programs.     

Non-manifold surface reconstruction from high-dimensional point cloud data

We present an algorithm capable of reconstructing a non-manifold surface embedded as a point cloud in a high-dimensional space. Our algorithm extends a previously developed incremental method and produces a non-optimal triangulation, but will work for non-orientable surfaces, and for surfaces with certain types of self-intersection. The self-intersections must be ordinary double curves and are fitted locally by intersecting planes using a degenerate quadratic surface. We present the algorithm in detail and provide many examples, including a dataset describing molecular conformations of cyclo-octane.     

CDT meets Hořava–Lifshitz gravity

The theory of causal dynamical triangulations (CDT) attempts to define a nonperturbative theory of quantum gravity as a sum over spacetime geometries. One of the ingredients of the CDT framework is a global time foliation, which also plays a central role in the quantum gravity theory recently formulated by Ho?ava. We show that the phase diagram of CDT bears a striking resemblance with the generic Lifshitz phase diagram appealed to by Ho?ava. We argue that CDT might provide a unifying nonperturbative framework for anisotropic as well as isotropic theories of quantum gravity.     

Minimal triangulations of graphs: A survey

Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.     

Convex combination maps over triangulations, tilings, and tetrahedral meshes

In a recent paper by the first author, a simple proof was given of a result by Tutte on the validity of barycentric mappings, recast in terms of the injectivity of piecewise linear mappings over triangulations. In this note, we make a short extension to the proof to deal with arbitrary tilings. We also give a simple counterexample to show that convex combination mappings over tetrahedral meshes are not necessarily one-to-one. Mathematics subject classifications (2000) 05C10, 05C85, 65D17, 58E20     

Design and analysis of planar shape deformation

Shape deformation refers to the continuous change of one geometric object to another. We develop a software tool for planning, analyzing and visualizing deformations between two shapes in . The deformation is generated automatically without any user intervention or specification of feature correspondences. A unique property of the tool is the explicit availability of a two-dimensional shape space, which can be used for designing the deformation either automatically by following constraints and objectives or manually by drawing deformation paths.